The Finite Element

The Finite Element Method for Boundary

Value ProblemsMathematics and Computations

http://taylorandfrancis.com

Karan S. Surana • J. N. Reddy

The Finite Element Method for Boundary

Value ProblemsMathematics and Computations

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Names: Surana, Karan S. | Reddy, J. N. (Junuthula Narasimha), 1945-Title: The finite element method for boundary value problems : mathematics and computations / Karan S. Surana and J.N. Reddy.Description: Boca Raton : CRC Press, 2017.Identifiers: LCCN 2016035534| ISBN 9781498780506 (hardback : alk. paper) | ISBN 9781315365718 (ebook)Subjects: LCSH: Boundary value problems--Numerical solutions. | Finite element method.Classification: LCC QA379 .S87 2017 | DDC 515/.62--dc23

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To

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Abha, Deepak, Rishi, and Yogini

My (JNR) loving grandchildren

Rohan, Asha, and Mira

Contents

Preface xix

About the Authors xxv

1 Introduction 1

1.1 General Comments and Basic Philosophy . . . . . . . . . . . 1

1.2 Basic Concepts of the Finite Element Method . . . . . . . . 3

1.2.1 Discretization . . . . . . . . . . . . . . . . . . . . . . 3

1.2.2 Local approximation . . . . . . . . . . . . . . . . . . 5

1.2.3 Integral forms and algebraic equations over an element 7

1.2.4 Assembly of element equations . . . . . . . . . . . . . 8

1.2.5 Computation of the solution . . . . . . . . . . . . . . 9

1.2.6 Post-processing . . . . . . . . . . . . . . . . . . . . . 9

1.2.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.8 k-version of the finite element methodand hpk framework . . . . . . . . . . . . . . . . . . . 11

1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Concepts from Functional Analysis 15

2.1 General Comments . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Sets, Spaces, Functions, Functions Spaces, and Operators . . 15

2.2.1 Hilbert spaces Hk(Ω) . . . . . . . . . . . . . . . . . . 18

2.2.2 Definition of scalar product in Hk(Ω) space . . . . . 19

2.2.3 Properties of scalar product . . . . . . . . . . . . . . 19

2.2.4 Norm of u in Hilbert space Hk(Ω) . . . . . . . . . . . 20

2.2.5 Seminorm of u in Hilbert space Hk(Ω) . . . . . . . . 20

2.2.6 Function spaces . . . . . . . . . . . . . . . . . . . . . 22

2.2.7 Operators . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.8 Types of operators . . . . . . . . . . . . . . . . . . . 24

2.2.9 Energy product . . . . . . . . . . . . . . . . . . . . . 27

2.2.10 Integration by parts (IBP) . . . . . . . . . . . . . . . 27

2.3 Elements of Calculus of Variations . . . . . . . . . . . . . . . 31

vii

viii CONTENTS

2.3.1 Concept of the variation of a functional . . . . . . . . 332.3.2 Euler’s equation: Simplest variational problem . . . . 342.3.3 Variation of a functional: some practical aspects . . . 412.3.4 Riemann and Lebesgue integrals . . . . . . . . . . . . 42

2.4 Examples of Differential Operators andtheir Properties . . . . . . . . . . . . . . . . . . . . . . . . . 442.4.1 Self-adjoint differential operators . . . . . . . . . . . 442.4.2 Non-self-adjoint differential operators . . . . . . . . . 582.4.3 Non-linear differential operators . . . . . . . . . . . . 63

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3 Classical Methods of Approximation 693.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.2 Basic Steps in Classical Methods of

Approximation based on Integral Forms . . . . . . . . . . . . 703.3 Integral forms using the Fundamental Lemma

of the Calculus of Variations . . . . . . . . . . . . . . . . . . 723.3.1 The Galerkin method . . . . . . . . . . . . . . . . . . 73

3.3.1.1 Self-adjoint and non-self-adjoint lineardifferential operators . . . . . . . . . . . . . . 74

3.3.1.2 Non-linear differential operators . . . . . . . 773.3.2 The Petrov–Galerkin and weighted-residual methods 78

3.3.2.1 Self-adjoint and non-self-adjoint lineardifferential operators . . . . . . . . . . . . . . 78

3.3.2.2 Non-linear differential operators . . . . . . . 803.3.3 The Galerkin method with weak form . . . . . . . . . 81

3.3.3.1 Linear differential operators . . . . . . . . . . 853.3.3.2 Non-linear differential operators . . . . . . . 86

3.3.4 The least-squares method . . . . . . . . . . . . . . . . 873.3.4.1 Self-adjoint and non-self-adjoint linear

differential operators . . . . . . . . . . . . . . 923.3.4.2 Non-linear differential operators . . . . . . . 93

3.3.5 Collocation method . . . . . . . . . . . . . . . . . . . 943.4 Approximation Spaces for Various Methods of Approximation 953.5 Integral Formulations of BVPs using

the Classical Methods of Approximations . . . . . . . . . . . 973.5.1 Self-adjoint differential operators . . . . . . . . . . . 983.5.2 Non-self-adjoint Differential Operators . . . . . . . . 1273.5.3 Non-linear Differential Operators . . . . . . . . . . . 141

3.6 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 1533.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

4 The Finite Element Method 193

CONTENTS ix

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

4.2 Basic steps in the finite element method . . . . . . . . . . . . 194

4.2.1 Discretization . . . . . . . . . . . . . . . . . . . . . . 194

4.2.2 Construction of integral forms over an element . . . . 198

4.2.2.1 Integral forms for GM, PGM, and WRM . . 199

4.2.2.2 Integral form for GM/WF . . . . . . . . . . 200

4.2.2.3 Integral form based on residual functional . . 202

4.2.3 The local approximation φeh of φ over an element . . 203

4.2.4 Element matrices and vectors resulting from theintegral form and the local approximation . . . . . . 204

4.2.4.1 Galerkin method, Petrov–Galerkin method,and weighted residual method . . . . . . . . 205

4.2.4.2 Galerkin method with weak form . . . . . . . 206

4.2.4.3 Least-squares process based on residualfunctional . . . . . . . . . . . . . . . . . . . . 210

4.2.5 Assembly of element equations: GM, PGM, WRM,GM/WF and LSP when A is linear . . . . . . . . . . 212

4.2.6 Consideration of boundary conditions in theassembled equations . . . . . . . . . . . . . . . . . . . 214

4.2.6.1 GM, PGM, WRM, and LSP based on theresidual functional . . . . . . . . . . . . . . . 214

4.2.6.2 GM/WF . . . . . . . . . . . . . . . . . . . . 215

4.2.7 Computation of the solution: finite element processesbased on all methods of approximation except LSPfor non-linear operators . . . . . . . . . . . . . . . . . 216

4.2.8 Assembly of element equations and their solution infinite element processes based on residual functional(LSP) when the differential operator A is non-linear . 217

4.2.9 Post processing of the solution . . . . . . . . . . . . . 220

4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

5 Self-Adjoint Differential Operators 223

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

5.1.1 GM/WF . . . . . . . . . . . . . . . . . . . . . . . . . 224

5.1.2 LSP based on residual functional . . . . . . . . . . . 225

5.2 One-dimensional BVPs in a single dependent variable . . . . 226

5.2.1 1D steady-state diffusion equation: finite elementprocesses based on GM/WF . . . . . . . . . . . . . . 226

5.2.1.1 Discretization . . . . . . . . . . . . . . . . . 227

5.2.1.2 Integral form using GM/WF (weak form) ofthe BVP for an element e with domain Ωe . 227

x CONTENTS

5.2.1.3 Approximation space Vh, test function spaceV and local approximation φeh . . . . . . . . 231

5.2.1.4 Local approximation φeh and mapping Ωe toΩξ . . . . . . . . . . . . . . . . . . . . . . . . 233

5.2.1.5 Element equations . . . . . . . . . . . . . . . 234

5.2.1.6 Assembly of element equations andcomputation of the solution . . . . . . . . . . 235

5.2.1.7 Inter-element continuity conditions on PVsor dependent variables . . . . . . . . . . . . . 237

5.2.1.8 Rules for assembling element matrices andvectors . . . . . . . . . . . . . . . . . . . . . 237

5.2.1.9 Inter-element continuity conditions on the sumof secondary variables . . . . . . . . . . . . . 242

5.2.1.10 Imposition of EBCs . . . . . . . . . . . . . . 243

5.2.1.11 Solving for unknown degrees of freedom . . . 243

5.2.1.12 Special case: numerical study . . . . . . . . . 244

5.2.1.13 Post-processing of solution . . . . . . . . . . 246

5.2.1.14 Analytical solution and comparison withfinite element solutions . . . . . . . . . . . . 246

5.2.2 1D steady-state diffusion equation . . . . . . . . . . . 251

5.2.2.1 Case (a): a = 1, L = 1, q(x) = xn, n is apositive integer . . . . . . . . . . . . . . . . . 252

5.2.2.2 Case (b): a = 1, L = 1, q(x) = sinnπx, n = 4 260

5.2.3 Least-squares finite element formulation . . . . . . . 265

5.2.3.1 Approximation space Vh . . . . . . . . . . . . 275

5.2.3.2 Numerical studies . . . . . . . . . . . . . . . 275

5.2.4 LSFEP using auxiliary variables and auxiliaryequations . . . . . . . . . . . . . . . . . . . . . . . . . 278

5.2.4.1 Approximation spaces for φeh and τ eh . . . . . 285

5.2.4.2 Numerical studies . . . . . . . . . . . . . . . 285

5.2.5 One-dimensional heat conduction withconvective boundary . . . . . . . . . . . . . . . . . . 289


5.2.5.2 Numerical study . . . . . . . . . . . . . . . . 297

5.2.6 1D axisymmetric heat conduction . . . . . . . . . . . 300


5.2.6.2 LSM based on residual functional . . . . . . 303

5.2.7 A 1D BVP governed by a fourth-order differentialoperator . . . . . . . . . . . . . . . . . . . . . . . . . 305


5.3 Two-dimensional boundary value problems . . . . . . . . . . 310

5.3.1 A general 2D BVP in a single dependent variable . . 310

CONTENTS xi

5.3.1.1 Definition of Ωe: element geometry . . . . . 314


5.3.1.3 Computation of the element matrix [Ke] andvector F e . . . . . . . . . . . . . . . . . . . 317

5.3.1.4 Details of secondary variable vector P e . . 317

5.3.2 2D Poisson’s equation: numerical studies . . . . . . . 320

5.3.2.1 Case (a): f = 1 with BCs φ(±1, y) = φ(x,±1) =0; GM/WF . . . . . . . . . . . . . . . . . . . 320

5.3.2.2 Case (b): BCs φ(±1, y) = φ(x,±1) = 1.0;GM/WF . . . . . . . . . . . . . . . . . . . . 327

5.3.3 Two-dimensional boundary value problemsin multi-variables: 2D plane elasticity . . . . . . . . . 328


5.3.3.2 Least-squares method using residualfunctional . . . . . . . . . . . . . . . . . . . . 337

5.4 Three-dimensional boundary value problems . . . . . . . . . 344

5.4.1 Three-dimensional boundary value problems in asingle dependent variable . . . . . . . . . . . . . . . . 344


5.4.1.2 Approximation space . . . . . . . . . . . . . 347

5.4.1.3 Local approximation T eh . . . . . . . . . . . . 347

5.4.1.4 Definition of Ωe: Element geometry . . . . . 348

5.4.1.5 Computations of element matrix [Ke] andvector F e . . . . . . . . . . . . . . . . . . . 350

5.4.1.6 Details of secondary variable vector P e . . 350

5.4.2 Three-dimensional boundary value problemsin multivariables . . . . . . . . . . . . . . . . . . . . . 353


5.4.2.2 Approximation spaces . . . . . . . . . . . . . 357

5.4.2.3 Local approximation . . . . . . . . . . . . . . 357

5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

6 Non-Self-Adjoint Differential Operators 363

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

6.2 1D convection-diffusion equation . . . . . . . . . . . . . . . . 365

6.2.1 Analytical solution . . . . . . . . . . . . . . . . . . . 365

6.2.2 The Galerkin method with weak form (GM/WF) . . 367

6.2.3 Least squares finite element formulation . . . . . . . 378

6.2.4 Least squares formulation: first order system . . . . . 380

6.3 2D convection-diffusion equation . . . . . . . . . . . . . . . . 390

6.3.1 Least squares finite element formulation based on theresidual functional . . . . . . . . . . . . . . . . . . . . 395

xii CONTENTS

6.3.2 Least squares finite element formulation of (6.113) byrecasting it as a system of first order PDEs . . . . . . 397

6.3.3 Convection dominated thermal flow (advection skewedto a square domain) . . . . . . . . . . . . . . . . . . . 401

6.3.4 Advection of a cosine hill in a rotating flow field . . . 4086.3.5 Thermal boundary layer . . . . . . . . . . . . . . . . 411

6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

7 Non-Linear Differential Operators 4197.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 4197.2 One dimensional Burgers equation . . . . . . . . . . . . . . . 422

7.2.1 The Galerkin method with weak form . . . . . . . . . 4237.2.2 LSP based on residual functional . . . . . . . . . . . 4287.2.3 LSP based on residual functional: first order system

of equations . . . . . . . . . . . . . . . . . . . . . . . 4297.3 Fully developed flow of Giesekus fluid between parallel plates

(polymer flow) . . . . . . . . . . . . . . . . . . . . . . . . . . 4427.4 2D steady-state Navier–Stokes equations . . . . . . . . . . . 450

7.4.1 LSP based on residual functional: first order systemof PDEs . . . . . . . . . . . . . . . . . . . . . . . . . 452

7.4.2 LSP based on residual functional: higher ordersystems of PDEs . . . . . . . . . . . . . . . . . . . . . 454

7.4.3 Slider bearing; flow of a viscous lubricant . . . . . . . 4557.4.4 A square lid-driven cavity . . . . . . . . . . . . . . . 4577.4.5 Asymmetric backward facing step . . . . . . . . . . . 4617.4.6 Flow past a circular cylinder . . . . . . . . . . . . . . 467

7.5 2D compressible Newtonian fluid flow . . . . . . . . . . . . . 4717.5.1 Carter’s plate . . . . . . . . . . . . . . . . . . . . . . 474

7.5.1.1 Mach 1 flow . . . . . . . . . . . . . . . . . . 4767.5.1.2 General consideration for higher Mach

number flows . . . . . . . . . . . . . . . . . . 4787.5.1.3 Mach 2 flow . . . . . . . . . . . . . . . . . . 4797.5.1.4 Mach 3 flow . . . . . . . . . . . . . . . . . . 4807.5.1.5 Mach 5 flow . . . . . . . . . . . . . . . . . . 481

7.5.2 Mach 1 flow past a circular cylinder . . . . . . . . . . 4837.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486

8 Basic Elements of Mapping and Interpolation Theory 4938.1 Mapping in one dimension . . . . . . . . . . . . . . . . . . . 493

8.1.1 Mapping of points . . . . . . . . . . . . . . . . . . . . 4948.1.2 Mapping of lengths . . . . . . . . . . . . . . . . . . . 4948.1.3 Behavior of dependent variable φ over Ωe . . . . . . . 495

8.2 Elements of interpolation theory over Ωξ = [−1, 1] . . . . . . 495

CONTENTS xiii

8.2.1 A polynomial approximation in one dimension . . . . 495

8.2.2 Lagrange interpolating polynomials in one dimension 497

8.2.3 p-version hierarchical functions in one dimension . . . 501

8.2.4 Higher order global differentiability approximationsin one dimension: p-version . . . . . . . . . . . . . . . 507

8.2.4.1 Local approximation of class C1(Ωe) . . . . . 508

8.2.4.2 Interpolations or local approximations of classC2(Ωe): . . . . . . . . . . . . . . . . . . . . . 511

8.2.4.3 Local approximations of class Ci(Ωe) . . . . 515

8.3 Mapping in two dimensions: quadrilateral elements . . . . . 516

8.4 Local approximation over Ωm: quadrilateral elements . . . . 520

8.4.1 C00 local approximations over Ωξη: polynomialapproach . . . . . . . . . . . . . . . . . . . . . . . . . 522

8.4.2 C00 Lagrange type local approximation usingtensor product . . . . . . . . . . . . . . . . . . . . . . 525

8.4.3 C00 p-version hierarchical local approximations basedon Lagrange polynomials . . . . . . . . . . . . . . . . 529

8.5 2D Cij(Ωe) p-version local approximations . . . . . . . . . . 532

8.5.1 2D interpolations of type C11(Ωe) with p-levelsof pξ and pη . . . . . . . . . . . . . . . . . . . . . . . 538

8.5.2 2D interpolations of type C22(Ωe) with p-levelsof pξ and pη . . . . . . . . . . . . . . . . . . . . . . . 540

8.5.3 2D Cij(Ωe) interpolations of p-levels pξ and pη . . . . 542

8.6 2D Cij(Ωe) approximations for quadrilateral elements . . . . 542

8.6.1 C11 HGDA for 2D distorted quadrilateral elements inxy space . . . . . . . . . . . . . . . . . . . . . . . . . 549



8.6.4 Derivation of Cij approximations for distortedquadrilateral elements . . . . . . . . . . . . . . . . . . 553

8.6.5 Limitations of 2D C11 global differentiability localapproximations for distorted quadrilateral elements . 554

8.7 Interpolation theory for 2D triangular elements . . . . . . . . 556

8.7.1 Langrange family C00 basis functions based on Pascaltriangle . . . . . . . . . . . . . . . . . . . . . . . . . . 556

8.7.2 Lagrange family C00 basis functions based onarea coordinates . . . . . . . . . . . . . . . . . . . . . 558

8.7.3 Higher degree C00 basis functions using areacoordinates . . . . . . . . . . . . . . . . . . . . . . . . 559

8.8 1D and 2D approximations based on Legendre polynomials . 566

xiv CONTENTS

8.8.1 Legendre polynomials . . . . . . . . . . . . . . . . . . 566

8.8.2 1D p-version C0 hierarchical approximation functions(Legendre polynomials) . . . . . . . . . . . . . . . . . 567

8.8.3 2D p-version C00 hierarchical interpolation functionsfor quadrilateral elements (Legendre polynomials) . . 567

8.8.4 2D Cij p-version interpolations functions for quadri-lateral elements (Legendre polynomials) . . . . . . . 568

8.8.5 2D C00 p-version interpolation functions for triangu-lar elements (Legendre polynomials) . . . . . . . . . . 568

8.8.6 2D Cij interpolation functions for triangular elements(Legendre polynomials) . . . . . . . . . . . . . . . . . 571

8.9 1D and 2D interpolations based on Chebyshev polynomials . 577

8.9.1 Chebyshev polynomials . . . . . . . . . . . . . . . . . 577

8.9.2 1D C0 p-version hierarchical interpolations based onChebyshev polynomials . . . . . . . . . . . . . . . . . 577

8.9.3 2D p-version C00 hierarchical interpolation functionsfor quadrilateral elements (Chebyshev polynomials) . 578

8.9.4 2D Cij p-version interpolation functions for quadri-lateral elements (Chebyshev polynomials) . . . . . . . 578

8.10 Serendipity family of C00 interpolations . . . . . . . . . . . . 578

8.10.1 Method of deriving serendipity interpolation functions 579

8.11 Interpolation functions for 3D elements . . . . . . . . . . . . 584

8.11.1 Hexahedron elements . . . . . . . . . . . . . . . . . . 584

8.11.1.1 Mapping of points . . . . . . . . . . . . . . . 584

8.11.1.2 Mapping of lengths . . . . . . . . . . . . . . 586

8.11.1.3 Mapping of volumes . . . . . . . . . . . . . . 586

8.11.1.4 Obtaining derivatives of φeh(ξ, η, ζ) withrespect to x, y, z . . . . . . . . . . . . . . . . 587

8.11.2 Local approximation for a dependent variableφ over Ωm . . . . . . . . . . . . . . . . . . . . . . . . 588

8.11.2.1 Hexahedron elements . . . . . . . . . . . . . 588

8.11.2.2 Higher degree approximations of φ over Ωm . 590

8.11.2.3 C000 Lagrange type local approximationsusing tensor product . . . . . . . . . . . . . . 592

8.11.2.4 C000 p-version 3D hierarchical localapproximations: using tensor product . . . . 597

8.11.2.5 3D Cijk(Ωe) p-version local approximations:Hexahedron elements . . . . . . . . . . . . . 599

8.11.2.6 3D Cijk(Ωe) p-version interpolations fordistorted hexahedron elements:27 node element . . . . . . . . . . . . . . . . 600

CONTENTS xv

8.11.2.7 Interpolation theory for 3D tetrahedronelements: basis functions of class C000(Ωe)based on Lagrange interpolations . . . . . . . 600

8.11.2.8 Lagrange family C000 interpolations basedon volume coordinates . . . . . . . . . . . . . 601

8.11.2.9 Higher degree C000 basis functions usingvolume coordinates . . . . . . . . . . . . . . 603

8.11.2.10 Four-node linear tetrahedron element (p-levelof one) . . . . . . . . . . . . . . . . . . . . . 604

8.11.2.11 A ten-node tetrahedron element (p-level of 2) 6048.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604

9 Linear Elasticity using the Principle of MinimumTotal Potential Energy 6099.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 6099.2 New notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 6109.3 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6119.4 Element equations . . . . . . . . . . . . . . . . . . . . . . . . 611

9.4.1 Local approximation of the displacement field . . . . 6119.4.2 Stresses and strains . . . . . . . . . . . . . . . . . . . 6129.4.3 Strain energy Πe

1 and potential energy of loads Πe2 . . 612

9.4.4 Total potential energy Πe for an element e . . . . . . 6149.5 Finite element formulation for 2D linear elasticity . . . . . . 617

9.5.1 Local approximation of u and v over Ωe or Ωξη . . . 6189.5.2 Stresses, strains and constitutive equations . . . . . . 6189.5.3 [B] matrix relating strains to nodal degrees of freedom 6199.5.4 Element stiffness matrix [Ke] . . . . . . . . . . . . . 6199.5.5 Transformations from (ξ, η) to (x, y) space . . . . . . 6209.5.6 Body forces . . . . . . . . . . . . . . . . . . . . . . . 6209.5.7 Initial strains (thermal loads) . . . . . . . . . . . . . 6219.5.8 Equivalent nodal loads F ep due to pressure acting

normal to the element faces . . . . . . . . . . . . . . 6229.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624

10 Linear and Nonlinear Solid Mechanics using the Principle ofVirtual Displacements 62510.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 62510.2 Principle of virtual displacements . . . . . . . . . . . . . . . 62610.3 Virtual work statements . . . . . . . . . . . . . . . . . . . . . 627

10.3.1 Stiffness matrix . . . . . . . . . . . . . . . . . . . . . 63310.4 Solution method . . . . . . . . . . . . . . . . . . . . . . . . . 635

10.4.1 Summary of solution procedure . . . . . . . . . . . . 63610.5 Finite element formulation for 2D solid continua . . . . . . . 637

xvi CONTENTS

10.6 Finite element formulation for 3D solid continua . . . . . . . 641

10.7 Axisymmetric solid finite elements . . . . . . . . . . . . . . . 644

10.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648

11 Additional Topics in Linear Structural Mechanics 651

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 651

11.2 1D axial spar or rod element in R1 (1D space) . . . . . . . . 651

11.2.1 Stresses and strains . . . . . . . . . . . . . . . . . . . 653

11.2.2 Total potential energy: Πe . . . . . . . . . . . . . . . 653

11.3 1D axial spar or rod element in R2 . . . . . . . . . . . . . . . 656

11.3.1 Coordinate transformation . . . . . . . . . . . . . . . 656

11.3.2 A two member truss . . . . . . . . . . . . . . . . . . 659

11.3.2.1 Computations . . . . . . . . . . . . . . . . . 660

11.3.2.2 Post-processing . . . . . . . . . . . . . . . . . 665

11.4 1D axial spar or rod element in R3 (3D space) . . . . . . . . 666

11.5 The Euler–Bernoulli beam element . . . . . . . . . . . . . . . 668

11.5.1 Derivation of the element equations (GM/WF) . . . . 670

11.5.2 Local approximation . . . . . . . . . . . . . . . . . . 671

11.6 Euler-Bernoulli frame elements in R2 . . . . . . . . . . . . . 675

11.7 The Timoshenko beam elements . . . . . . . . . . . . . . . . 677

11.7.1 Element equations: GM/WF . . . . . . . . . . . . . . 678

11.8 Finite element formulations in R2 and R3 . . . . . . . . . . . 681

11.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681

12 Convergence, Error Estimation, and Adaptivity 683

12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 683

12.2 h-, p-, k-versions of FEM and theirconvergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684

12.2.1 h-version of FEM and h-convergence . . . . . . . . . 685

12.2.2 p-version of FEM and p-convergence . . . . . . . . . 686

12.2.3 hp-version of FEM and hp-convergence . . . . . . . . 686

12.2.4 k-version of FEM and k-convergence . . . . . . . . . 687

12.3 Convergence and convergence rate . . . . . . . . . . . . . . . 689

12.3.1 Convergence behavior of computations . . . . . . . . 690

12.3.2 Convergence rates . . . . . . . . . . . . . . . . . . . . 692

12.4 Error estimation and error computation . . . . . . . . . . . . 693

12.5 A priori error estimation . . . . . . . . . . . . . . . . . . . . 694

12.5.1 Galerkin method with weak form (GM/WF):self-adjoint operators . . . . . . . . . . . . . . . . . . 694

12.5.2 GM/WF for non-self adjoint and non-linear operators 697

12.5.3 Least-squares method based on residual functional:self-adjoint and non-self-adjoint operators . . . . . . 698

CONTENTS xvii

12.5.4 Least-squares method based on residual functional fornon-linear operators . . . . . . . . . . . . . . . . . . . 699

12.5.5 Integral forms based on other methods ofapproximation . . . . . . . . . . . . . . . . . . . . . . 702

12.5.6 General remarks . . . . . . . . . . . . . . . . . . . . . 702

12.5.7 A priori error estimates: GM/WF and LSP . . . . . 703

12.5.7.1 Model problem 1: GM/WF . . . . . . . . . . 703

12.5.7.2 Model problem 2: LSP . . . . . . . . . . . . 704

12.5.7.3 Proposition and proof . . . . . . . . . . . . . 706


12.5.7.5 Convergence rates . . . . . . . . . . . . . . . 714


12.5.7.7 General Remarks . . . . . . . . . . . . . . . . 719

12.6 Model problems . . . . . . . . . . . . . . . . . . . . . . . . . 719

12.6.1 Model problem 1: Self-adjoint operator, 1D diffusionequation . . . . . . . . . . . . . . . . . . . . . . . . . 720

12.6.1.1 GM/WF . . . . . . . . . . . . . . . . . . . . 721

12.6.1.2 LSP, higher-order system (no auxiliaryequation) . . . . . . . . . . . . . . . . . . . . 727

12.6.2 Model problem 2: Non-self-adjoint operator,1D convection-diffusion equation . . . . . . . . . . . . 730

12.6.2.1 LSP: First order system . . . . . . . . . . . . 731

12.6.2.2 GM/WF . . . . . . . . . . . . . . . . . . . . 737

12.6.2.3 LSP: Higher order system (without auxiliaryequation) . . . . . . . . . . . . . . . . . . . . 738

12.6.3 Model problem 3: Non-linear operator, 1D Burgersequation . . . . . . . . . . . . . . . . . . . . . . . . . 741

12.6.3.1 LSP: Higher-order system (without auxiliaryequation) . . . . . . . . . . . . . . . . . . . . 743

12.6.3.2 GM/WF . . . . . . . . . . . . . . . . . . . . 747

12.7 A posteriori error estimation and computation . . . . . . . . 747

12.7.1 A posteriori error estimation . . . . . . . . . . . . . . 747

12.7.2 A posteriori error computation . . . . . . . . . . . . . 749

12.8 Adaptive processes in finite element computations . . . . . . 751

12.8.1 Adaptive processes for 1D convection-diffusionequation: non-self adjoint operator . . . . . . . . . . 752

12.8.1.1 Adaptivity in the pre-asymptotic range:uniform h-refinement . . . . . . . . . . . . . 752

12.8.1.2 Adaptivity in the pre-asymptotic range: adap-tiveh-refinement . . . . . . . . . . . . . . . . . . 753

xviii CONTENTS

12.8.1.3 Adaptivity in the pre-asymptotic range: gradedh-rediscretizations . . . . . . . . . . . . . . . 754

12.8.1.4 General Remarks . . . . . . . . . . . . . . . . 75712.8.1.5 Adaptivity in the onset of asymptotic and

asymptotic ranges: uniform p-refinement . . 75812.8.1.6 Adaptivity in the onset of asymptotic and

asymptotic ranges: adaptive p-refinement . . 75812.8.1.7 Adaptivity in the onset of asymptotic and

asymptotic ranges: adaptive h-refinement . . 76012.8.1.8 Adaptivity using higher geometric ratios for

h-rediscretization at Pe = 1000 and Pe = 106 76012.8.2 Adaptive processes for 1D Burgers equation:

non-linear operator . . . . . . . . . . . . . . . . . . . 76112.8.3 Adaptive processes for 1D diffusion equation:

self-adjoint operator . . . . . . . . . . . . . . . . . . . 76512.8.4 General Remarks . . . . . . . . . . . . . . . . . . . . 767

12.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767

Appendix A: Numerical Integration using Gauss Quadrature 771A.1 Gauss quadrature in R1, R2 and R3 . . . . . . . . . . . . . . 771

A.1.1 Line integrals over Ωm = Ωξ = [−1, 1] . . . . . . . . . 771A.1.2 Area integrals over Ωm = Ωξη = [−1, 1]× [−1, 1] . . . 772A.1.3 Volume Integrals over Ωm = Ωξηζ = [−1, 1]×[−1, 1]×

[−1, 1] . . . . . . . . . . . . . . . . . . . . . . . . . . 773A.2 Gauss quadrature over triangular domains . . . . . . . . . . 775

INDEX 779

Preface

Since there are already many textbooks and monographs on the finite el-ement method, it is perhaps natural to ask “why another book?” This ques-tion can be answered if one examines the published material on the subject.Broadly speaking, the books on the subject can be classified into two cate-gories: those that present the finite element method as a study in appliedmathematics and those that approach the subject using specific applications.Finite element books on linear elasticity, stress analysis, heat transfer, fluidmechanics, and so on are examples of application based approach. Bothtypes of writings have their own strengths and weaknesses from the point ofview of the students wanting to learn the subject. The applied mathematicsapproach requires more rigorous mathematical background and preparationand as a consequence graduate students in engineering and sciences shy awayfrom learning the subject through this approach. Secondly, these writingsoften lack application aspects of the subject that are generally helpful forengineering and science students. The writings that are highly focused inpresenting the subject through specific applications obviously result in lossof generality and as a consequence the students often learn the subject asa technique for a specific class of problems. For example, a finite elementbook on linear elasticity may generally focus on minimization of total poten-tial energy and as a consequence the students may never realize the muchbroader impact of the subject on all BVPs in the other areas of mechanicsand applied sciences. Distinct demarcation in writings and teaching of thesubject for linear processes, non-linear processes, solids, liquids and gassesoften leaves the students confused and unclear not only regarding the math-ematical foundations of the subject, but also its much broader impact inapplications in all areas of engineering and sciences.

This book is intended to bridge the gap between the applied mathemat-ics and strictly application-oriented books. The material in this book ispresented in a mathematically rigorous fashion but with sufficient examples,applications, and illustrations in various areas of engineering, sciences, andmathematical physics so that students are able to grasp the mathematicalfoundation of the subject as well as its versatility of applications in all ar-eas of engineering, sciences, and mathematical physics. The book is aimedfor a first semester graduate study of the finite element method for bound-ary values problems (BVPs). The finite element method is introduced andpresented as a method of approximation for obtaining numerical solutionsof differential and partial differential equations describing time-independent

xix

xx PREFACE

processes (BVPs) regardless of their origin or field of application. In order toaddress the totality of all BVPs rigorously and in an application-independentfashion, the differential operators appearing in all BVPs are classified math-ematically into three categories: self-adjoint, non-self adjoint, and non-linearoperators, and their properties are established. These are then utilized withvarious methods of approximation such as the Galerkin method, Petrov-Galerkin method, weighted residual method, Galerkin method with weakform, least squares processes, and other methods from which the details ofthe finite element processes are derived. A correspondence is established be-tween the methods of approximation and hence finite element method andthe elements of the calculus of variations. This is then utilized for vari-ous methods of approximation for the three classes of differential operatorsto determine which methods of approximation yield unconditionally stablecomputational processes.

Chapter 1 provides a brief introduction of the subject of the mathematicsof computations and the finite element method for boundary value problems.Concepts of discretization, local approximations, integral forms, element al-gebraic equations, assembly of element equations, computations of solutions,and post-processing of solutions are introduced. An introduction to the k-version of the finite element method and hpk framework for computationsof the solutions of the boundary value problems is also presented. Basicelements from applied mathematics: spaces, scalar product spaces, scalarproduct and its significance, function spaces, differential operators and theirmathematical classifications, and energy product are presented in chapter 2.Chapter 2 also contains elements of calculus of variations and functional anal-ysis: concept of variation of a functional, Euler’s equations, correspondencebetween extrema of functionals and solutions of boundary value problems,fundamental and other lemmas in calculus of variations and their proofs,Riemann and Lebesgue integrals, properties of self adjoint, non-self adjoint,and non-linear differential operators including examples. Concepts and defi-nitions of variationally consistent (VC) and variationally inconsistent (VIC)integral forms are introduced to establish when the integral forms yield un-conditionally stable computational processes.

Chapter 3 contains classical methods of approximation based on funda-mental lemma such as the Galerkin method, Galerkin method with weakform, Petrov-Galerkin method, and method of weighted residuals (GM,GM/WF, PGM, WRM), and least squares method based on residual func-tional (LSM) for all three classes of differential operators. Many theoremsand their proofs related to VC and VIC integral forms from these methodsare presented for the three classes of differential operators. Applicationsand model problems are considered to illustrate various concepts. Seriousshortcomings of classical methods of approximation for practical applica-

xxi

tions are discussed. Chapter 4 introduces the finite element model detailsfor GM, GM/WF, PGM, WRM, and LSM for all three classes of differentialoperators. Specific details and formulations for model problems in R1, R2,and R3, proofs of VC and VIC integral forms for the model problems, C0

solutions and solutions of higher classes in hpk framework for finite elementformulations and processes for self adjoint, non-self adjoint, and non-lineardifferential operators are presented in chapters 5, 6, and 7, respectively.

Chapter 8 contains basic elements of mapping and interpolation theory.Details of mapping of points, lengths, areas, and volumes are discussed.Local approximations are presented for elements in R1, R2, and R3 usingLagrange, Legendre, and Chebyshev polynomials. Local approximations ofclass C0 and higher classes, p-version hierarchical local approximations ofclass C0 and higher classes are presented in R1, R2, and R3. Area and volumecoordinates are introduced and utilized for triangular and tetrahedral familyof elements in R2 and R3 to derive local approximations of various classes.

Chapter 9 presents finite element formulations in linear solid and struc-tural mechanics, derived using principle of minimum total potential energy,formulated directly from the physics of deformation without utilizing theunderlying differential equations. A general derivation applicable to finiteelement formulations in R1, R2, and R3 is presented first and then followed byspecific examples in R2 such as plane stress. Chapter 10 contains derivationsof finite element formulations using principle of virtual work. This approachis specially meritorious for finite deformation and finite strain reversible pro-cesses. A general derivation in R3 is presented first that is specialized forapplications in R1 and R2. Some additional finite element formulations re-lated to axial deformation of rods (or spars) in R1, R2, and R3, use of elementlocal coordinate systems, and finite element formulations for Euler–Bernoulliand Timoshenko beams are presented in chapter 11. Appendix A containsdetails related to Gauss quadrature in R1, R2, and R3. The computer pro-gram, Finesse (“Finite element system”), used to solve the problems in thisbook is available free of cost from the first author upon request.

The material in this book is self-contained and requires no supplementaryreading or any other reference material. The students learning the subjectthrough this book are expected to have two semesters of calculus, an un-dergraduate course in differential and partial differential equations, a coursein linear algebra and an undergraduate level course in numerical methods.An advanced course in partial differential equations and a course in calculusof variations are helpful but are not prerequisite to learning the materialpresented in this book. This book is a result of the evolution of the first au-thor’s thirty years of teaching and research of the subject in the Departmentof Mechanical Engineering at the University of Kansas. The author’s ownresearch work in mathematics of computations and the finite element subject

xxii PREFACE

and continuum mechanics has contributed heavily to this unique approach ofpresenting the mathematical details of the foundation of the finite elementmethod with simplicity while maintaining its versatility and transparencyfor applications. The first author has successfully utilized this material ineducating graduate students on the subject as well as preparing them forpost-graduate studies and research.

Both authors over the last twenty years have been engaged in joint re-search grants, publications and collaborative research efforts that have re-sulted in many new concepts such as operator classifications, the k-versionof finite element method, variationally consistent integral forms, and so onthat form the foundation of much of the material in this book. Authors’long friendship and research collaborations have been extremely enjoyableand fruitful in bringing focus, depth, clarity, and in developing this uniqueapproach of presenting the mathematics and computations related to thefinite element method for boundary value problems presented in this book.The DEPSCoR/AFOSR grant to the first author and the joint researchgrants to the authors from the U.S. Army (ARO) related to k-version offinite element method, operator classifications, VC and VIC integral forms,and unconditionally stable computational processes resulted in a significantnumber of joint fundamental publications that form the core of the mate-rial in many chapters of this book. The authors are truly grateful to manyof their graduate students whose Ph.D. and M.S. theses in many areas ofcomputational mathematics and finite element method have contributed im-mensely in bringing the subject matter to its present level of maturity.

The first author is extremely thankful to his Ph.D. student Mr. TylerStone who prepared the first draft of the manuscript of this book singlehandedly, performed many numerical studies contained in the book, andalso helped in many subsequent versions. Thanks are also due to Dr. DanielNunez, a former Ph.D. student of the first author, who encouraged andsupported such writing endeavors and contributed heavily in many por-tions of chapters 6, 7, and 8 including numerical studies for model prob-lems contained in these chapters. A very special thanks to Mr. AaronD. Joy, the first author’s current Ph.D. student who has typeset the bookand has typed and retyped many portions of the book, reorganized and inmany cases redid the graphs and illustrations to bring the manuscript ofthe book to its present level. His interest and knowledge of the subject,hard work, and commitment to this book project have been instrumental inthe completion of the book. This book would not have been possible with-out the research grant from DEPSCoR/AFOSR to the first author and thejoint research grants: W911NF-09-1-0548 (FED0065623), W-911NF-11-1-0471 (FED0061541), and W911NF-12-1-0463 from the U.S. Army ResearchOffice (ARO) to the authors that led to research in various areas of com-

xxiii

putational mathematics and finite element processes. Sincere thanks to Dr.Joseph Myers, Division Chief, Mathematical Sciences Division, InformationScience Directorate, ARO, for his interest and support of some of the re-search results included in this book.

This book contains so many equations, derivations, and mathematicaldetails that it is hardly possible to avoid some typographical and other errors.Authors would be grateful to those readers who are willing to draw attentionto the errors using the emails: [email protected] or [email protected].

Karan S. Surana, Lawrence, KSJ. N. Reddy, College Station, TX

mailto:[email protected]

mailto:[email protected]

About the Authors

Karan S. Surana, born in India, went to undergraduate school at BirlaInstitute of Technology and Science (BITS), Pilani, India, and received aB.E. degree in Mechanical Engineering in 1965. He then attended the Uni-versity of Wisconsin, Madison, where he obtained M.S. and Ph.D. degrees inMechanical Engineering in 1967 and 1970, respectively. He worked in indus-try, in research and development in various areas of computational mechanicsand software development, for fifteen years: SDRC, Cincinnati (1970–1973),EMRC, Detroit (1973–1978); and McDonnell Douglas, St. Louis (1978–1984). In 1984, he joined the Department of Mechanical Engineering facultyat University of Kansas, where he is currently the Deane E. Ackers UniversityDistinguished Professor of Mechanical Engineering.

His areas of interest and expertise are computational mathematics, com-putational mechanics, and continuum mechanics. He is author of over 350research reports, conference papers, and journal articles. He has served asadvisor and chairman of 50 M.S. students and 22 Ph.D. students in variousareas of Computational Mathematics and Continuum Mechanics. He hasdelivered many plenary and keynote lectures in various national and inter-national conferences and congresses on computational mathematics, compu-tational mechanics, and continuum mechanics. He has served on interna-tional advisory committees of many conferences and has co-organized mini-symposia on k-version of the finite element method, computational meth-ods, and constitutive theories at US National Congresses of ComputationalMechanics organized by the US Association of Computational Mechanics(USACM). He is a member of International Association of ComputationalMechanics (IACM) and USACM, and a fellow and life member of ASME.

Dr. Surana’s most notable contributions include: large deformation finiteelement formulations of shells, the k-version of the finite element method,operator classification and variationally consistent integral forms in methodsof approximations, and ordered rate constitutive theories for solid and fluentcontinua. His most recent and present research work is in non-classical inter-nal polar continuum theories and non-classical Cosserat continuum theoriesfor solid and fluent continua and associated ordered rate constitutive theo-ries. He is author of recently published continuum mechanics textbook, Ad-vanced Mechanics of Continua, CRC/Taylor & Francis, Boca Raton, Florida,2015.

xxv

xxvi ABOUT THE AUTHORS

J. N. Reddy was born in Telangana, India, and obtained his B,E.(Mech) from Osmania University, Hyderabad, in 1968. He obtained M.S.from Oklahoma State University (1970) and Ph.D. from University of Al-abama in Huntsville (1973). He worked at Lockheed Missiles and SpaceCompany for a short period in 1974 and served on the faculty in the Schoolof Aerospace, Mechanical, and Nuclear Engineering at University of Okla-homa (1975–1980), in the Department of Engineering Science and Mechanicsat Virginia Polytechnic Institute and State University (1980–1992). Since1992 he has been in the Department of Mechanical Engineering at TexasA&M University, currently holding the titles University Distinguished Pro-fessor, Regents Professor, and the Oscar S. Wyatt Endowed Chair Professor.Professor Reddy is internationally-recognized for his research on mechanicsof composite materials and for computational methods. The shear defor-mation plate and shell theories that he developed bear his name (Reddythird-order shear deformation theory and Reddy layerwise theory) in theliterature. The finite element formulations and models he developed havebeen implemented into commercial software like ABAQUS, NISA, and Hy-perXtrude. He is the author of numerous journal papers and 20 textbooks,several of them with multiple editions. Dr. Reddy is one of the originaltop 100 ISI Highly Cited Researchers in Engineering around world with over19,200 citations with h-index of over 66 as per Web of Science; the numberof citations is over 47,000 with h-index of 89 and i10-index of 405 (i.e., 405papers are cited at least 10 times) as per Google Scholar.

Most significant awards and honors he received to date are: 1992 Worces-ter Reed Warner Medal and 1995 Charles Russ Richards Memorial Award ofthe American Society of Mechanical Engineers (ASME); 1997 Archie HigdonDistinguished Educator Award from the Mechanics Division of the AmericanSociety of Engineering Education; 1998 Nathan M. Newmark Medal from theAmerican Society of Civil Engineers (ASCE); 2000 Excellence in the Fieldof Composites and 2004 Distinguished Research Award from the AmericanSociety for Composites (ASC); 2003 Computational Solid Mechanics awardfrom the US Association of Computational Mechanics; 2014 The IACM O.C.Zienkiewicz Award from the International Association of Computational Me-chanics; 2014 Raymond D. Mindlin Medal from the Engineering MechanicsInstitute of ASCE; and 2016 William Prager Medal of the Society of En-gineering Science. He is an elected member of the US National Academyof Engineering for contributions to composite structures and to engineeringeducation and practice and elected as a Foreign Fellow of the Indian Na-tional Academy of Engineering. He is a fellow of many professional societies(e.g., AIAA, ASC, ASCE, ASME, AAM, USACM, IACM), and serves onthe editorial boards of two dozen journals. A more complete resume can befound at http://www.tamu.edu/acml/.

http://www.tamu.edu/acml/

1

Introduction

1.1 General Comments and Basic Philosophy

The physical processes encountered in all branches of sciences and en-gineering can be classified into two categories: time-dependent processesand stationary processes. Time-dependent processes describe evolutions inwhich quantities of interest change as time elapses. If the quantities of in-terest cease to change in an evolution, then the evolution is said to have astationary state. Not all evolutions reach stationary states. The evolutionswithout a stationary states are often referred to as unsteady processes. Sta-tionary processes are those in which the quantities of interest do not dependon time. For a stationary process to be valid or viable it must correspond tothe stationary state of an evolution. Every process in nature is an evolution,but nonetheless it is sometimes convenient to consider their stationary state.In this book, we consider only time-independent or stationary processes.

A mathematical description of most stationary processes in sciences andengineering often leads to a system of ordinary or partial differential equa-tions. The mathematical descriptions of the stationary processes are referredto as boundary value problems (BVPs). Since stationary processes are inde-pendent of time, the partial differential equations describing their behavioronly involve dependent variables and spatial coordinates as independent vari-ables. On the other hand, mathematical descriptions of evolutions leads topartial differential equations in dependent variables and space coordinates aswell as time as independent variables and are referred to as initial value prob-lems (IVPs). The numerical solutions of the BVPs using the finite elementmethod is the subject of study in this book.

In case of simple physical systems, the BVPs may be simple enough topermit analytical solutions, however most physical processes of interest maybe quite complicated and their mathematical descriptions (BVPs) may becomplex enough not to permit analytical solutions. In such case, one couldundertake simplification of the mathematical description to a point that an-alytical solutions are possible. In this approach, the simplified forms maynot describe the actual behavior and sometimes this simplification may notbe possible. In the second alternative, we abandon the possibility of ana-lytical solutions all together as viable means of solving complex practicalproblems and instead resort to numerical methods for obtaining numerical

1

2 INTRODUCTION

solutions of BVPs. The finite element method is one such method of solvingBVPs numerically and constitutes the subject matter for this book. Beforeproceeding with the details of the FEM for BVPs, it is perhaps fitting to dis-cuss some of the commonly used numerical methods for obtaining numericalsolutions of the BVPs. These are:

1. finite difference method

2. finite volume method

3. finite element method

4. meshless and element free methods

5. boundary element method

6. and other methods, including hybrid methods that are a combination ofabove methods

A comprehensive and detailed description, merits and shortcomings ofall of these methods are not considered in this book but can be found inmany published works. In this book we consider only details of FEM asa technique for obtaining numerical solutions of the BVPs. Regardless ofthe method used for obtaining numerical solutions of the BVPs, all of themethods have one feature in common: in each case, the calculated numericalsolution is an approximation of the true or theoretical solution of the BVP.This is perhaps the reason that all numerical methods for BVPs are referredto as methods of approximation.

In the development of the numerical methods for BVPs one must exerciseutmost care in ensuring that the methods are general and are not problemdependent, as the host of BVPs in various areas of engineering, sciences, andmathematical physics is so vast that the study, development, and implemen-tation of problem dependent computational methods is a never-ending andexhausting undertaking with total lack of generality. It is for this reason thatwe must consider totality of all possible BVPs in all areas of engineering, sci-ences, and mathematical physics and perhaps classify them mathematicallyinto groups so that specialized, accurate and efficient numerical methodsonly need to be developed for each of these groups. In engineering and phys-ical sciences, it has been customary and perhaps more appealing to classifyBVPs as: elliptic, parabolic, hyperbolic, mixed type etc. due to the factthat such classification may be more meaningful in revealing the physicsof the processes described by the BVPs. However, if we intend to borrowvarious tools from different branches of applied mathematics to pursue thedevelopment of a comprehensive and general mathematical and computa-tional framework to solve BVPs numerically, then the classification of BVPsdepending upon the strict mathematical nature of the differential operatorsinvolved in their description is certainly the most prudent choice. All BVPs

1.2. BASIC CONCEPTS OF THE FINITE ELEMENT METHOD 3

can be classified mathematically into three categories, those described by:(1) self-adjoint differential operators, (2) non-self-adjoint differential oper-ators, and (3) non-linear differential operators. This classification is vitalin understanding the mathematical properties of these operators and devel-opment of general and problem independent numerical methods to addresstheir solutions. In this book, we consider FEM for the BVPs described bythese three classes of differential operators. This approach addresses numer-ical solutions of the totality of all BVPs using finite element method.

When addressing numerical methods for BVPs in addition to the factthat all such methods are methods of approximation there is one more fea-ture that is common in all methods: the BVPs described by the differentialoperators must somehow be converted into a system of algebraic equations.The manner in which this is accomplished differs in different methods of ap-proximation. The nature of the coefficient matrix in the resulting algebraicsystems depends upon: a) the method of approximation and b) the natureof the differential operators. A solution of the algebraic equation providesthe approximate solution. Thus, in principle this process is rather simplebut the ingenuity of one method of approximation over the others lies in themanner in which this is accomplished.

1.2 Basic Concepts of the Finite Element Method

Symbolically we can represent all BVPs by writing Aφ − f = 0 in Ω,where A is the differential operator, φ are the dependent variables and f isthe nonhomogeneous part. Here Ω is the domain of definition of the BVP.It is a subset of R1 ≡ R, R2 or R3, that is, it is a collection of x; x, y orx, y, z (in the continuum sense) for which the BVP is valid. In addition,the BVP description has boundary conditions on the part or the whole ofthe boundary Γ of the domain Ω. We define Ω as the closure Ω = Ω ∪ Γ.This symbolic representation of the BVPs is helpful in two ways. First,it makes the presentation of the development of methodologies for theirsolutions compact and concise and secondly with this representation we canconcentrate on the mathematical properties of the operator A as opposed togetting entangled with the physics.

1.2.1 Discretization

When addressing numerical solutions of BVPs using the finite elementmethod, the domain of definition Ω over which the BVP is defined is subdi-vided into smaller domains called subdomains. Now, the original domain ofdefinition Ω can be visualized as the assembly of these subdomains intercon-nected with each other through their common boundaries. The subdomainsare obviously of finite sizes. Each subdomain is referred to as a finite element.

4 INTRODUCTION

The assembly of these subdomains, that is, the finite elements is referred toas a finite element mesh or discretization. The precise nature and the shapesof these finite elements depend upon the number of independent variablesin the PDEs, that is, 1-D, 2-D or 3-D boundary value problems and also tosome extent on our choice. For example, for 1-D BVPs the finite elementsare simply line segments interconnected at the two ends with adjacent linesegments to constitute the entire domain of definition of the 1-D BVP whichis also a line segment. For 2-D BVPs the domain of definition is an area andhence in this case many choices exist for choosing a finite element shape:triangular with linear sides, rectangular with linear sides, triangular withdistorted sides, quadrilateral with distorted sides and others. For 3-D BVPsthe finite elements are three-dimensional as well. Tetrahedron and hexahe-dron shapes with undistorted and distorted edges and faces are commonlyused.

When all finite elements of a discretization are of the same shape and size,the discretization is called a uniform discretization. On the other hand, whenthe elements of the discretization vary in size, the discretization is called non-uniform or graded. Regardless of the type of discretization, the consequenceof discretization is a finite element mesh. For each finite element of thediscretization we generally identify a finite number of points on the boundaryof the element (sometimes in the interior of the element as well). Thesepoints are called “nodes.” An element must be connected to the neighborsat the node points as well as common boundaries. Thus, in this process wehave identified a finite number of points for the whole discretization of thedomain Ω. The collection of all of these points for the whole discretization issometimes referred to as grid points. The numerical values of the dependentvariables (and/or their derivatives) at these grid points are the quantities ofinterest.

In summary, discretization of the domain of definition of Ω of the BVPsin the finite element processes yields subdomains, that is, finite elements, afinite element mesh, which is a collection of all finite elements, node pointson the boundaries and interiors of the elements and the grid points. Sym-

bolically we denote ΩT =M⋃e

Ωe as the discretization of Ω, the closure of the

domain of definition of the BVP in which Ωe = Ωe ∪ Γe is the domain ofdefinition of an element e, Γe is the closed boundary of the element e and Mis the total number of elements in the discretization ΩT . Figure 1.1(a) showsan axial rod fixed at one end and subjected to an axial load P at the otherend. The area of cross-section of the rod is A and the modulus of elasticityof the rod material is E. The mathematical idealization of the rod and thetwo three-element discretizations employing two-node and three-node lineelements are shown in Figs. 1.1(b)-(d). Figure 1.2(a) shows a thin plate in


1 32

21 3 4

1 32

21 3 4

L

A,E

y

x

P

P

P

P

y

y

a typical element e

xx

xe

y

a typical element e

xxe

(b) Mathematical idealization

(a) Physical system

(c) Discretization ΩT using two-node elements

(d) Discretization ΩT using three-node elements

xe1

xe+1

xe+2

ΩT

ΩT

Ω

Ωe

Ωe

Figure 1.1: Axial rod with end load P

tension (plane stress problem) with thickness t, modulus of elasticity E andPoisson’s ratio ν. The finite element discretizations of the plate using three-node and six-node triangular elements and nine-node quadrilateral elementsare shown in Figs. 1.2(b)-(d). Similarly, for the three-dimensional domainsof definition of the BVPs one could construct discretizations using tetrahe-dron or hexahedron with desired numbers of nodes on the edges, faces andthe interiors of elements.

1.2.2 Local approximation

For each finite element of the discretization one must define the behaviorsof the dependent variables. This is accomplished by using basis functions

6 INTRODUCTION

x

σx

a typical element e

Ωe

ΩT

y

(c) Discretization using 6-node triangular elements

x

σxΩe

a typical element e

ΩT

y

(b) Discretization using 3-node triangular elements

x

σx

a typical element e

Ωe

ΩT

(a) Physical system

x

σxΩ

t, E, ν

y

y

(d) Discretization using 9-node quadrilateral elements

Figure 1.2: Thin plate in tension


or local approximation functions and the dependent variable and possiblytheir derivative values at the nodes of the elements, often referred to asthe degrees of freedom. One constructs a linear combination using the localapproximation functions and the nodal degrees of freedom in which the localapproximation functions are known but the nodal degrees of freedom areunknown. Thus, each dependent variable has a local approximation overeach element of the discretization with its own local approximation functionsand the nodal degrees of freedom. We can write φeh = [N ]φe in which φehis the local approximation of the dependent variable φ over Ωe, [N ] is thecollection of all local approximation functions for the dependent variables φand φe is the collection of all nodal degrees of freedom for the dependentvariable φ. Such local approximations are constructed for each dependentvariable φ in the description of the BVP. In the following, for simplicity, weconsider φ to be the only dependent variable.

In the process of constructing φeh, there are two important points to note:first, φeh is constructed locally over the domain Ωe of an element e, that iswithout regard to the connecting neighboring elements to it, secondly, φehis an approximation of the theoretical solution φ over Ωe locally. Hence,the reason for referring to φeh as a local approximation of φ. The choices of[N ] and φe are not arbitrary and bear great consequences on the resultingfinite element processes. The subjects of interpolation theory and mappingin applied mathematics are vital in this respect. The basic concepts andrequired details from both of these areas are covered in chapter 8. Here weonly remark that the approximation φh of φ over ΩT is given by φh =

⋃e φ

eh

and is naturally dependent on φeh over Ωe. φh is referred to as the globalapproximation of φ over the discretization ΩT .

If one chooses monomials to construct local approximation φeh, then thefunctions in [N ], that is, local approximation functions are algebraic polyno-mials. By choosing appropriate degree of these polynomials, one can controlthe differentiability of local approximations φeh over Ωe. However, the globaldifferentiability of φh over ΩT is dependent on the differentiability of φehoverthe elements as well as their inter-element behaviors. The choice of thenodal degrees of freedom for Ωe plays a crucial role in controlling the globaldifferentiability of φh over ΩT .

1.2.3 Integral forms and algebraic equations over an element

Now we have the mathematical model of the BVP, Aφ − f = 0 in Ω, adiscretization ΩT of Ω = Ω∪Γ, and a local approximation φeh of φ over Ωe inwhich the degrees of freedom φe for φeh are unknown for each element of thediscretization. Determining numerical values of the degrees of freedom φeand thereby φeh for each element of ΩT is our objective. First, we note thatusing Aφ− f = 0 and φeh over Ωe one cannot solve for φe due to the fact

8 INTRODUCTION

that solutions of BVPs must consider ΩT so that boundary conditions canbe applied without which the solution is non-unique. However, we can takeadvantage of the fact that ΩT =

⋃e Ωe, and since ΩT is the discretization of

Ω, it suggests that perhaps we must examine φn, the approximation of φ overΩ and establish how we must proceed to determine the unknown constantsin φn. This is the subject of classical methods of approximations for BVPs.In these methods of approximation, we consider approximation φn of φ overΩ, that is, global approximation of φ over the non-discretized domain Ω.The finite element method is simply an application of these methods overdiscretization ΩT with local approximations φeh over Ωe. Let us first considerclassical methods of approximation. In all such methods we eventually havean integral from corresponding to Aφ − f = 0 over Ω. This is valid basedon the fundamental Lemma of the calculus of variations. These integralforms contain a function called a test function. When the approximationφn is substituted in the integral form and when we choose as many testfunctions as the number of basis functions in the approximation φn, theintegral form results into a system of as many algebraic equations as thenumber of basis functions or the number of unknowns in the approximationφn. The solution of these algebraic equations yields the desired numericalvalues of the constants in the approximation φn. The restrictions on thechoices of basis functions and the test functions are subject of detailed studyin the various methods of approximation.

In the finite element method, we construct the integral form over Ωe, thedomain of an element e. This is justified by the fact that integral forms overΩ results in functionals that are scalars and hence the functional over ΩT

(the discretization of Ω) can be written as the sum of the functionals over theindividual elements. The construction of the integral form over an elemente results in a system of algebraic equation for the element. Thus, we have asystem of algebraic equations for each element of the discretization contain-ing nodal dofs of the element and the right hand side. (details postponedfor later). These are generally referred to as element equations, discretizedequations of equilibrium, element stiffness equations etc. depending uponthe specific discipline of engineering, sciences, and mathematical physics.

1.2.4 Assembly of element equations

The fact that the functionals resulting from the integral form over ΩT

can be written as the sum of the functionals over the individual elements,and since the construction of the integral form over an element e results in asystem of algebraic equations in the element dofs, suggests that the algebraicequations of the elements must be summed or assembled in order to obtaina system of algebraic equations that is valid for the discretization ΩT . Therules of assembling the element equations are derived using the facts that: (1)


the functional for the whole discretization ΩT is the sum of the functionalsfor the elements and (2) the dofs at a node on the common boundariesbetween the elements are unique. Precise details of how to accomplish theassembly will be considered in the subsequent chapters. The outcome of theassembly process is a set of algebraic equations that are valid for the wholediscretization in which the dofs at the grid points are the unknowns.

1.2.5 Computation of the solution

The assembled equations in Section 1.2.4 must be subjected to boundaryconditions (transformed in terms of the dofs at the nodes on the boundaries)and then solved for the remaining unknown degrees of freedom. If the alge-braic system consists of a system of linear simultaneous equations (when Ain Aφ− f = 0 is linear), the elimination methods provide the most straight-forward and efficient means of finding the solution. When the system ofalgebraic equations are non-linear (i.e., A in Aφ− f = 0 is non-linear), iter-ative methods such as fixed-point method, Newton’s method, etc. must beused to find the solution. In either case we have the numerical values of thedofs at all of the grid points of the discretization.

1.2.6 Post-processing

Once we have the numerical values of the degrees of freedom at the gridpoints and thus at the node points of each element, the element local approx-imation provides an analytical expression for the behaviors of the dependentvariables over each element. Using such descriptions one could easily calcu-late the derivatives of the dependent variables or any other desired quantitieselementwise, hence for the whole discretization. This phase of the computa-tions is referred to as post-processing of the solution as it is the step after thecomputation of the solution at the grid points of the whole discretization.

1.2.7 Remarks

1. The most crucial steps in the finite element process are:

(a) Discretization. The finite element mesh.

(b) Local approximation. The interpolation theory is crucial for a goodunderstanding of this area

(c) Algebraic equations for an element. The methods of approximation andcalculus of variations are essential in this regard.

2. The finite element method is a piecewise application of the classical meth-ods of approximation in which we consider the entire domain of definitionof the BVP without discretization, a piece being a finite element of the

10 INTRODUCTION

discretization. Thus, in order to gain a good understanding of the finite ele-ment method one must have a thorough knowledge of the classical methodsof approximation.

3. The accuracy of the finite element method depends upon:

(a) The nature of the discretization, that is, the number of elements andtheir sizes and locations. If he is the measure of the size of element e, thenwe define

h = maxehe (1.1)

h is referred to as the characteristic length of the discretization.

(b) The nature of the local approximation, that is, the behaviors of thedependent variables over each element of the discretization. If we use mono-mials to construct the local approximation functions, then the local ap-proximation functions are algebraic polynomials of some degree. If p is thedegree of local approximation and if this is the same for all elements of thediscretization, then we can say that the accuracy of the finite element solu-tion also depends upon p. We refer to degree p of the polynomials defininglocal approximation functions as p-level.

4. There are three basic sources of errors in the finite element processes.Errors can be due to:

(a) Approximation of the shape of the domain of definition Ω of the BVP.This can be completely eliminated by choosing appropriate element shapeswith curved sides, edges and distorted faces so that any desired geometry ofΩ can be discretized accurately.

(b) Local approximations over each Ωe may not be an accurate description ofthe true behaviors of the dependent variables. This error is generally alwayspresent in all finite element solutions but can be reduced by reducing h andincreasing p.

(c) Numerical computations. Such errors are generally a results of the lackof desired precision during computations due to inadequate word size of thecomputer and due to the use of faulty or inadequate algorithms. Obviously,such errors can also be eliminated completely from the numerically computedsolutions.

Thus, the subject of study of error in the finite element process is predomi-nantly the study of errors in the computed solution due to local approxima-tion, thus it is dependent on h and p and of course the characteristics of thetheoretical solution of the BVP.

5. In all finite element processes, as more degrees of freedom are added,the solution accuracy improves, that is, the errors reduce. The degrees of


freedom can be added to a finite element computational process in one ofthree ways:

(a) One could progressively reduce h, that is, refine the mesh thereby addingmore elements, hence, more grid points and therefore more degrees of free-dom. This process is know as the h-version of the finite element methodand the process of improvement in the calculated solution and eventuallyapproaching the theoretical solution is referred to as h-convergence. In thisprocess one generally chooses a value of p and keeps it fixed while h is reducedso the mesh is refined.

(b) In the other approach, one could keep a fixed discretization, that is, notchange h but increase p-level, thereby adding more dofs at the nodes andhence, the grid points. This process is known as the p-version of the finiteelement method. The process of reducing error in the computed solution byonly increasing p-levels for the elements of the discretization while keepingh constant and eventually converging to the theoretical solution is known asp-convergence.

(c) The finite element processes in which h is reduced and p is increasedsimultaneously are known as the hp-version of the finite element methodwhich leads to hp-convergence.

1.2.8 k-version of the finite element methodand hpk framework

Now since we understand the basic concept of the finite element method(even though in the most rudimentary way), it is perhaps fitting to askwhat features the numerically computed solutions must have. The answer,of course, is that it should be as close to the theoretical solution in as manyaspects as possible. This is rather vague but can be made more concrete ifwe examine the features of the theoretical solutions of the BVPs. When thetheoretical solutions of BVPs are analytic, then we observe that the deriva-tives of the theoretical solution up to certain order (say j) are continuousover Ω, the domain of definition of the BVP, and we say that such solu-tions are of class Cj(Ω) in which j can be infinity. However, j can neverbe less than the highest orders of the derivatives of the dependent variablesapprearing in the GDEs of the BVP. We say that the theoretical solutionhas global differentiability of order j. In order to incorporate this featureof the theoretical solution in the finite element computational processes, theglobal approximation φh of φ over ΩT , the discretization of Ω must be ofclass Cj . Since φh =

⋃e φ

eh, the local approximations φeh must yield global

differentiability of order j for ΩT . The global differentiability of φh overΩT depends upon: (1) The global differentiability of φh over each Ωe whichis the same as the differentiability of φeh over Ωe and can be controlled by

12 INTRODUCTION

choosing appropriate p-levels of the local approximation functions [N ] andcan be greater than j. That is, φh over each Ωe can be of class Cp in whichp can be greater than j. (2) At the inter-element boundaries the globaldifferentiability of φh is controlled by the basis functions and the dofs ofthe mating elements. The basis functions for the elements and the dofs atthe nodes of the elements must be specifically chosen to ensure that at themating boundaries φh has the desired global differentiability.

In the currently used finite element processes, h and p are considered asthe only independent parameters in the computational processes. The localapproximations are generally chosen to be of class C0, that is in such casesφh is of class Cp over each Ωe but of class C0 at the inter-element boundariesand as a consequence, φh over ΩT remains of class C0 regardless of p-level.Secondly, the change in h (mesh refinement) obviously does not influenceglobal differentiability of φh as each φeh remains of class C0 regardless of itssize.

Surana and coworkers [1–3] have shown that global differentiability of ap-proximations in finite element processes can not be increased (or decreased)by changing h or p, hence global differentiability is an independent parameterin all finite element processes in addition to h and p. The global differentia-bility of φh is intrinsic in local approximations φeh, hence the local approxima-tions φeh must be constructed specifically for a desired global differentiabilityat the inter-element boundaries. In order to give the idea of global differen-tiability a more concrete mathematical form, we proceed as follows. Let thelocal approximation [N ] or Ni belong to an approximation space Vh whichis a subspace of Hk,p (a bigger space). At this point, it suffices to say that kis the order of the space and this space contains basis functions of degree p(p-level) and of global differentiability k − 1. That is, when the basis func-tions from Vh ⊂ Hk,p are used in constructing local approximations φeh, weare ensured global differentiability of φh of order k − 1 at the inter-elementboundaries and of order p > (k − 1) over each Ωe. Thus, with the basisfunctions from Vh, φh has global differentiability of order k − 1. Hence, theorder of the approximation space is an independent parameter in additionto h and p used currently. Hence, the k-version of the finite element methodin addition to the h-version and p-version and associated k, hk, pk and hpkprocesses in addition to h, p and hp processes used currently. Thus, whenone looks at the mathematical framework for finite element processes, h, pand k become independent parameters as opposed to h and p used presently.To emphasize this, we say hpk mathematical framework as opposed to hpframework. This terminology emphasizes the fact that global differentiabil-ity of approximation has been considered as an independent parameter inthe development and construction of the mathematical framework for thefinite element processes.

1.3. SUMMARY 13

The material presented in this book uses hpk framework. During the de-velopments of the details of the finite element processes, clear distinctions aremade between hpk framework and hp framework with local approximationsof class C0 so that the reader clearly sees the benefits of k as an independentparameter in the design and development of the finite element processes.Obviously, hp framework with C0 local approximations is a subset of hpkframework.

1.3 Summary

In the material presented in the various sections of this chapter, it is per-haps clear that in the development of the mathematical foundation and com-putational infrastructure for finite element processes we utilize concepts andprinciples from various areas of applied mathematics such as theory of differ-ential operators, calculus of variations, theory of functions or real analysis,functional analysis, interpolation theory and approximation theory. Basicunderstanding of these areas of applied mathematics helps in developing aclearer and deeper understanding of the mathematical concepts involved inthe finite element method. This book covers only the essential elements fromthese areas but in sufficient details to gain a good theoretical and workingknowledge of the subject of finite element method for BVPs.

[1–23]

References for additional reading[1] K. S. Surana, A. R. Ahmadi, and J. N. Reddy. The k-version of finite element method

for self-adjoint operators in BVP. Int. J. Comp. Eng. Sci., 3(2):155–218, 2002.

[2] K. S. Surana, A. R. Ahmadi, and J. N. Reddy. The k-version of finite element methodfor non-self-adjoint operators in BVP. Int. J. Comp. Eng. Sci., 4(4):737–812, 2003.

[3] K. S. Surana, A. R. Ahmadi, and J. N. Reddy. The k-version of finite element methodfor non-linear operators in BVP. Int. J. Comp. Eng. Sci., 5(1):133–207, 2004.

[4] G. Hellwig. Differential Operators of Mathematical Physics. Addison-Wesley Pub-lishing Co., 1967.

[5] G. Mikhlin. Variational Methods in Mathematical Physics. Pergamon Press, 1964.

[6] C. Johnson. Numerical Solutions of Partial Differential Equations by Finite ElementMethod. Cambridge University Press, 1987.

[7] M. Gelfand and S. V. Fomin. Calculus of Variations. Dover Publications, 2000.

[8] J. N. Reddy. Applied Functional Analysis and Variational Methods in Engineering.McGraw Hill Company, 1986.

[9] M. Becker. The Principles and Applications of Variational Methods. MIT Press, 1964.

[10] M. Forray. Variational Calculus in Science and Engineering. McGraw-Hill, 1968.

[11] K. Rektorys. Variational Methods in Mathematics, Science and Engineering. Reidel,1977.

14 REFERENCES FOR ADDITIONAL READING

[12] R. S. Schechter. The Variational Methods in Engineering. McGraw-Hill, 1967.

[13] K. Washizu. Variational Methods in Elasticity and Plasticity. Pergamon Press, 3ndedition, 1982.

[14] R. Weinstock. Calculus of Variations with Applications to Physics and Engineering.McGraw-Hill, 1952.

[15] W. Rudin. Real and Complex Analysis. McGraw-Hill, 1966.

[16] J. T. Oden and L. Demkowicz. Applied Funtional Analysis. CRC-Press, 1996.

[17] W. R. Clough. The finite element method in plane stress analysis. In Proceedingsof 2nd Conference on Electronic Computation, pages 345–378. Journal of StructuresDivision, ASCE, 1960.

[18] A. Hrenikoff. Solution of problems in elasticity by the frame work method. J. AppliedMath., Trans of the ASME, 8:169–175, 1941.

[19] J. T. Oden. Finite Elements of Nonlinear Continua. McGraw-Hill, New York, 1972.

[20] J. T. Oden and J. N. Reddy. An Introduction to the Mathematical Theory of FiniteElements. John Wiley, New York, 1976.

[21] J. N. Reddy. An Introduction to the Finite Element Method. McGraw Hill Inc., NewYork, 3rd edition, 2006.

[22] G. Strang and G. Fix. An Analysis of the Finite Element Method. Prentice Hall, NewJersey, 1973.

[23] M. J. Turner, R. W. Clough, H. C. Martin, and L. J. Topp. Stiffness and deflectionanalysis of complex structures. J. Aero. Sci., 23:805–823, 1956.

2

Concepts from FunctionalAnalysis

2.1 General Comments

In this chapter we present some basic elements from various areas of lin-ear algebra and functional analysis such as sets, function spaces, theory offunctions, differential operators and calculus of variations. These elementsprovide the necessary concepts and basic principles for developing a math-ematical framework and associated computational infrastructure for finiteelement processes. The foundation of the finite element method for bound-ary value problems (BVPs) relies heavily on these ideas. We begin withsome basic definitions. Many theorems and lemmas are given, many of themwith proofs. Proofs that are not included here may be found in referencescited at the end of the chapter.

The material presented here is not meant to be a crash course in variousareas of applied mathematics but its intention is that of review so that thereaders are refreshed. References are cited for those who may be interestedin further reading on these topics.

2.2 Sets, Spaces, Functions, Functions Spaces, andOperators

The material presented here contains basic definitions, theorems (manyof them without proofs) related to sets, spaces, functions, function spacesand operators. The objective here is to help readers refresh the materialessential for the further development of the concepts and principles relatedto the mathematics of computations and finite element subject for BVPs.

Definition 2.1 (Set). A set is a collection of objects that share a certaincommon feature or property. Sets could be open or closed. In an open set,the boundary points (limit points) are not included in the set. For a closedset, boundary points are part of the set.

15

16 CONCEPTS FROM FUNCTIONAL ANALYSIS

Notation for sets of elements from the real number field, R:

a < x < b ⇒ x ∈ (a, b), open set; open on left as well as right

a ≤ x < b⇒ x ∈ [a, b), closed on left but open on right

a < x ≤ b⇒ x ∈ (a, b], open on left but closed on right

a ≤ x ≤ b⇒ x ∈ [a, b], closed set; closed on left as well as right

Definition 2.2 (Space). A set could also be called a space, except thatspaces are more restricted (i.e., obey certain rules) than sets. A space has abasis and hence dimension, but a set may not.

Definition 2.3 (Linear space). A set S is called a linear space if thefollowing rules of addition and multiplication by a scalar are satisfied by theelements of the set:

(i) w = u+ v ∈ S ∀u, v ∈ S (defines the sum of u and v).

(ii) αu =∈ S ∀u ∈ S, α ∈ R (defines the product of α and u where R isthe space of real numbers).

(iii) In addition, sums and products obey the following laws:

(a) u+ v = v + u (commutative)(b) (u+ v) + w = u+ (v + w) (associative)(c) For any u ∈ S there exists a unique element z ∈ S independent of

u such that u+ z = u (existence of the zero element, z = 0)(d) For any u ∈ S there exists a unique element w ∈ S that depends on

u such that u+w = z (existence of the negative element, w = −u)(e) 1 · u = u(f) α(βu) = (αβ)u ∀α, β ∈ R(g) (α+ β)u = αu+ βu ∀α, β ∈ R(h) α(u+ v) = αu+ αv ∀α ∈ R

Definition 2.4 (Linear relation). For u1, u2, . . . , un ∈ S, a linear space,and α1, α2, . . . , αn ∈ R an expression of the form

α1u1 + α2u2 + . . .+ αnun =

n∑i=1

αiui ∈ S

is called a linear combination of u1, u2, . . . , un. An expression of the form∑ni=1 αiui = 0 is called a linear relation.

Definition 2.5 (Linear dependence and independence). A linear re-lation among ui ∈ S

α1u1 + α2u2 + · · ·+ αnun = 0

2.2. SETS, SPACES, FUNCTIONS, FUNCTIONS SPACES, AND OPERATORS 17

with all αi = 0 (i = 1, 2, . . . , n) is called a trivial relation among ui (i =1, 2, . . . , n). A set of elements ui ∈ S (i = 1, 2, . . . , n) is called linearlydependent if there exists a nontrivial relation among them (i.e., at least oneαi is nonzero); otherwise, the elements are said to be linearly independent. Inother words a set of elements is linearly independent when a linear relation∑n

i=1 αiui = 0 holds with all αi = 0 (i = 1, 2, . . . , n).

Definition 2.6 (Finite-dimensional space). A linear space S is said tobe finite-dimensional or more precisely n-dimensional if there are n linearlyindependent elements in S, and n+ 1st element in S is linearly dependent.The n linearly independent elements of an n-dimensional space form a basis.

Theorem 2.1. If S is an n-dimensional linear space and if ui ∈ S (i =1, 2, . . . , n) are linearly independent, then every element u ∈ S can be writtenas a linear combination of the n elements

u = α1u1 + α2u2 + . . .+ αnun, αi ∈ R

where αi (i = 1, 2, . . . , n) are uniquely determined by u.

Definition 2.7 (Function). A function defines a rule, law or mapping.Consider f(x), x ∈ Ω = [a, b]. The function f defining a rule, maps everyelement of the set Ω called its domain of definition to another set R calledits range of definition; that is, f(x) maps set Ω to set R or we simply writef : Ω→ R.

Definition 2.8 (Metric space). A linear space M of elements u, v, w, . . .is said to be a metric space if to each pair of elements u, v ∈ M therecorresponds a number ρ(u, v) called the distance between u and v with thefollowing properties:

(i) ρ(u, v) = ρ(v, u)

(ii) ρ(u,w) ≤ ρ(u, v) + ρ(v, w) triangle inequality

(iii) ρ(u, v) ≥ 0 and ρ(u, v) = 0⇔ u = v

Example: n-dimensional Euclidean space.

Definition 2.9 (Normed space). A linear space S is said to be a normedspace if for every u ∈ S there corresponds a real number ||u||, called thenorm of u, which satisfies the following properties:

(i) ||αu|| = |α| ||u|| ∀α ∈ R (homogeneous)

(ii) ||u+ v|| ≤ ||u||+ ||v|| triangle inequality

(iii) ||u|| ≥ 0 and ||u|| = 0⇔ u = 0 (positive-definite)


Definition 2.10 (Banach space). A Banach space B is a complete normedspace (the notion of completeness is defined later).

Definition 2.11 (Inner product space). A linear space S with elementsu, v, w, . . . is said to be an inner product space if for every pair of elementsu, v ∈ S there corresponds a real number (u, v), called the scalar product ofu and v, which satisfies the following properties:

(i) (u, v) = (v, u) (symmetry)

(ii) (αu+ βv,w) = α(u,w) + β(v, w) ∀α, β ∈ R

(iii) (u, u) ≥ 0 and (u, u) = 0⇔ u = 0

An inner product space becomes a normed space when the norm is definedwith respect to the inner product.

Definition 2.12 (Hilbert space). A Hilbert space is a complete innerproduct space (completeness is defined later).

In the definition of Hilbert spaces we generally also specify the differen-tiability of the functions contained in them. This is doen in section 2.2.1.

Remarks.

1. The scalar product of u and v given by (u, v) has not been defined yet andcould have any convenient definition as long as it satisfies the propertiesdefined above.

2. In the solutions of BVPs, we are interested in Hilbert spaces of functions.

3. We only consider Hilbert spaces that are separable (defined later).

2.2.1 Hilbert spaces Hk(Ω)

A Hilbert space of square-integrable functions defined over Ω is denotedby H0(Ω) = L2(Ω). A Hilbert space Hk(Ω) is a space of functions thatpossesses continuous derivatives up to order k − 1 defined over a set Ω (i.e.,the kth derivatives exist and are square-integrable in Lebesgue sense). Ob-viously, Hk ⊂ Hk−1 ⊂ · · · ⊂ H2 ⊂ H1 ⊂ H0 ≡ L2; that is, H0(Ω) is thelargest and least restrictive of all of the spaces H i(Ω) (i = 1, 2, . . . , k) andH1, H2, . . . ,Hk (k > 2) are progressively more and more restricted spacesthan H0 = L2. Thus, space Hk(Ω) is more restrictive than space Hn(Ω)when k > n.


2.2.2 Definition of scalar product in Hk(Ω) space

In the theory of differential operators and hence finite element method,the following definition of scalar product (·, ·) is meaningful. If u and v aretwo elements of a Hilbert space H0 = L2, then we could define (called theL2-inner product)

(u, v)0 =

∫Ω

uv dΩ, ∀u, v ∈ L2(Ω)

and Ω is the domain of definition of the functions u and v. We note thatthe Hilbert space H1 contains functions that are continuous but the firstderivatives may be discontinuous. However, if the first derivatives are squareintegrable, the definition of the scalar product

(u, v)1 =

∫Ω

(uv +

du

dx

dv

dx

)dΩ, ∀u, v ∈ H1(Ω)

is meaningful (assuming u = u(x), v = v(x), . . .). Likewise, the scalar prod-uct (u, v) in Hk(Ω) can be defined as

(u, v)k =k∑i=0

∫Ω

diu

dxidiv

dxidΩ ∀u, v ∈ Hk(Ω)

The extension of the above definition for functions of more than one variableis rather straight forward.

2.2.3 Properties of scalar product

One could easily verify that the definition of (u, v) in section 2.2.2 satisfiesall of the properties of the scalar product, i.e.

(i) (u, v) = (v, u)

(ii) (a1u1 + a2u2, b1v1 + b2v2) = a1b1(u1, v1) + a1b2(u1, v2) + a2b1(u2, v1) +a2b2(u2, v2) ∀a, b ∈ R

(iii) (u, u) =k∑i=0

∫Ω

(diudxi

)2dΩ ≥ 0 and (u, u) = 0⇔ diu

dxi= 0; i = 0, . . . , k


2.2.4 Norm of u in Hilbert space Hk(Ω)

Based on the definition of the scalar product (u, v)k over Hk(Ω) Hilbertspace in section 2.2.2, we define the norm of u in Hk(Ω) by

||u||Hk = ||u||k =

k∑i=0

∫Ω

(diu

dxi

)2

dΩ

12

≥ 0

||u||Hk = ||u||k = 0⇔ diu

dxi= 0; i = 0, . . . , k ∀x ∈ Ω

Thus

||u||H0 = ||u||0 = ||u||L2=

∫Ω

u2 dΩ

12

≥ 0 ; ||u||H0 = 0⇔ u = 0 ∀x ∈ Ω

||u||H1 = ||u||1 =

∫Ω

(u2 +

(du

dx

)2)dΩ

12

≥ 0 ; ||u||H1 =0⇔ u=0=du

dx∀x ∈ Ω

and so on.

2.2.5 Seminorm of u in Hilbert space Hk(Ω)

Seminorm of u in Hk(Ω) is defined by

|u|Hk = |u|k =

∫Ω

(dku

dxk

)2

dΩ

12

≥ 0

Thus|u|H0 = ||u||H0 = ||u||L2

=

∫Ω

u2 dΩ

12

≥ 0

where equality implies that u = 0 ∀x ∈ Ω.

Theorem 2.2 (Cauchy-Schwarz inequality). Let u and v be arbitrary ele-ments of an inner product space. Then

|(u, v)| ≤ (u, u)12 (v, v)

12

Proof. For v = 0, the inequality is obviously satisfied. Suppose v 6= 0. Thenfor arbitrary α ∈ R, define

f(α) = (u+ αv, u+ αv)

= α2(v, v) + 2α(u, v) + (u, u) > 0


Choose α = α1 = − (u,v)(v,v) , then

f(α1) =(u, v)2

(v, v)− 2

(u, v)2

(v, v)+ (u, u)

= −(u, v)2

(v, v)+ (u, u) > 0

Hence,(u, v)2 ≤ (u, u)(v, v)

and|(u, v)| ≤ (u, u)

12 (v, v)

12

This completes the proof.

Theorem 2.3 (Triangle inequality). Assume that u, v ∈ H0(Ω), then

||u+ v||0 ≤ ||u||0 + ||v||0

Proof.

||u+ v||20 = ||u||20 + ||v||20 + 2(u, v)

≤ ||u||20 + ||v||20 + 2|(u, v)|

Using Cauchy-Schwarz inequality for |(u, v)|

||u+ v||20 ≤ ||u||20 + ||v||20 + 2 ||u||0 ||v||0

= (||u||0 + ||v||0)2

Taking square root of both sides

||u+ v||0 ≤ ||u||0 + ||v||0This completes the proof.

Theorem 2.4 (Friedrichs inequality). If u ∈ H1(0, l) and u(0) = 0, then l∫0

|u(x)|2dx

12

≤ l

l∫0

|u′(x)|2dx

12

Proof. Since

u(x) = u(x)− u(0) =

x∫0

u′(t) dt


using Cauchy-Schwarz inequality

|u(x)|2 =

∣∣∣∣∣∣x∫

0

1u′(t) dt

∣∣∣∣∣∣2

≤

x∫0

1 dt

x∫0

|u′(t)|2dt

≤ l l∫0

|u′(x)|2dx

Integrating both sides

l∫0

|u(x)|2dx ≤ ll∫

0

l∫0

|u′(x)|2dx

dx

≤ l2l∫

0

|u′(x)|2dx

Taking square root of both sides l∫0

|u(x)|2

12

≤ l

l∫0

|u′(x)|2dx

12


Theorem 2.5 (Continuity of scalar product). If limn→∞

un = u and limn→∞

vn =

v, thenlimn→∞

(un, vn) = (u, v)

2.2.6 Function spaces

When the elements of a space are functions, we refer to such spaces asfunction spaces. In the theory of differential operators, function spaces con-tain desired functions. In the definition of the scalar product (section 2.2.2)u, v ∈ H where H can be a Hilbert space containing functions.

Definition 2.13 (Fundamental sequence). A sequence u1, u2, . . . of ele-ments, ui ∈ M , is said to be a fundamental sequence if for any ε > 0 thereexists a positive number N(ε) such that

ρ(un, um) < ε, ∀n,m > N(ε)

Theorem 2.6. A convergent sequence of elements u1, u2, . . . is a funda-mental sequence. (Note: Not every fundamental sequence is a convergentsequence.)


Definition 2.14 (Complete Hilbert space). A Hilbert space H is saidto be complete if every fundamental sequence in it converges.

Definition 2.15 (Separable Hilbert space). A Hilbert space H is said tobe separable if there exists a sequence of elements u1, u2, . . . ∈ H such thatfor every u ∈ H and every ε > 0 we can find an element ul in the sequencefor which ||u− ul|| < ε holds.

Definition 2.16 (Orthogonality). Two elements u, v ∈ H are said to beorthogonal if

(u, v) = 0

The element u ∈ H is said to be orthogonal to a subspace S ⊆ H if u isorthogonal to every v ∈ S.

Theorem 2.7 (Dense subspace). The subspace S ⊆ H is dense in H ifand only if there is no element in H except null element which is orthogonalto S.

2.2.7 Operators

• An operator also defines a rule, law or mapping.

• An operator acts on functions and hence must be defined for a set or aspace of functions called the domain of definition of the operator.

• An operator is denoted by a symbol together with the functions on whichit acts.

• For each element in its domain of definition, an operator produces anotherelement. The collection of these elements is called the range of the operator.

Thus, if V ⊂ H is the linear space containing functions u, v, w, . . . constitut-ing the domain of definition of an operator A then the collection of elementsAu,Av,Aw, . . . is the range of A, denoted AV , and we have

A : V → AV, ∀u ∈ V

That is, operator A maps V into AV (∀u ∈ V , A(u) ∈ AV ). Even thoughV is a linear space containing functions u, v, w, . . ., the nature of space AVdepends upon the specific nature of the operator A, which can be linear ornonlinear.


2.2.8 Types of operators

Many different types of operators are encountered in mathematical physics,engineering, and sciences. In this book we are interested in differential oper-ators. Differential operators define operation of differentiation on functionsconstituting its domain of definition. Differential operators transform ele-ments of higher-order spaces to elements in lower order spaces.

Definition 2.17 (Linear operator). Let A be an operator, A : V ⊂ H →H. Then A is linear if the following relation holds:

A(a1u1+· · ·+anun) = a1Au1+· · ·+anAun ∀u1, . . . , un ∈ V,∀a1, . . . , an ∈ R

Example 2.1. Consider the differential equation

∂2φ

∂x2+∂2φ

∂y2− f(x, y) = 0, (x, y) ∈ Ω

Define operator A as A = ∂2

∂x2+ ∂2

∂y2so that we have Aφ−f = 0 over Ω. This

compact notation is very useful in the mathematical developments associatedwith the finite element processes. We note that since the differential equationis linear, the operator A is not a function of φ; hence A is a linear operator:

A(αu+ βv) =

(∂2

∂x2+

∂2

∂y2

)(αu+ βv)

= α

(∂2u

∂x2+∂2u

∂y2

)+ β

(∂2v

∂x2+∂2v

∂y2

)= αA(u) + βA(v)

Example 2.2. Consider the differential equation

φdφ

dx− k d

2φ

∂x2= 0 ∀x ∈ Ω

Defining operator A as A = φ ddx − k

d2

dx2, we see that A is a function of φ.

Hence, the differential operator A is not linear:

A(αφ1 + βφ2) = (αφ1 + βφ2)d

dx(αφ1 + βφ2)− k d

2

dx2(αφ1 + βφ2)

= (αφ1 + βφ2)

(αdφ1

dx+ β

dφ2

dx

)− kαd

2φ1

dx2− kβ d

2φ2

dx2

6= αA(φ1) + βA(φ2)

Definition 2.18 (Adjoint of an operator). Let A be a linear (differential)operator and let V ⊂ H be its domain of definition. If

(Au, v) = (u,A∗v) + 〈Au, v〉Γ


holds, then A∗ is called the adjoint of A. Here 〈Au, v〉Γ is called the concomi-tant, and it is only a symbol that represents the boundary termsthat are obtained in the process of moving operator A from u tooperator A∗ on v. Thus, the concomitant is a collection of boundary termsobtained as a consequence of integration by parts. This is amply illustratedin examples presented shortly.

Definition 2.19 (Symmetric operator). Let A be a linear operator withits domain of definition V ⊂ H, then A is symmetric if

(Au, v) = (u,Av), ∀u, v ∈ V

If we use the definition of the scalar product in section 2.2.2, the Symmetryof A implies ∫

Ω

(Au)v dΩ =

∫Ω

u(Av) dΩ, ∀u, v,∈ V

for Au,Av ∈ V and A∗ = A, that is, for symmetric operators the adjoint ofthe operator is the same as the operator.

Definition 2.20 (Self-adjoint operator). If an operator A is linear andsymmetric, then it is self-adjoint. Thus, for self-adjoint operators we have

A(αu+ βv) = αAu+ βAv, ∀u, v ∈ V ⊂ H, ∀α, β ∈ R(Au, v) = (u,Av), ∀u, v ∈ V ⊂ H

Definition 2.21 (Non-self-adjoint operator). If an operator A is linearbut not symmetric, then it is non-self-adjoint. Thus, for non-self-adjointoperators we have

A(αu+ βv) = αAu+ βAv, ∀u, v ∈ V ⊂ H, ∀α, β ∈ R= (Au, v) 6= (u,Av), ∀u, v ∈ V ⊂ H

For non-self-adjoint operators the adjoint A∗ of the operator is not the sameas the operator A.

Definition 2.22 (Non-linear operator). If an operator A is neither linearnor symmetric, then it is non-linear. Thus, for non-linear operators we have

A(αu+ βv) 6= αAu+ βAv, ∀u, v ∈ V ⊂ H, ∀α, β ∈ R(Au, v) 6= (u,Av), ∀u, v ∈ V ⊂ H

Since linearity of the operator is essential for symmetry, it suffices to say thatif the operator A is not linear, then it is also not symmetric. The definitionof the adjoint A∗ of A in such cases is not meaningful.


Definition 2.23 (Positive-definite operator). A linear symmetric oper-ator (hence, self-adjoint) A is positive-definite if,

(Au, u) > 0 ∀u ∈ V ⊂ H\0

Definition 2.24 (Positive bounded below operator). A linear symmet-ric operator A is positive positive bounded below if,

(Au, u) ≥ γ2 ||u||2 ∀u ∈ V ⊂ H

where γ is a positive constant.

Definition 2.25 (Functional). For each function in its field or domainof definition V if an operator A generates a function identically equal toa constant then this operator is known as a functional. Functionals area restricted class of operators. Functionals encountered in the theory ofdifferential operators are integrals corresponding to the differential operators.

Definition 2.26 (Linear functionals). A functional L(u) with a singleargument u is linear if

(i) its field of definition V ⊂ H is linear.

(ii) for ui; i = 1, . . . , n ∈ V ⊂ H and ai; i = 1, 2, . . . , n ∈ R,

L(a1u1 + a2u2 + · · ·+ anun) = a1L(u1) + a2L(u2) + · · ·+ anL(un)

holds.

Definition 2.27 (Bilinear functionals). A functional B(u, v) with twoarguments u and v is bilinear if

(i) its field of definition V ⊂ H is linear.

(ii) it is linear in u as well as v,

B(a1u1 + a2u2 + · · ·+ anun, v) = a1B(u1, v) + a2B(u2, v)

+ · · ·+ anB(un, v)

B(u, b1v1 + b2v2 + · · ·+ bnvn) = b1B(u, v1) + b2B(u, v2)

+ · · ·+ bnB(u, vn)

holds for all ui ∈ V ⊂ H, ai ∈ R, vi ∈ V ⊂ H, and bi ∈ R fori = 1, 2, . . . , n.

Definition 2.28 (Symmetric functional). A functional B(u, v) is sym-metric if

B(u, v) = B(v, u)


Definition 2.29 (Quadratic functional). If A is a linear and symmetricoperator with domain of definition V1 ⊂ H1 and if L is a linear functionalwith domain of definition V2 ⊂ H2, then functionals of the form

F (u) = (Au, u) + L(u) + c ∀u ∈ V1 ∩ V2, c being a constant

is called a quadratic functional.

2.2.9 Energy product

IfA is a positive-definite differential operator with its domain of definitionV ⊂ H, then

(Au, u) =

∫Ω

uA(u) dΩ ∀u ∈ V ⊂ H

is proportional to the amount of energy stored in the system described byA. We generalize the definition the energy product. Consider (Au, v), whichis meaningful if u ∈ V ⊂ H and v is an arbitrary function with finite norm,||v|| = (v, v)1/2 < ∞. v is called a test function. Now we consider (Au, v)in which u, v ∈ V ⊂ H and v satisfies the same boundary conditions as u.Under these conditions, the scalar product (Au, v) is called energy productof functions u and v and we denote

[u, v] = (Au, v) =

∫Ω

vA(u) dΩ ∀u, v ∈ V ⊂ H

Obviously, the energy product possesses the same properties as the scalarproduct of functions.

2.2.10 Integration by parts (IBP)

Integration by parts is an important calculus tool that allows one totransfer differentiation from one function to another in an integral represen-tation. In the following we consider line, surface and volume integrals thatcorrespond to R ≡ R1, R2, and R3.

Example 2.3. (Line integrals in 1D) Consider the linear differential oper-ator A : S ⊂ H1(a, b)→ H0(a, b) defined by

Aφ = −dφdx, x ∈ (a, b)

Then we have by integration by parts the relation (here Ω = (a, b) and Γ = aand b)

(Aφ,ψ) = −∫ b

a

dφ

dxψ dx =

∫ b

aφdψ

dxdx+ [φψ]ba

= (φ,A∗ψ) + 〈Aφ,ψ〉Γ


Clearly, A∗ = d/dx and 〈Aφ,ψ〉Γ = [φψ]ba. Thus, d/dx is the adjoint of−d/dx, and A = −d/dx is not self-adjoint. 〈Aφ,ψ〉Γ is called concomitant.

Example 2.4. (Line integrals in 1D) Now consider the linear differentialoperator A : S ⊂ H2(a, b)→ H0(a, b) defined by

Aφ =d2φ

dx2, x ∈ (a, b)

Then using the integration by parts twice we obtain the relation (Γ = a andb)

(Aφ,ψ) =

∫ b

a

d2φ

dx2ψ dx =

∫ b

aφd2ψ

dx2dx+

[dφ

dxψ − φdψ

dx

]ba

= (φ,A∗ψ) + 〈Aφ,ψ〉Γ

Clearly, A∗ = d2/dx2 and the concomitant is defined by

〈Aφ,ψ〉Γ =

[dφ

dxψ − φdψ

dx

]ba

Thus, A = A∗ = d2/dx2 is self-adjoint. If S is defined to be a linear spaceof elements φ and ψ satisfying the conditions φ(a) = 0 and φ(b) = 0 as wellas ψ(a) = 0 and ψ(b) = 0, then we have

(Aφ,ψ) = (φ,Aψ), φ, ψ ∈ S

Example 2.5. (Area integrals in 2D) Consider the linear differential oper-ator A = ∂

∂x + ∂∂y , A : S ⊂ H1(Ω)→ H0(Ω) (Ω ⊂ R2), and

(Aφ,ψ) =

∫Ω

(∂φ∂x

+∂φ

∂y

)ψ dΩ

= −∫Ω

φ

(∂ψ

∂x+∂ψ

∂y

)dx dy +

∮Γ

(nx + ny)φψ dΓ

= (φ,A∗ψ) + 〈Aφ,ψ〉Γ

where Γ is the closed contour constituting the boundary of domain Ω and(nx, ny) are the direction cosines of the unit exterior normal to the boundaryΓ (see Fig. 2.1). We have transferred all of the differentiation from φ to ψusing integration by parts. The concomitant is defined by

〈Aφ,ψ〉Γ =

∮Γ

(nx + ny)φψ dΓ


x

y

Ω

nx

ny

n2x + n2

y = 1

Γ

Figure 2.1: Domain of definition of a 2D BVP

Example 2.6. (Area integrals in 2D) Consider the Laplace operator A =

∇2 = ∂2

∂x2+ ∂2

∂y2and

(Aφ,ψ) =

∫Ω

(∂2φ

∂x2+∂2φ

∂y2

)ψ dΩ

=

∫Ω

φ

(∂2ψ

∂x2+∂2ψ

∂2y

)dx dy

+

∮Γ

(∂φ

∂xnx +

∂φ

∂yny

)ψ dΓ−

∮Γ

φ

(∂ψ

∂xnx +

∂ψ

∂yny

)dΓ

= (φ,A∗ψ)Ω + 〈Aφ,ψ〉Γ

where two orders of differentiations are transferred from φ to ψ (using thegradient theorem), and the concomitant < Aφ,ψ >Γ is given by

〈Aφ,ψ〉Γ =

∮Γ

(∂φ

∂nψ − φ∂ψ

∂n

)dΓ

where the normal derivative ∂∂n in 2-D is defined by

∂

∂n= nx

∂

∂x+ ny

∂

∂y

The direction cosines (nx, ny) and Γ have the same meaning as in Example2.5. Thus, the adjoint of A is

A∗ =∂2

∂x2+

∂2

∂y2= ∇2 = A

Thus, the operator is self-adjoint.


Example 2.7. (Volume integrals in 3D) Consider the operator A = ∂∂x +

∂∂y + ∂

∂z and

(Aφ,ψ) =

∫Ω

(∂φ

∂x+∂φ

∂y+∂φ

∂z

)ψ dΩ

= −∫Ω

φ

(∂ψ

∂x+∂ψ

∂y+∂ψ

∂z

)dx dy dz

+

∮Γ

(nx + ny + nz)φψ dΓ

= (φ,A∗ψ)Ω + 〈Aφ,ψ〉Γwhere one order of differentiation is transferred from φ to ψ with respect tox, y and z in each of the terms in the integrand using integration by parts.Thus, the adjoint of A is A∗ = − ∂

∂x −∂∂y −

∂∂z 6= A and the concomitant

〈Aφ,ψ〉Γ is given by

〈Aφ,ψ〉Γ =

∮Γ(nx + ny + nz)φψ dΓ

Example 2.8. (Volume integrals in 3D) Consider the operator A = ∂2

∂x2+

∂2

∂y2+ ∂2

∂z2, and

(Aφ,ψ) =

∫Ω

(∂2φ

∂x2+∂2φ

∂y2+∂2φ

∂z2

)ψ dΩ

=

∫Ω

φ

(∂2ψ

∂x2+∂2ψ

∂2y+∂2ψ

∂2z

)dx dy dz

+

∮Γ

(∂φ

∂nψ − φ∂ψ

∂n

)dΓ

= (φ,A∗ψ)Ω + 〈Aφ,ψ〉Γwhere two orders of differentiation are transferred from φ to ψ with respectto x, y and z in each of the three terms in the integrand and the normalderivative is defined by

∂

∂n= nx

∂

∂x+ ny

∂

∂y+ nz

∂

∂z

Thus, the adjoint of A is A∗ = ∂2

∂x2+ ∂2

∂y2+ ∂2

∂z2= A and the concomitant is

given by

〈Aφ,ψ〉Γ =

∮Γ

(∂φ

∂nψ − φ∂ψ

∂n

)dΓ

2.3. ELEMENTS OF CALCULUS OF VARIATIONS 31

2.3 Elements of Calculus of Variations

In an earlier section we introduced the concept of ’functionals’. In the ab-stract sense, we mean a mapping or a correspondence that assigns a definitereal number to each function belonging to some class or space. Function-als are variable quantities and play a very important role in mathematicalphysics, sciences, and engineering. Calculus of variations is a branch of ap-plied mathematics that deals with extrema of functionals, that is, maximum,saddle points, and minimum. First, we need to establish a connection be-tween the differential equations and their solutions, which is the subject ofinterest to us, and the functionals and their extrema, that is calculus of vari-ations. There are four basic lemmas that are important in this regard. Westate these and provide their proofs.

Lemma 2.1 (Fundamental lemma). If α(x) ∈ H1(a, b) and

b∫a

α(x)h(x)dx = 0

holds ∀h(x) ∈ H1(a, b), then α(x) = 0 ∀x ∈ (a, b).

Proof. We construct the proof of this lemma by contradiction. Suppose α(x)is nonzero, say positive at some point [x1, x2] in [a, b]. If we let

h(x) = (x− x1)(x2 − x)

for some x ∈ [x1, x2] and h(x) = 0 otherwise, then h(x) obviously satisfiesthe conditions of the lemma. However,

b∫a

α(x)h(x)dx =

x2∫x1

α(x)(x− x1)(x2 − x)dx > 0

since the integrand is positive (except at x1 and x2, where it is zero), whichcontradicts the lemma. Conversely, since h(x) is arbitrary we can choose itto be equal to α(x). Then an integral of a positive integrand is zero only ifthe integrand itself is equal to zero, proving that α(x) = 0.

Lemma 2.2. If α(x) ∈ H2(a, b) and

b∫a

α(x)h′(x)dx = 0

holds ∀h(x) ∈ H2(a, b) such that h(a) = h(b) = 0, then

α′(x) = 0 or α(x) = c, ∀x ∈ (a, b)


where c is a constant.

Proof. Considerb∫a

α(x)h′(x)dx = 0

Transfer of differentiation from h(x) to α(x) using integration by parts gives

b∫a

α(x)h′(x)dx = −b∫a

α′(x)h(x)dx+(α(x)h(x)

)∣∣∣ba

= 0

Since h(a) = h(b) = 0,(α(x)h(x)

)∣∣∣ba

= 0 and we have

b∫a

α′(x)h(x)dx = 0

Using lemma 2.1, we have α′(x) = 0, which implies that α(x) is a constant.This completes the proof of the lemma.

Lemma 2.3. If α(x) ∈ H3(a, b) and

b∫a

α(x)h′′(x)dx = 0

holds ∀h(x) ∈ H3(a, b) such that h(a) = h(b) = h′(a) = h′(b) = 0, then

α′′(x) = 0 or α(x) = c0 + c1x, ∀x ∈ (a, b)

where c0 and c1 are constants.

Proof. Considerb∫a

α(x)h′′(x)dx = 0

Transferring the differentiation from h(x) to α(x) using integration by partstwice, we obtain

b∫a

α(x)h′′(x)dx = −b∫a

α′′(x)h(x)dx+[α(x)h′(x)

]ba−[α′(x)h(x)

]ba

= 0


Since h(a) = h(b) = h′(a) = h′(b) = 0, the boundary terms vanish and weobtain

b∫a

α(x)h′′(x)dx =

b∫a

α′′(x)h(x)dx = 0

Using lemma 2.1, we have α′′(x) = 0 which yields α(x) = c0 + c1x. Thiscompletes the proof of the lemma.

Lemma 2.4. If α(x) ∈ H1(a, b) and β(x) ∈ H2(a, b) and

b∫a

[α(x)h(x) + β(x)h′(x)

]dx = 0

holds ∀h(x) ∈ H2(a, b) with the property h(a) = h(b) = 0, then

α(x)− β′(x) = 0 ∀x ∈ (a, b)

Proof. Transferring differentiation from h(x) to β(x) using integration byparts once, we arrive at

b∫a

[α(x)h(x) + β(x)h′(x)

]dx =

b∫a

[α(x)− β′(x)

]h(x) dx+

[β(x)h(x)

]ba

= 0

Since h(a) = h(b) = 0, we obtain

b∫a

[α(x)h(x) + β(x)h′(x)

]dx =

b∫a

[α(x)− β′(x)

]h(x)dx = 0

Using lemma 2.1, we obtain α(x)− β′(x) = 0 ∀x ∈ (a, b) must hold. Henceα(x)− β′(x) = 0. This completes the proof of the lemma.

2.3.1 Concept of the variation of a functional

Variation means change or in the sense of calculus, differential. Let I(y)with y = y(x) be a functional defined over some normed linear space and let

∆I(h) = I(y + h)− I(y)

be the increment in I corresponding to increment h = h(x) of the dependentvariable y = y(x). If y is fixed, then ∆I(h) is a functional of h, in general anon-linear functional. Suppose that

∆I(h) = Φ(h) + ε ||h||


where Φ(h) is a linear functional and ε→ 0 as ||h|| → 0. Then the functionalI is said to be differentiable and the principal linear part of the increment∆I(h), that is, the linear functional Φ(h) which differs from ∆I(h) by aninfinitesimal of order higher than 1 relative to ||h|| is called the variation ofI(y) denoted by δI or δI(h).

Theorem 2.8. The variation of a differentiable functional is unique (seereference [1] for proof).

Theorem 2.9. A necessary condition for the differentiable functional I(y)to have an extremum for y = y∗ is that its variation vanishes for y = y∗;that is, δI(h) = 0 for y = y∗ for all admissible h (see reference [1] for proof).

2.3.2 Euler’s equation: Simplest variational problem

Let F (x, y, y′) be a function with continuous first and second derivativeswith respect to all of its arguments. Then among all functions y(x) whichare differentiable for a ≤ x ≤ b and satisfy boundary conditions y(a) = Aand y(b) = B, find the function y(x) for which the functional

I(y) =

b∫a

F (x, y, y′) dx

has an extremum. Details are provided in the following.Let us consider an increment h(x) in y(x) and h′(x) in y′(x); that is, let

y(x) change to y(x) + h(x) and y′(x) to y′(x) + h′(x). In order for the newvalues of y(x) to satisfy boundary conditions we must have h(a) = h(b) = 0(homogeneous part of the boundary conditions). Due to increments in y(x)and y′(x), there must be a change in I(y). Let ∆I be the incremental changein I. Then

∆I = I(y(x) + h(x)

)− I(y(x)

)or

∆I =

b∫a

(F (x, y(x) + h(x), y′(x) + h′(x))

)dx−

b∫a

F (x, y, y′) dx

=

b∫a

(F (x, y(x) + h(x), y′(x) + h′(x))− F (x, y, y′)

)dx

Expanding F (x, y(x) + h(x), y′(x) + h′(x)) in Taylor series about y, y′ (xbeing fixed)

∆I =

b∫a

(F (x, y, y′) +Fy(x, y, y

′)h+Fy′(x, y, y′)h′+O(h2, (h′)2)−F (x, y, y′)

)dx


or

∆I =

b∫a

(Fy(x, y, y

′)h+ Fy′(x, y, y′)h′

)dx+

b∫a

O(h2, (h′)2) dx

where O(h2, (h′)2) is a measure of the remaining terms in the Taylor se-ries expansion that are quadratic and of higher degrees in h and h′. Let∫ ba O(h2, (h′)2 dx = ε ||h|| in which as ε→ 0, ||h|| → 0. Hence

∆I =

b∫a

(Fy(x, y, y

′)h+ Fy′(x, y, y′)h′

)dx+ ε ||h||

we note that the integral in the above expression is a linear functional in hand h′ and satisfies the definition of the variation of I. Therefore, we have

δI =

b∫a

(Fy(x, y, y

′)h+ Fy′(x, y, y′)h′

)dx

Based on Theorem 2.9, a necessary condition for the functional I(y) to havean extremum at y = y∗ is that δI must vanish at y = y∗. Thus

δI =

b∫a

(Fy(x, y, y

′)h+ Fy′(x, y, y′)h′

)dx = 0

for all admissible h(x). Using Lemma 2.4 and comparing with the aboveintegral we find that

α(x) = Fy and β(x) = Fy′

and β(x) must be differentiable and α(x) = β′(x) must hold which gives

Fy =d

dx(Fy′)

or

Fy −d

dx(Fy′) = 0

must hold for a y(x) obtained from δI = 0. This is a differential equationand is known as Euler’s equation. Thus, a y(x) obtained from δI = 0 givesa unique extremum of I(y) and also satisfies Euler’s equation which is adifferential equation. We note that based on lemma 2.4 Fy′ is differentiable,a necessary condition for the second term in Euler’s equation.


Remarks. The Euler’s equation can also be derived in an alternate mannerusing δI = 0. Consider

δI =

b∫a

(Fy(x, y, y

′)h(x) + Fy′(x, y, y′)h′(x)

)dx = 0

Transfer one order of differentiation with respect to x from h to Fy′ in thesecond term in the integrand.

δI =

b∫a

(Fy(x, y, y

′)h(x)−d(Fy′)

dxh(x)

)dx+

[Fy′h(x)

]ba

= 0

Since h(a) = h(b) = 0,[Fy′h(x)

]ba

= 0 and we have

δI =

b∫a

(Fy(x, y, y

′)−d(Fy′)

dx

)h(x)dx

Using lemma 2.1 we obtain

Fy(x, y, y′)−

d(Fy′)

dx= 0

which is Euler’s equation.

Theorem 2.10. Let I(y) be a functional of the form

I(y) =

b∫a

F (x, y, y′) dx (2.1)

defined on some set of functions y(x), which have continuous first derivativesin [a, b] and satisfy boundary conditions y(a) = A and y(b) = B. Then anecessary condition for I(y) to have an extremum is that δI = 0 must holdand a y(x) determined from δI = 0 must satisfy the Euler’s equation

Fy −d

dx(Fy′) = 0 (2.2)

or conversely if y(x) is a solutions to (2.2) then it yields an extremum of thefunctional (2.1) and δI = 0 holds for this y(x).

Remarks.


(1) For the first time we observe a correspondence between solutions of BVPsand the associated functionals and their extrema.

(2) Section 2.3.2 and Theorem 2.10 suggest that one could use the followingapproach for obtaining solutions of boundary value problems:

If Aφ− f = 0 is defined over Ω, then the problem of finding a solutionof Aφ−f = 0 is equivalent to finding the extremum of a functional I(φ)associated with Aφ − f = 0 such that Euler’s equation resulting fromδI = 0 is the differential equation Aφ− f = 0.

(3) We note that Theorem 2.10 only provides necessary conditions. Suffi-cient conditions that ensure the uniqueness of the solutions are yet tobe established.

(4) From the definition of δI, we observe that δI involves differentiation ofthe integrand of I with respect to dependent variables only.

The following theorem generalizes the concepts presented here.

Theorem 2.11 (Variationally Consistent (VC) integral form of aboundary value problem). Let Aφ−f = 0 in Ω be a system of differentialor partial differential equations with some boundary conditions describing aboundary value problem.

(i) Existence of a functional I(φ)

Let there exist a functional I(φ) (an integral) corresponding to the BVPAφ− f = 0. This is generally by construction.

(ii) Necessary Condition

If I(φ) is differentiable in φ, then the integral form given by δI = 0gives the necessary condition from which we determine a function φthat yields an extremum of I(φ). Let the Euler’s equation resultingfrom δI = 0 be Aφ − f = 0, then a φ obtained from δI = 0 is also asolution of Aφ− f = 0, hence a solution of the BVP.

(iii) Sufficient condition or extremum principle

The second variation of the functional, δ2I, provides extremum prin-ciple or sufficient condition. A unique extremum principle ensures aunique φ from δI = 0, hence a unique extremum of I(φ) and a uniquesolution of the Euler’s equation resulting from δI(φ) = 0 which is theBVP Aφ − f = 0. δ2I > 0, = 0, < 0 correspond to minimum, saddlepoint, and maximum of I(φ).

If for a BVP Aφ−f = 0 (i), (ii), and (iii) exist or hold then the integralform δI = 0 is called variationally consistent (VC) integral form associatedwith the BVP Aφ−f = 0. We also state this as a definition in the following.


Definition 2.30 (Variationally consistent (VC) integral form of aBVP). A variationally consistent integral form corresponding to the BVPAφ− f = 0 consists of

(1) Existence of a functional I(φ) corresponding to the BVP Aφ − f = 0.This is generally by construction (or is assumed).

(2) Necessary condition for the existence of an extremum of I(φ) is givenby δI(φ) = 0. The integral form δI(φ) = 0 is used to determine φ. TheEuler’s equation resulting from δI(φ) = 0 must be the BVP Aφ−f = 0.

(3) δ2I > 0, = 0, < 0 (minimum, saddle point, maximum of I(φ)) is thesufficient condition or extremum principle. Extremum principle ensuresthat a φ obtained from δI(φ) = 0 is unique. Extremum principle alsoestablishes whether φ from δI(φ) = 0 minimizes or maximizes I(φ) oryields a saddle point of I(φ).

When all these three elements are present in an integral formulation of theBVP Aφ − f = 0, then the integral form (resulting from δI(φ) = 0 orotherwise) is called a variationally consistent integral form of the BVP Aφ−f = 0 (or simply VC integral process). VC integral form or process yieldsunique extremum of the functional I(φ) corresponding to Aφ− f = 0, hencea unique solution of the BVP Aφ − f = 0 (the Euler’s equation resultingfrom δI(φ) = 0).

Definition 2.31 (Variationally inconsistent integral form (VIC) ofa BVP). If an integral form of a BVP (resulting from δI(φ) = 0 or other-wise) is not variationally consistent, then it is variationally inconsistent. Avariationally inconsistent integral form or process violates one or more of thethree requirements needed for variational consistency of the integral form.

Remarks.

(1) Thus, we see that a variationally consistent integral form of a BVPAφ − f = 0 emerges as a method of obtaining a unique solution of theBVP Aφ− f = 0.

(2) The necessary condition (the integral form resulting from δI(φ) = 0or otherwise) provides a system of algebraic equations from which thesolution φ is determined.

(3) The sufficient condition or unique extremum principle ensures that a φobtained from the integral form (δI(φ) = 0 or otherwise) is unique, hencethis φ yields a unique extremum of I(φ) as well as a unique solution ofthe Euler’s equation which is the BVP under consideration.

(4) Variationally consistent integral forms yield symmetric coefficient ma-trices in the algebraic systems and the coefficient matrices are positive-definite, hence have real, positive eigenvalues and real eigenvectors (ba-


sis). Such coefficient matrices are invertible , hence yield unique valuesof the unknowns in the corresponding algebraic systems.

(5) When the integral form is variationally inconsistent, a unique extremumprinciple does not exist. In such cases the coefficient matrix in the alge-braic system resulting from the integral form is not symmetric, hence isnot ensured to be positive-definite. A unique solution of the unknownsin such algebraic systems is not ensured. A consequence of the non-positive-definite coefficient matrix in the algebraic system is that suchcoefficient matrices may have zero or negative eigenvalues or the eigen-values and eigenvectors may be complex. In summary, variationallyinconsistent integral forms must be avoided at all cost due to the factthat when using such integral forms a unique solution of the BVP is notensured. In other words when obtaining solution of BVPs, variationallyconsistent integral forms are essential to ensure unique solutions of theBVPs.

(6) The theorem stated above can be applied to any BVP provided we canshow existence of a functional I(φ) corresponding to the BVP Aφ−f = 0such that δI = 0 and δ2I are necessary and sufficient conditions for theexistence of extremum of I(φ). A φ yielding unique extremum of I(φ)is also a unique solution of Aφ− f = 0.

(7) When the operator A in Aφ − f = 0 has some specific properties, thenwe can obtain a special form of Theorem 2.11 (shown later).

(8) Consider

δI =

b∫a

(Fy(x, y, y′)h+ Fy′(x, y, y

′)h′) dx

where

I =

b∫a

F (x, y, y′) dx

we note that h(a) = h(b) = 0 must hold when y(a) = A and y(b) = Bare the boundary conditions; that is, h satisfies the homogeneous partof the boundary conditions on y(x) and ε → 0 as ||h|| → 0, other thanthese, h is arbitrary. Thus, h is a virtual change in y. We define h = δy(variation of y) for fixed x. Similarly, we define h′ = δy′ (variation of y′)also for fixed x. We note that for fixed x, δy and δy′ are not functionsof x. Substituting h = δy and h′ = δy′ in δI we can write

δI =

b∫a

Fy(x, y, y′) dx

δy +

b∫a

Fy′(x, y, y′) dx

δy′


Differentiation of F (·) with respect to y and y′ can be taken outside theintegral as the integral is with respect to x:

δI =∂

∂y

b∫a

F (x, y, y′) dx

δy +∂

∂y′

b∫a

F (x, y, y′) dx

δy′

or

δI =∂I

∂yδy +

∂I

∂y′δy′

Thus, the variation of a functional I(x, y, y′) requires differentiation ofI with respect to dependent variables y and y′ and variations in y andy′ (i.e. δy, δy′) for fixed position coordinates x. This relationship isistrumental in relating δI to dI, the differential of I. Treating I(·) as afunction of x, y, and y′ we can write the following for the differential ofI

dI =∂I

∂ydy +

∂I

∂y′dy′ +

(∂I

∂xdx

)If the positions x are fixed, then the term in the bracket in the expressionfor dI becomes zero and we obtain

dI =∂I

∂ydy +

∂I

∂y′dy′

Comparing with δI we conclude that the variational operator δ acts onI as a differential operator with respect to the dependent variables.

In the following we state two important theorems related to positive-definite differential operators and provide their proofs. These theorems areimportant in relation to the methods of approximation in chapters 3 and 4.

Theorem 2.12. If the linear differential operator A in Aφ − f = 0 ispositive-definite, then Aφ− f = 0 can not have more than one solution.

Proof. We construct the proof of this theorem by contradiction. Let φ1

and φ2 be the two solutions of Aφ − f = 0. Then, Aφ1 − f = 0 andAφ2 − f = 0. We can write Aφ1 − Aφ2 = A(φ1 − φ2) = 0 due to linearityof A. Using φ1 − φ2 = φ, we can write Aφ = 0. Taking scalar product, weobtain (Aφ, φ) = 0. Since the operator is positive-definite (Aφ, φ) > 0 and(Aφ, φ) = 0 if and only if (iff) φ = 0, which implies φ = φ1 − φ2 = 0 orφ1 = φ2. This proves that Aφ − f = 0 can only have one solution, that is,the solution of Aφ− f = 0 is unique.

Theorem 2.13 (Minimal functional theorem.). Let Aφ− f = 0 have asolution where A is a positive-definite operator. We construct the quadraticfunctional I(φ) = 1

2(Aφ, φ) − (f, φ). Then, out of all values given to the


quadratic functional I(φ) by all possible functions φ from the domain ofdefinition of A, the least is the value given to I(φ) by a φ that constitutesthe solution to Aφ − f = 0. Conversely, if there exists a function φ in thedomain of definition of A that gives the minimum value to I(φ), then thisfunction is the solution of BVP Aφ− f = 0.

Proof. Let φ0 be the solution of Aφ − f = 0 which is unique by virtue oftheorem 2.12 so that Aφ0 − f = 0. Consider I(φ) = 1

2(Aφ, φ) − (f, φ) andreplace f by Aφ0.

I(φ) =1

2((Aφ, φ)− 2(Aφ0, φ))

Add and subtract 12(Aφ0, φ0) to write I(φ) as

I(φ) =1

2

((Aφ, φ)− (Aφ0, φ)− (Aφ0, φ) + (Aφ0, φ0)− (Aφ0, φ0)

)=

1

2

((A(φ− φ0), φ

)−((Aφ0, φ)− (Aφ0, φ0)

)− (Aφ0, φ0)

)=

1

2

((A(φ− φ0), φ

)− (Aφ0, φ− φ0)− (Aφ0, φ0)

)Since A is positive-definite, it is symmetric, hence

(Aφ0, φ− φ0) =(φ0, A(φ− φ0)

)=(A(φ− φ0), φ0

)Substituting in I(φ)

I(φ) =1

2

((A(φ− φ0), φ

)−(A(φ− φ0), φ0

)− (Aφ0, φ0)

)or

I(φ) =1

2

((A(φ− φ0), φ− φ0

)− (Aφ0, φ0)

)Since A is positive-definite

(A(φ − φ0), φ − φ0

)> 0 and (Aφ0, φ0) > 0,

hence I(φ) assumes its least value when and only when φ = φ0, that is, asolution φ0 of Aφ− f = 0 yields minimum value of I(φ) and

min I(φ) = −1

2(Aφ0, φ0)


2.3.3 Variation of a functional: some practical aspects

Consider a functional I = I(x, u, u′) in which x is the independent co-ordinate and u is the dependent variable. For fixed values of coordinatex, I depends upon u and u′. Let v be an arbitrary change in u; that is,


let v = δu (variation of u). The symbol variational symbol δ is a differen-tial operator with respect to dependent variables. The following propertiesregarding v = δu hold.

(a) δu represents an admissible change in u for fixed position coordinates i.efixed x. Boundary conditions, loads, and their points of application donot change due to admissible change δu.

(b) If u is specified at some points in the domain (usually the boundary ofthe domain) then v = δu = 0 at such points because the specified valuesof u are fixed, hence cannot be changed or varied. Thus, if u = u0

on some boundary Γ, then v = δu0 = 0 on Γ, i.e v = δu satisfies thehomogeneous form of the boundary conditions on u which is u = 0. Inother words, v = δu vanishes on Γ where u is specified and is arbitraryeverywhere else. So v = δu can be thought of as virtual change in u. vis called test function. Hence, the methods or the techniques based onthis approach are also referred to as the methods based on the principleof virtual work.

(c) As shown earlier the variational operator δ acts as a differential operatorwith respect to dependent variables.

(d) Thus, the laws of variations of sums, products, ratios and powers offunctionals are completely analogous to the corresponding laws of dif-ferentiations. If F and G are two functionals then

(i) δ(F ±G) = δF ± δG(ii) δ(FG) = (δF )G+ F (δG)

(iii) δ(FG

)= GδF−F δG

G2

(iv) δ(F )n = nFn−1δF(v) Variational and differential operators can commute, that is, change

positions, and the same is true for variational and integral oper-ators. This is obviously due to the fact that variation is differ-entiation with respect to dependent variables whereas the integralor differential operators contain operations of integration or differ-entiation with respect to independent variables, that is, positioncoordinates. Thus

d

dx(δu) = δ

(du

dx

)

δ

b∫a

u(x) dx =

b∫a

δu(x) dx

2.3.4 Riemann and Lebesgue integrals

In finite element processes we encounter definite integrals over the dis-cretized domains of definition of the differential operators. These integrals


b

f(x)

xa c

Figure 2.2: f(x) versus x

b

f(x)

xa

fu

c

fl

Figure 2.3: f(x) versus x

must be expressed as the sum of the integrals over the subdomains (finiteelements). In doing so, the continuity of the integrand (or lack of it) overthe whole domain (discretization) is crucial in understanding what theseintegrals mean and or represent.

Consider a simple definite integral in one dimension

I =

b∫a

f(x) dx (2.3)

In the strict sense of calculus of continuous and differentiable functions, theintegral in (2.3) is valid if and only if f(x) is continuous for all x ∈ [a, b].When this is the case, the above integral is called Riemann.

Consider f(x) versus x shown in Fig. 2.2. The figure shows two differentbehaviors of f(x) versus x. In both cases, f(x) is continuous and, hence,(2.3) is a Riemann integral in both cases. In this case we can write 2.3 as,

I =

b∫a

f(x) dx =

c∫a

f(x) dx+

b∫c

f(x) dx (2.4)

All integrals in (2.4) are Riemann and (2.4) holds precisely in the strict senseof calculus of continuous and differentiable functions.

Consider f(x) versus x shown in Fig. 2.3; f(x) is continuous for anyx ∈ [a, c) and x ∈ (c, b]. However, at x = c, f(x) is discontinuous, that is,f(x) changes from fl to fu at x = c; that is, there is a jump in f(x). In thiscase the integral in (2.3) is not valid in the Riemann sense and we cannotexpress (2.3) as a sum of integrals over the subintervals [a, c] and [c, b]. Wenote that change in f(x) from fl to fu is at a point which is a set of measurezero. Thus, if we decide to ignore the integral of f(x) over a set of measurezero then we can write (2.4) in this case also. Thus, the behavior of f(x) atc is ignored. Such integrals in which the discontinuous integrand behavior


over sets of measure zero are neglected are called Lebesgue. In summary,for f(x) versus x in Fig. 2.2 the integrals in (2.3) and (2.4) are Riemannwhereas for f(x) versus x in Fig. 2.3 the integrals are in the Lebesgue sense.

Remarks.

(1) Riemann integrals are based on calculus of continuous and differentiablefunctions.

(2) Lebesgue integrals are based on theory of distributions.

(3) Use of Lebesgue integrals over Riemann integrals must be done withcare. If such an assumption disturbs the physics then the consequencesmay be serious. We will learn more about these and their use in thefinite element processes in later chapters.

2.4 Examples of Differential Operators andtheir Properties

In this section we consider specific examples of boundary value prob-lems to examine the mathematical properties of the differential operators.This will permit us to classify them into three categories: self-adjoint, non-self-adjoint and non-linear. The linearity and symmetry of the differentialoperators (or lack there of) are two important properties that permit theirmathematical classification into these three categories.

2.4.1 Self-adjoint differential operators

The self-adjoint differential operators are linear and symmetric. In thissection we consider some examples of BVPs in which we show that thedifferential operators are self-adjoint under certain restrictions.

Example 2.9 (2D Poisson’s Equation). Consider the BVP

− ∂

∂x

(∂φ

∂x

)− ∂

∂y

(∂φ

∂y

)+Q(x, y) = 0 ∀(x, y) ∈ Ω ⊂ R2 (2.5)

with boundary conditions

φ = φ0 on Γ1 (2.6)

∂φ

∂xnx +

∂φ

∂yny = g(x, y) on Γ2 (2.7)

where φ = φ(x, y) is the dependent variable, Q(x, y) is the source term, Ωis the domain of definition, φ0 is a specified value on Γ1, g(x, y) on Γ2 is aknown function, and Γ = Γ1 ∪ Γ2 is a closed contour defining the boundary

2.4. EXAMPLES OF DIFFERENTIAL OPERATORS AND THEIR PROPERTIES 45

x

y

Ω

nx

ny

n2x + n2

y = 1

n

Γ2

Γ1

Figure 2.4: Domain Ω, boundaries Γ1 and Γ2, and unit exterior normal n to boundaryΓ2

of Ω such that Ω = Γ ∪ Ω; nx and ny are the direction cosines of the unitexterior normal to the boundary Γ2. Figure 2.4 shows the details.

First, we write (2.5) symbolically as

Aφ− f = 0 ∀(x, y) ∈ Ω ⊂ R2 (2.8)

in which

A = − ∂

∂x

(∂

∂x

)− ∂

∂y

(∂

∂y

)(2.9)

andf = −Q(x, y) (2.10)

In the following we determine if the differential operator A defined by (2.9)is linear and symmetric or not.

Linearity of A

Let V ⊂ H be a subspace containing functions defined over Ω that areadmissible in (2.5). Then, to establish linearity of the differential operatorA we must show that

A(αu+ βv) = α(Au) + β(Av) ∀u, v ∈ V, ∀α, β ∈ R. (2.11)

The proof is straight forward. Substitute A from (2.9) in the left side of(2.11), expand and regroup

A(αu+ βv) = −(∂2

∂x2+

∂2

∂y2

)(αu+ βv)

= α

(−∂

2u

∂x2− ∂2u

∂y2

)+ β

(−∂

2v

∂x2− ∂2v

∂y2

)= α(Au) + β(Av)


Hence, the operator A defined by (2.9) is linear.

Symmetry of A

Let φ, v ∈ V . Then to establish the symmetry of the differential operatorA defined by (2.9), we must show that

(Aφ, v) = (φ,Av) (2.12)

The scalar products in (2.12) are over Ω. First, we note that based onfundamental Lemma 2.1 with α = Aφ− f and h = v,∫

Ω

(Aφ− f)v dΩ = (Aφ, v)Ω − (f, v)Ω = 0 is valid (2.13)

in which dΩ = dx dy and the restriction that v = 0 where φ is specified.Thus, based on (2.6) v = 0 on Γ1. v is commonly known as a test function.We note that v = δφ (variation of φ) is also an admissible choice, but v doesnot necessarily have to be δφ. This property of v is important in establish-ing symmetry of the differential operator A. Consider (2.12) in which v = 0on Γ1 due to (2.6). In order to show whether (2.12) holds or not, we canproceed in two ways. The intermediate details are somewhat different, butthe end result and conclusion is not effected.

Method I

We can transfer all of the differentiation from φ to v in the scalar producton the left side of (2.12) using integration by parts, thereby obtaining

(Aφ, v) = (φ,A∗v) + 〈Aφ, v〉Γ

in which 〈Aφ, v〉Γ is the concomitant containing terms resulting from inte-gration by parts. Thus, for symmetry of A we must show that A = A∗ and〈Aφ, v〉Γ = 0. Establishing 〈Aφ, v〉Γ = 0 requires the use of boundary con-ditions (2.6) and (2.7) and their variations. This approach is most generaland is recommended.

Method II

This approach is meaningful when the differential operator has deriva-tives of even order as in (2.9). In (Aφ, v), we transfer half the order ofdifferentiation (one order in this case) from φ to v to obtain

(Aφ, v) = B(φ, v) + 〈Aφ, v〉Γ


Likewise, in (φ,Av) we transfer half the order of differentiation (one orderin this case also) from v to φ to obtain

(φ,Av) = B˜ (φ, v) + 〈φ,Av〉Γ

So symmetry of A in (2.12) reduces to showing that the following relationholds:

B(φ, v) + 〈Aφ, v〉Γ = B˜ (φ, v) + 〈φ,Av〉Γ (2.14)

Thus, A is symmetric if

〈Aφ, v〉Γ − 〈φ,Av〉Γ = 0, B(φ, v) = B˜ (φ, v) (2.15)

If we can show that (2.15) holds using boundary conditions (2.6) and (2.7)and their variations, the symmetry of the operator A is established.

Details of Method I

Consider (Aφ, v) in (2.12):

(Aφ, v)Ω =

∫Ω

[− ∂

∂x

(∂φ∂x

)− ∂

∂y

(∂φ∂y

)]v dx dy (2.16)

Transfer one order of differentiation from φ to v with respect to x and yusing integration by parts

(Aφ, v)Ω =

∫Ω

[∂φ

∂x

∂v

∂x+∂φ

∂y

∂v

∂y

]dx dy −

∮Γ

(∂φ

∂xnx +

∂φ

∂yny

)v dΓ (2.17)

Transfer one more order of differentiation from φ to v in the first integral inthe right side of (2.17)

(Aφ, v)Ω =

∫Ω

[− ∂

∂x

(∂v∂x

)− ∂

∂y

(∂v∂y

)]φdx dy

−∮Γ

(∂φ

∂xnx +

∂φ

∂yny

)v dΓ +

∮Γ

(∂v

∂xnx +

∂v

∂yny

)φdΓ (2.18)

or (Aφ, v)Ω = (φ,A∗v)Ω + 〈Aφ, v〉Γ where A∗ = − ∂∂x

(∂∂x

)− ∂

∂y

(∂∂y

)is the

adjoint of A, which is obviously the same as A and the concomitant 〈Aφ, v〉Γis given by

〈Aφ, v〉Γ = −∮Γ

(∂φ

∂xnx +

∂φ

∂yny

)v dΓ +

∮Γ

(∂v

∂xnx +

∂v

∂yny

)φdΓ (2.19)


Thus, A is symmetric if 〈Aφ, v〉Γ in (2.19) can be shown to be zero. We useboundary conditions (2.6) and (2.7) and their variations to simplify (2.19).First, we note that integrals over Γ can be written as the sum of the integralsover Γ1 and Γ2. This is necessitated due to the fact that BCs are only givenon Γ1 and Γ2

〈Aφ, v〉Γ = −∫Γ1

(∂φ

∂xnx +

∂φ

∂yny

)v dΓ−

∫Γ2

(∂φ

∂xnx +

∂φ

∂yny

)v dΓ

+

∫Γ1

φ

(∂v

∂xnx +

∂v

∂yny

)dΓ +

∫Γ2

φ

(∂v

∂xnx +

∂v

∂yny

)dΓ (2.20)

The expressions in (2.20) can be simplified using

φ = φ0 and v = 0 on Γ1

∂φ

∂xnx +

∂φ

∂yny = g and

∂v

∂xnx +

∂v

∂yny = 0 on Γ2

(2.21)

Substituting from (2.21) in (2.20) we get

〈Aφ, v〉Γ = −∫Γ2

vg dΓ +

∫Γ1

φ0

(∂v

∂xnx +

∂v

∂yny

)dΓ (2.22)

Remarks.

(1) When g = 0 and φ0 = 0, 〈Aφ, v〉Γ = 0 and the operator A is symmetricand, hence, self-adjoint.

(2) When g 6= 0 or φ0 6= 0, the operator A is not symmetric in the strictmathematical sense. However, we note that 〈Aφ, v〉Γ in (2.20) only con-tains v and known functions or quantities φ0 and g on boundaries Γ1

and Γ2. In the classical methods of approximation presented in chapter3 and the finite element processes in the subsequent chapters, we shallsee that 〈Aφ, v〉Γ only contributes to the right hand side vector and doesnot influence the coefficient matrix. That is, when (2.19) is not zeroand when 〈Aφ, v〉Γ is not a function of φ, the operator A is as good asbeing symmetric with regard to methods of approximations and finiteelement processes. Thus, the most important properties in the proof ofsymmetry is the fact that A = A∗ must hold and 〈Aφ, v〉Γ must not bea function of φ.

(3) In this case the adjoint of A is the same as A, A∗ = A. The determinationof adjoint A∗ requires that we transfer all of the differentiation from φto v which is only possible in this method labelled as method I.


Details of Method II

Here, we consider both (Aφ, v) and (φ,Av) in (2.12). First, consider(Aφ, v)

(Aφ, v)Ω =

∫Ω

[− ∂

∂x

(∂φ∂x

)− ∂

∂y

(∂φ∂y

)]v dx dy (2.23)

Transfer one order of differentiation with respect to x and y from φ to vusing integration by parts

(Aφ, v)Ω =

∫Ω

(∂v

∂x

∂φ

∂x+∂v

∂y

∂φ

∂y

)dx dy −

∮Γ

(∂φ

∂xnx +

∂φ

∂yny

)v dΓ (2.24)

or

(Aφ, v)Ω = B(φ, v) + 〈Aφ, v〉Γ (2.25)

Now consider (φ,Av)Ω

(φ,Av)Ω =

∫Ω

[− ∂

∂x

(∂v∂x

)− ∂

∂y

(∂v∂y

)]φdx dy (2.26)

Transfer one order of differentiation with respect to x and y from v to φusing integration by parts

(φ,Av)Ω =

∫Ω

(∂φ

∂x

∂v

∂x+∂φ

∂y

∂v

∂y

)dx dy −

∮Γ

φ

(∂v

∂xnx +

∂v

∂yny

)dΓ (2.27)

or

(φ,Av)Ω = B˜ (φ, v) + 〈φ,Av〉Γ (2.28)

The symmetry of A requires that (2.25) and (2.28) be equal:

B(φ, v) + 〈Aφ, v〉Γ = B˜ (φ, v) + 〈φ,Av〉Γ (2.29)

From (2.24) and (2.27) we note that B(φ, v) = B˜ (φ, v) holds, hence, sym-

metry of A requires that

〈Aφ, v〉Γ − 〈φ,Av〉Γ = 0 must hold

or

−∮Γ

(∂φ

∂xnx +

∂φ

∂yny

)v dΓ +

∮Γ

φ

(∂v

∂xnx +

∂v

∂yny

)dΓ = 0 (2.30)


which is the same as the concomitant in Method I. Thus

〈Aφ, v〉Γ − 〈φ,Av〉Γ = −∫Γ2

vg dΓ +

∫Γ1

φ0

(∂v

∂xnx +

∂v

∂yny

)dΓ (2.31)

must hold for the symmetry of A. Since this is identical to Method I, theremarks presented there hold precisely here as well, except that in this ap-proach it is not possible to determine the adjoint A∗ of A due to the factthat we have not transferred all of the differentiation from φ to v.

Example 2.10 (1D Diffusion Equation). Consider the BVP

− d

dx

(a(x)

dφ

dx

)− c(x)φ+ x2 = 0 ∀x ∈ Ω = (0, 1) ⊂ R (2.32)


φ(0) = φ0 (2.33)

a(1)dφ

dx

∣∣∣1

= q1 (2.34)

where φ is the dependent variable, a(x) and c(x) are known coefficients andφ0 and q1 are given. First, we write (2.32) symbolically as

Aφ− f = 0 ∀x ∈ Ω (2.35)

where

A = − d

dx

(a(x)

d

dx

)− c(x) (2.36)

and

f = −x2 (2.37)

In the following we determine if the differential operator A is linear andsymmetric or not.

Linearity of A

Let V ⊂ H be a subspace containing functions that are admissible in(2.32) over Ω. Then, ∀u, v ∈ V and ∀α, β ∈ R we must show that

A(αu+ βv) = α(Au) + β(Av) (2.38)

holds to establish the linearity of the differential operator A defined by (2.36).To prove (2.38), we substitute for A from (2.36) in the left side of (2.38),


expand and regroup

A(αu+ βv) = − d

dx

(a(x)

d(αu+ βv)

dx

)− c(x)(αu+ βv)

= α

(− d

dx

(a(x)

du

dx

)− c(x)u

)+ β

(− d

dx

(a(x)

dv

dx

)− c(x)v

)= α(Au) + β(Av)

(2.39)


Symmetry of A

Let φ, v ∈ V . Then to establish the symmetry of the differential operatorA in (2.36) we must show that

(Aφ, v)Ω = (φ,Av)Ω (2.40)

First, we note that based on fundamental Lemma 2.1 with α = Aφ− f andv = h ∫

Ω


in which dΩ = dx and the restriction that v = 0 at x = 0 where φ = φ0. Asin example 1, v is the test function and the choice v = δφ is admissible aswell. In this case also, the differential operator has even order derivatives ofφ. Hence, we can use Method I as well as Method II to determine symmetryof A.

Details of Method I

Consider (Aφ, v) in (2.40)

(Aφ, v)Ω =

∫Ω

(− d

dx

(a(x)

dφ

dx

)− c(x)φ

)v dx (2.42)

Transfer one order of differentiation with respect to x from φ to v in the firstterm of the integrand in (2.42) using integration by parts

(Aφ, v)Ω =

∫Ω

(dv

dxa(x)

dφ

dx− c(x)φv

)dx−

[v(a(x)

dφ

dx

)]1

0

(2.43)


Transfer one more order of differentiation with respect to x from φ to v inthe first term of the integrand in (2.43) using integration by parts

(Aφ, v)Ω =

∫Ω

(− d

dx

(a(x)

dv

dx

)φ− c(x)φv

)dx

−[v(a(x)

dφ

dx

)]1

0

+

[φ(a(x)

dv

dx

)]1

0

(2.44)

or(Aφ, v)Ω = (φ,A∗v)Ω + 〈Aφ, v〉Γ (2.45)

where A∗ = − ddx

(a(x) d

dx

)− c(x) is the adjoint of A which is clearly same as

the differential operator A and the concomitant 〈Aφ, v〉Γ is given by

〈Aφ, v〉Γ = −[v(a(x)

dφ

dx

)]1

0

+

[φ(a(x)

dv

dx

)]1

0

(2.46)

Thus, A is symmetric if 〈Aφ, v〉Γ = 0 in (2.46). We use boundary conditions(2.33) and (2.34) and their variations to simplify (2.46). Expanding termson the right side of (2.46)

〈Aφ, v〉Γ = −v(1)a(1)dφ

dx

∣∣∣1

+ v(0)a(0)dφ

dx

∣∣∣0

+ φ(1)(a(1)dv

dx

∣∣∣∣1

− φ(0)a(0)dv

dx

∣∣∣∣0

(2.47)The expression on the right hand side of (2.47) can be simplified using

φ(0) = φ0 ⇒ v(0) = 0

a(1)dφ

dx

∣∣∣1

= q1 ⇒ a(1)dv

dx

∣∣∣∣1

= 0(2.48)

Substituting (2.48) into (2.47)

〈Aφ, v〉Γ = −v(1)q1 − φ0a(0)dv

dx

∣∣∣∣0

(2.49)

Remarks.

1. When q1 = 0 and φ0=0, 〈Aφ, v〉Γ = 0 and, hence, the operator A issymmetric and, thus, self-adjoint.

2. When q1 6= 0 or φ0 6= 0, the operator A is not symmetric in the strictmathematical sense. However, we note that 〈Aφ, v〉Γ in (2.49) only constainsv and known φ0 and q1. In the methods of approximation presented inchapter 3 and the finite element processes subsequently, we shall see that〈Aφ, v〉Γ only contributes to the right hand side vector and does not influencethe coefficient matrix. That is, when (2.45) holds with A∗ = A and when


〈Aφ, v〉Γ is not a function of φ, the operator A is as good as being symmetricfrom the point of view of the methods of approximation and finite elementprocesses.

3. The adjoint A∗ of A is the same as A. Determination of the adjoint A∗

requires that we transfer all of the differentiation from φ to v which is onlypossible in this method.


Here we consider both (Aφ, v)Ω and (φ,Av)Ω in (2.40). First, consider(Aφ, v)Ω

(Aφ, v)Ω =

∫Ω

[− d

dx

(a(x)

dφ

dx

)− c(x)φ

]v dx (2.50)

Transfer one order of differentiation with respect to x from φ to v in the firstterm of the integrand in (2.50) using integration by parts

(Aφ, v)Ω =

∫Ω

(dv

dxa(x)

dφ

dx− c(x)φv

)dx−

[v(a(x)

dφ

dx

)]1

0

(2.51)

or(Aφ, v)Ω = B(φ, v) + 〈Aφ, v〉Γ (2.52)

Now consider (φ,Av)

(φ,Av)Ω =

∫Ω

φ

[− d

dx

(a(x)

dv

dx

)− c(x)v

]dx (2.53)

Transfer one order of differentiation with respect to x from v φ in the firstterm of the integrand in (2.54) using integration by parts

(φ,Av)Ω =

∫Ω

(dφ

dxa(x)

dv

dx− c(x)vφ

)dx−

[φ(a(x)

dv

dx

)]1

0

(2.54)

or(φ,Av)Ω = B˜ (φ, v) + 〈φ,Av〉Γ (2.55)

Symmetry of A requires that (2.52) and (2.55) be equal

B(φ, v) + 〈Aφ, v〉Γ = B˜ (φ, v) + 〈φ,Av〉Γ (2.56)

From (2.78) and (2.54) we note that B(φ, v) = B˜ (φ, v) holds. Hence, sym-


〈Aφ, v〉Γ − 〈φ,Av〉Γ = 0 must hold (2.57)


or

−[v(a(x)

dφ

dx

)]1

0

+

[φ(a(x)

dv

dx

)]1

0

= 0 (2.58)

which is the same as the concomitant in Method I and hence following thedetails presented in Method I we obtain

〈Aφ, v〉Γ − 〈φ,Av〉Γ = −v(1)q1 − φ0a(0)dv

dx

∣∣∣∣0

= 0 (2.59)

Thus, for the symmetry of A (2.59) must hold. Since (2.59) is the same as(2.49) in Method I, the remarks presented there hold here as well. In thismethod, determination of A∗ is not possible in this method as it requirestransferring all of the differentiation from φ to v.

Example 2.11 (1D Beam Equation). Consider the BVP

d2

dx2

(b(x)

d2φ

dx2

)+Q(x) = 0 ∀x ∈ Ω = (0, L) ⊂ R (2.60)


φ(0) = φ0;dφ

dx

∣∣∣0

= q0 (2.61)(bd2φ

dx2

)∣∣∣L

= ML;

[d

dx

(bd2φ

dx2

)]L

= QL (2.62)

where φ is the dependent variable, Q(x) is the source terms and φ0,q0,ML

and QL are known data. First, we write (2.60) symbolically as

Aφ− f = 0 ∀x ∈ Ω (2.63)

where

A =d2

dx2

(a(x)

d2

dx2

)(2.64)

andf = −Q(x) (2.65)


Linearity of A


A(αu+ βv) = α(Au) + β(Av) (2.66)


holds to establish the linearity of the differential operator A defined by (2.64).To prove (2.66), we substitute for A from (2.64) in the left side of (2.66),expand and regroup,

A(αu+ βv) =d2

dx2

(b(x)

d2(αu+ βv)

dx2

)= α

[d2

dx2

(b(x)

d2u

dx2

)]+ β

[d2

dx2

(b(x)

d2v

dx2

)]= α(Au) + β(Av)

(2.67)


Symmetry of A


(Aφ, v)Ω = (φ,Av)Ω (2.68)

First, we note that based on fundamental Lemma 2.1 with α = Aφ− f andv = h ∫

Ω

(Aφ− f)v dΩ = (Aφ, v)Ω − (f, v)Ω = 0 (2.69)

in which dΩ = dx and the restriction that v = 0 at x = 0 where φ = φ0.As in previous examples, v is the test function and the choice v = δφ isadmissible as well. In this case also, the differential operator has even orderderivatives of φ. Hence, we can use Method I as well as Method II todetermine symmetry of A.

Details of Method I

Consider (Aφ, v)Ω in (2.68),

(Aφ, v)Ω =

∫Ω

d2

dx2

(b(x)

d2φ

dx2

)v dx (2.70)

Transfer all orders of differentiation from the dependent variable φ to thetest function v by using integration by parts. This results in four boundary


terms:

(Aφ, v)Ω =

∫Ω

φ

[d2

dx2

(b(x)

d2v

dx2

)]dx

+

[d

dx

(b(x)

d2φ

dx2

)v

]L0

−[dv

dxb(x)

d2φ

dx2

]L0

+

[d2v

dx2b(x)

dφ

dx

]L0

−[d

dx

(b(x)

d2v

dx2

)φ

]L0

(2.71)

or

(Aφ, v)Ω = (φ,A∗v)Ω + 〈Aφ, v〉Γ (2.72)

where A∗ = d2

dx2(a(x) d2

dx2) is the adjoint of A which is clearly the same as A

and the concomitant 〈Aφ, v〉Γ is given by

〈Aφ, v〉Γ =

[d

dx

(b(x)

d2φ

dx2

)v

]L0

−[dv

dxb(x)

d2φ

dx2

]L0

+

[d2v

dx2b(x)

dφ

dx

]L0

−[d

dx

(b(x)

d2v

dx2

)φ

]L0

(2.73)

Expanding terms on the right side of (2.73)

〈Aφ, v〉Γ =

[d

dx

(b(x)

d2φ

dx2

)v

]x=L

−[d

dx

(b(x)

d2φ

dx2

)v

]x=0

−[dv

dxb(x)

d2φ

dx2

]x=L

+

[dv

dxb(x)

d2φ

dx2

]x=0

+

[d2v

dx2b(x)

dφ

dx

]x=L

−[d2v

dx2b(x)

dφ

dx

]x=0

−[d

dx

(b(x)

d2v

dx2

)φ

]x=L

+

[d

dx

(b(x)

d2v

dx2

)φ

]x=0

(2.74)

The terms on the right hand side of (2.74) can be simplified using

φ(0) = φ0 ⇒ v(0) = 0

dφ

dx

∣∣∣∣0

= q0 ⇒dv

dx

∣∣∣∣0

= 0(bd2φ

dx2

)∣∣∣∣L

= ML ⇒(bd2v

dx2

)∣∣∣∣L

= 0

d

dx

(bd2φ

dx2

)∣∣∣∣L

= QL ⇒d

dx

(bd2v

dx2

)∣∣∣∣L

= 0

(2.75)



〈Aφ, v〉Γ = QLv(L)−ML

(dvdx

)∣∣∣∣L

− q0

(b(x)

d2v

dx2

)∣∣∣∣0

+ φ0

[d

dx

(b(x)

d2v

dx2

)]0

(2.76)

Remarks.

(1) When φ0 = 0, q0 = 0, ML = 0 and QL = 0 the differential operator issymmetric and, hence self-adjoint.

(2) When φ0 6= 0 or q0 6= 0 or ML 6= 0 or QL 6= 0, the operator A isnot symmetric in the strict mathematical sense. However, we note that〈Aφ, v〉Γ in (2.76) only constains v and known φ0, q0, ML and QL. In themethods of approximation presented in chapter 3 and the finite elementprocesses subsequently, we shall see that 〈Aφ, v〉Γ only contributes to theright hand side vector and does not influence the coefficient matrix. Thatis, when (2.45) holds with A∗ = A and when 〈Aφ, v〉Γ is not a functionof φ, the operator A is as good as being symmetric from the point ofview of the methods of approximation and finite element processes.

(3) The adjoint A∗ of A is the same as A. Determination of the adjoint A∗

requires that we transfer all of the differentiation from φ to v which isonly possible in this method.


Here we consider both (Aφ, v)Ω and (φ,Av)Ω in (2.68). First, consider(Aφ, v)

(Aφ, v)Ω =

∫Ω

[d2

dx2

(b(x)

d2φ

dx2

)v

]dx (2.77)

Transfer two orders of differentiation with respect to x from φ to v in (2.77)using integration by parts

(Aφ, v)Ω =

∫Ω

(d2v

dx2b(x)

d2φ

dx2

)dx+

[d

dx

(b(x)

d2φ

dx2

)v

]L0

−[dv

dxb(x)

d2φ

dx2

]L0

(2.78)or

(Aφ, v)Ω = B(φ, v) + 〈Aφ, v〉Γ (2.79)

in which

〈Aφ, v〉Γ =

[d

dx

(b(x)

d2φ

dx2

)v

]L0

−[dv

dxb(x)

d2φ

dx2

]L0

(2.80)


Now consider (φ,Av)Ω

(φ,Av)Ω =

∫Ω

[d2

dx2

(b(x)

d2v

dx2

)φ

]dx (2.81)

Transfer two orders of differentiation with respect to x from v to φ in (2.81)using integration by parts

(φ,Av)Ω =

∫Ω

(d2φ

dx2b(x)

d2v

dx2

)dx+

[d

dx

(b(x)

d2v

dx2

)φ

]L0

−[dφ

dxb(x)

d2v

dx2

]L0

(2.82)or

(φ,Av)Ω = B˜ (φ, v) + 〈φ,Av〉Γ (2.83)

in which

〈φ,Av〉Γ =

[d

dx

(b(x)

d2v

dx2

)φ

]L0

−[dφ

dxb(x)

d2v

dx2

]L0

(2.84)

Symmetry of A requires that (2.79) and (2.83) be equal

B(φ, v) + 〈Aφ, v〉Γ = B˜ (φ, v) + 〈φ,Av〉Γ (2.85)

From (2.78) and (2.54) we note that B(φ, v) = B˜ (φ, v) holds. Hence, sym-


〈Aφ, v〉Γ − 〈φ,Av〉Γ = 0 (2.86)

Substituting (2.80) and (2.80) into (2.86) and comparing the resulting ex-pression with the concomitant in Method I, we note that they are exactly thesame. Thus, the remaining details and remarks are the same as in Method I.In this method, the determination of A∗ is not possible for the same reasonsas in previous examples.

2.4.2 Non-self-adjoint differential operators

The non-self-adjoint differential operators are linear but not symmetric.For these operators, the adjoint A∗ of the operator is never the same asthe operator A itself. This is a fundamental difference between self-adjointand non-self-adjoint operators. In this section we consider two examples ofnon-self-adjoint operators.

Example 2.12 (1D convection diffusion equation). Consider the BVP

dφ

dx− kd

2φ

dx2= 0 ∀x ∈ Ω = (0, 1) ⊂ R (2.87)



φ(0) = 1 and φ(1) = 0 (2.88)

where φ(x) is the dependent variable and k is a known diffusion coefficientand is constant. First, we write (2.87) symbolically as

Aφ− f = 0 ∀x ∈ Ω (2.89)

where

A =d

dx− k d

2

dx2(2.90)

and

f = 0 (2.91)


Linearity of A


A(αu+ βv) = α(Au) + β(Av) (2.92)

holds to establish the linearity of the differential operator A defined by (2.90).To prove (2.92), we substitute for A from (2.90) in the left side of (2.92),expand and regroup

A(αu+ βv) =d

dx(αu+ βv)− kd

2(αu+ βv)

dx2

= α

(du

dx− kdu

2

dx2

)+ β

(dv

dx− k dv

2

dx2

)= α(Au) + β(Av)

(2.93)


Symmetry of A


(Aφ, v)Ω = (φ,Av)Ω (2.94)


As in the case of previous examples, here also we note that based on funda-mental Lemma 2.1 with α = Aφ− f and v = h∫

Ω


with the restriction that v = 0 where φ is specified. v is the test functionand v = δφ is admissible. Since in this case the differential operator containsboth odd and even order derivatives, we can only consider Method I in whichall of the differentiation is transfered from φ to v in (Aφ, v)Ω.

Consider (Aφ, v)Ω in (2.94)

(Aφ, v)Ω =

∫Ω

(dφ

dxv − kd

2φ

dx2v

)dx (2.96)

Transfer one order of differentiation from φ to v in the first term and andtwo orders in the second term from φ to v in the integrand of (2.96) usingintegration by parts

(Aφ, v)Ω =

∫Ω

(−dvdx− k d

2v

dx2

)φdx+[vφ]10−

[v(kdφ

dx

)]1

0

+

[kdv

dxφ

]1

0

(2.97)

or

(Aφ, v)Ω = (φ,A∗v)Ω + 〈Aφ, v〉Γ (2.98)

where A∗ = − ddx − k

d2

dx2is the adjoint of A. Clearly, A∗ is not the same

as the differential operator A, hence, the differential operator A cannot besymmetric regardless of whether the concomitant 〈Aφ, v〉Γ is zero or not. Wepresent the remaining details of simplifying 〈Aφ, v〉Γ as these are helpful inthe methods of approximation and the finite element processes presented insubsequent chapters. Comparing (2.98) with (2.97) we see that

〈Aφ, v〉Γ = [vφ]10 −[v(kdφ

dx

)]1

0

+

[kdv

dxφ

]1

0

(2.99)

The terms on the right side of (2.99) can be simplified using the boundaryconditions and v(0) = 0, v(1) = 0.

〈Aφ, v〉Γ = −(kdv

dx

)∣∣∣∣0

φ(0) = −(kdv

dx

)∣∣∣∣0

(2.100)

Clearly, the differential operatorA is not symmetric asA∗ 6= A and 〈Aφ, v〉Γ 6=0, hence the operator A is non-self-adjoint.


Example 2.13 (2D convection diffusion equation). Consider the BVP

u∂φ

∂x+ v

∂φ

∂y− k∂

2φ

∂x2− k∂

2φ

∂y2= f ∀(x, y) ∈ Ω ⊂ R2 (2.101)


φ = φ0 on Γ1 (2.102)

k∂φ

∂xnx + k

∂φ

∂yny = q on Γ2 (2.103)

where Γ = Γ1 ∪ Γ2 is the closed contour constituting the boundary of thedomain Ω such that Ω = Γ∪Ω. φ(x, y) is the dependent variable and, u andv are known and are constants and the diffusion coefficient k is also knownand is constant. First, we write (2.101) symbolically as

Aφ− f = 0 ∀x ∈ Ω ⊂ R2 (2.104)

where

A = u∂

∂x+ v

∂

∂y− k ∂

2

∂x2− k ∂

2

∂y2(2.105)

In the following we determine if the differential operator A defined by(2.105) is linear and symmetric or not.

Linearity of A

Let V ⊂ H be a subspace containing functions that are admissible in(2.101) over Ω. Then, ∀w1, w2 ∈ V and ∀α, β ∈ R we must show that

A(αw1 + βw2) = α(Aw1) + β(Aw2) (2.106)

holds to establish the linearity of the differential operator A defined by(2.105). To prove (2.106), we substitute for A from (2.105) in the left handside of (2.106), expand and regroup

A(αw1 + βw2) = u∂

∂x(αw1 + βw2) + v

∂

∂y(αw1 + βw2)

− k ∂2

∂x2(αw1 + βw2)− k ∂

2

∂y2(αw1 + βw2)

= α

(u∂

∂xw1 + v

∂

∂yw1 − k

∂2

∂x2w1 − k

∂2

∂y2w1

)+ β

(u∂

∂xw2 + v

∂

∂yw2 − k

∂2

∂x2w2 − k

∂2

∂y2w2

)= α(Aw1) + β(Aw2)

(2.107)



Symmetry of A

Let φ,w ∈ V . Then to establish the symmetry of the differential operatorA in (2.105) we must show that

(Aφ,w)Ω = (φ,Aw)Ω. (2.108)

As in the case of previous examples, here also we note that based on funda-mental Lemma 2.1 with α = Aφ− f and w = h∫

Ω

(Aφ− f)w dΩ = (Aφ,w)Ω − (f, w)Ω = 0 (2.109)

with the restriction that w = 0 where φ is specified. w is the test functionand w = δφ is admissible. Since in this case the differential operator containsboth odd and even order derivatives, we can only consider Method I in whichall of the differentiation is transferred from φ to w in (Aφ,w)Ω.

Consider (Aφ,w)Ω in (2.108)

(Aφ,w)Ω =

∫Ω

(u∂φ

∂x+ v

∂φ

∂y− k∂

2φ

∂x2− k∂

2φ

∂y2

)w dxdy (2.110)

Transfer all of the differentiation from φ to w in the integrand of (2.110)using integration by parts

(Aφ,w)Ω =

∫Ω

(−u∂w

∂x− v∂w

∂y− k∂

2w

∂x2− k∂

2w

∂y2

)φdx dy

+

∮Γ

((unx + vny)φw

)dΓ

−∮Γ

(k∂φ

∂xnx + k

∂φ

∂yny

)w dΓ

+

∮Γ

(k∂w

∂xnx + k

∂w

∂yny

)φdΓ (2.111)

or

(Aφ, v)Ω = (φ,A∗v)Ω + 〈Aφ, v〉Γ (2.112)

where A∗ = −u ∂∂x − v ∂

∂y − k ∂2

∂x2− k ∂2

∂y2is the adjoint of A. Clearly, A∗

is not same as A, hence the differential operator is not symmetric. Theconcomitant 〈Aφ, v〉Γ can be simplified using boundary conditions and theproperties of w. We leave this as an exercise.


2.4.3 Non-linear differential operators

The non-linear differential operators are not linear and hence cannot besymmetric. In the following we consider two examples. Here, we only needto check for the linearity of the operator.

Example 2.14 (1D Burgers equation). Consider the BVP

φdφ

dx− kd

2φ

dx2= 0 ∀x ∈ Ω = (0, 1) ⊂ R (2.113)


φ(0) = 1 and φ(1) = 0 (2.114)

where φ(x) is the dependent variable and k = 1Re is a known coefficient (Re

is Reynolds number). First, we write (2.113) symbolically as

Aφ− f = 0 ∀x ∈ Ω ⊂ R (2.115)

where

A = φd

dx− k d

2

dx2(2.116)

andf = 0 (2.117)

We only check for linearity of A in the following. Let V ⊂ H be a subspacecontaining functions that are admissible in (2.113) over Ω. Then, ∀u, v ∈ Vand ∀α, β ∈ R we must show that

A(αu+ βv) = αA(u) + β A(v). (2.118)

We substitute for A from (2.116) in the left hand side of (2.118)

A(αu+ βv) = (αu+ βv)d

dx(αu+ βv)− kd

2(αu+ βv)

dx2(2.119)

For the terms on the right side of (2.118) we can write,

αA(u) = α

(udu

dx− 1

Re

d2u

dx2

)(2.120)

and

β A(v) = β

(vdv

dx− 1

Re

d2v

dx2

)(2.121)

Comparing (2.119) with the sum of (2.120) and (2.121), we clearly see that(2.118) does not hold. Hence, the operator A is not linear [because of thefirst term in (2.116)] and, therefore, cannot be symmetric either.


Example 2.15 (2D Burgers equation). Consider the BVP

φ∂φ

∂x+ φ

∂φ

∂y− 1

Re

(∂2φ

∂x2+∂2φ

∂y2

)= f ∀(x, y) ∈ Ω (2.122)


φ = φ0 on Γ1 (2.123)

k∂φ

∂xnx + k

∂φ

∂yny = q on Γ2 (2.124)

where Re is the Reynolds number (a constant), φ0 and q are known data andΓ = Γ1 ∪ Γ2 is the closed contour constituting the boundary of the domainΩ such that Ω = Γ ∪ Ω. First, we write (2.122) symbolically as

Aφ− f = 0 ∀x ∈ Ω (2.125)

where

A = φ∂

∂x+ φ

∂

∂y− 1

Re

(∂2

∂x2+

∂2

∂y2

)∀(x, y) ∈ Ω (2.126)

Following previous examples, it is rather obvious that A is not linear[because of the first two terms in (2.126)] and, hence, also not symmetric.

Remarks.

(1) We note that when the description of a BVP contains product terms inthe dependent variables and/or their derivatives, the operator is boundto be non-linear.

(2) When a differential operator is not linear it cannot be symmetric aslinearity of an operator is an essential property for the symmetry of theoperator.

2.5 Summary

In this chapter basic elements of applied mathematics that are pertinentin understanding the mathematical details of the methods of approxima-tions and the finite element method are presented. Sets, spaces, functions,function spaces, operators, differential operators, Hilbert spaces, scalar prod-ucts in Hilbert spaces, properties of scalar product, norm of a function, andmathematical classifications of differential operators are defined with exam-ples. Details of the integration by parts in R, R2, and R3 are presented withexamples. Basic elements of the calculus of variations, derivation of thevariation of a functional, necessary condition for extrema of a functional,and associated Euler’s equation are presented and derived. Riemann andLebesgue integrals are defined and illustrated. Examples of differential op-erators: self-adjoint, non-self-adjoint, and non-linear appearing in BVPs arepresented and their properties are established.

2.5. SUMMARY 65

Problems

In Problems 2.1 to 2.3 show whether the differential operator is linear or not.

2.1 Consider the dimensionless form of the one dimensional steady state convection dif-fusion equation

dφ

dx− 1

Pe

d2φ

dx2= 0 ∀x ∈ Ω = (0, 1) ⊂ R

with φ(0) = 1, φ(1) = 0

where Pe is called the Peclet number and is known.

2.2 Consider one dimensional axial deformation of a rod (with variable properties).

d

dx

(a(x)

du

dx

)= 0 ∀x ∈ Ω = (0, 1) ⊂ R

with u(0) = 0,[a(x)

du

dx

]x=1

= P1

2.3 Consider the dimensional form of the one dimensional steady state Burgers equation.

φdφ

dx− 1

Re

d2φ

dx2= 0 ∀x ∈ Ω = (0, 1) ⊂ R

with φ(0) = 1, φ(1) = 0

where Re is the Reynolds number.

In Problems 2.4 to 2.7 show whether the functionals are bilinear and symmetric or not.

2.4 Consider the following functional:

B(φ, v) =

2∫1

a(x)dφ

dx

dv

dxdx+ b(2)φ(2) v(2)

where a(·) and b(·) are given functions of x.

2.5 Consider the functional

B(φ, v) =

1∫−1

a(x)d2φ

dx2d2v

dx2dx+

1∫−1

c(x)φv dx

where a(·), c(·) are known functions of x.

2.6 The functional B(·, ·) is given by

B(φ, v) =

1∫−1

φdφ

dx

dv

dxdx

2.7 Consider the following functional

B(φ, v) =

1∫−1

dφ

dxv dx+

1

Pe

1∫−1

dφ

dx

dv

dxdx

where Pe is the Peclet number.

In problems 2.8 to 2.11 determine


(a) the differential operator(b) if the differential operator is linear or not.(c) if the differential operator is symmetric or not. If not, then under what conditions

would it be symmetric.(d) the adjoint of the operator.

2.8 Consider the following BVP:

− d

dx

(a(x)

du

dx

)+ cu = q(x) ∀x ∈ Ω = (0, 1) ⊂ R

with u(0) = u0,

[a(x)

du

dx+ β(u− u∞)

]x=1

= Q1

Here a(x) and q(x) are known functions of x and β, c, u∞, u0, and Q1 are known constants.

2.9 Consider the BVP

− d

dx

(b(x)

dφ

dx

)= q(x) ∀x ∈ Ω = (0, L) ⊂ R

with φ(0) = 0,

[b(x)

dφ

dx+ kφ

]x=L

= PL

where b(x) and q(x) are known functions of x and k and PL are known constants.


d2

dx2

(b(x)

d2ψ

dx2

)+

d

dx

(c(x)

dψ

dx

)+ α(x)ψ = f(x) ∀x ∈ Ω = (0, L) ⊂ R

with

ψ(0) = ψ0, ψ(L) = ψL[b(x)

d2ψ

dx2

]x=0

= M0,[b(x)

d2ψ

dx2

]x=L

= ML

where b, c, α and f are known functions of x and ψ0, ψL, M0, and ML are known constants.

2.11 Consider the BVP

− d

dx

(φdφ

dx

)+ f(x) = 0 ∀x ∈ Ω = (0, 1) ⊂ R

withdφ

dx

∣∣∣∣x=0

= q0 and φ(1) =√

2

where f(·) is a known function of x and q0 is a constant.

[1–5,5, 5–9,9, 9, 10,10,11,11–20]

References for additional reading[1] M. Gelfand and S. V. Fomin. Calculus of Variations. Dover Publications, 2000.

[2] W. Rudin. Real and Complex Analysis. McGraw-Hill, 1966.

[3] F. B. Hildebrand. Methods of Applied Mathematics. Prentice-Hall, New York, 1965.

[4] J. N. Reddy and M. L. Rasmussen. Advanced Engineering Analysis. John Wiley, NewYork, 1982.


[6] E. C. Titchmarsh. The Theory of Functions. Oxford University Press, 2nd edition,1939.

REFERENCES FOR ADDITIONAL READING 67

[7] I. S. Sokolnikoff and R. M. Redheffer. Mathematics of Physics and Modern Engineer-ing. McGraw-Hill, 2nd edition, 1966.

[8] J. T. Oden and L. Demkowicz. Applied Funtional Analysis. CRC-Press, 1996.

[9] K. S. Surana, A. R. Ahmadi, and J. N. Reddy. The k-version of finite element methodfor self-adjoint operators in BVP. Int. J. Comp. Eng. Sci., 3(2):155–218, 2002.





[14] M. Becker. The Principles and Applications of Variational Methods. MIT Press, 1964.

[15] M. Forray. Variational Calculus in Science and Engineering. McGraw-Hill, 1968.


[17] R. S. Schechter. The Variational Methods in Engineering. McGraw-Hill, 1967.


[19] R. Weinstock. Calculus of Variations with Applications to Physics and Engineering.McGraw-Hill, 1952.


3

Classical Methods ofApproximation

3.1 Introduction

We consider an abstract boundary value problem Aφ − f = 0 definedover a domain Ω. Let the differential operator contain derivatives up toorder 2m of the dependent variables φ. We seek an approximation φn ofφ over Ω. In doing so, if the domain of definition Ω is not discretized,then we have what are called “classical methods” of approximation for theBVPs. On the other hand, if the domain of definition Ω is discretized intoΩT consisting of subdomains, then the entire domain Ω can be thought of asan assembly of the subdomains and the approximation of the solution maybe constructed over each subdomain. Then behavior of the solution over ΩT

in this process is obviously dependent on the behavior of the solution overeach subdomain as well as inter-subdomain boundaries. The finite elementmethod is a method in which one does precisely so. Thus, the finite elementmethod can be viewed as application of classical methods of approximationfor each subdomain of ΩT , i.e. piecewise application. Hence, herein lies themotivation for studying classical methods of approximation.

In the classical methods of approximation the domain of definition Ω ofa BVP Aφ − f = 0 is not discretized while seeking an approximation φnof φ. In the methods utilizing discretization of Ω, such as finite difference,finite volume, and finite element methods the domain of definition Ω ofa BVP is obviously discretized in some way. There are some methods ofapproximation in which one constructs an integral form using the descriptionAφ − f = 0 of the BVP over Ω. This integral form is then utilized to findthe approximation φn of φ. Such methods in the published literature arecalled “variational methods.” We simply refer to them as “methods basedon integral forms.” Then, there are other methods of approximation inwhich one does not construct an integral form using Aφ−f = 0 over Ω. Suchmethods in the published work are referred to as “non-variational methods.”We simply refer to them as “methods not based on integral forms.” We keepin mind that a method of approximation based on an integral form may ormay not be a variational method. This of course depends upon whether

69

70 CLASSICAL METHODS OF APPROXIMATION

the integral form at hand satisfies the criteria to be called variational. Inthis chapter we consider the following classical methods of approximationin which one either constructs an integral form directly using Aφ − f = 0based on fundamental lemma of calculus of variations in Ω or the integralform eventually results from the first variation of a functional (set to zero)constructed using Aφ− f = 0 in Ω.

1. the Galerkin method (GM)

2. the Petrov–Galerkin method (PGM)

3. the method of weighted-residuals (WRM)

4. the Galerkin method with weak form (GM/WF)

5. Based on variation of residual functional: least-squares method/process(LSM/LSP)

Methods 1 though 4 are very closely related in the sense that the funda-mental lemma of calculus of variations forms the basis for them. This is alsothe case in method 4 but in this case integration by parts plays a crucialrole. Method 5 is based on construction of a functional and its variation.In the following sections of this chapter we consider details of the variousmethods listed above for self-adjoint, non-self-adjoint and nonlinear differ-ential operators. Our aim is to determine which methods of approximationyield variationally consistent integral forms for which differential operators.

3.2 Basic Steps in Classical Methods ofApproximation based on Integral Forms

Step 1: One considers Aφ − f = 0 in Ω and constructs an integral eitherdirectly using the fundamental Lemma of the calculus of variationsor by taking the first variation of a functional associated with Aφ−f = 0 in Ω and setting it to zero.

Step 2: In all methods of approximation considered in the following sections,the theoretical solution φ(x) is approximated by φn(x) in the form

φn(x) = N0(x) +

n∑i=1

CiNi(x) (3.1)

where N0(x) and Ni(x) are known functions and Ci (i = 1, 2, . . . , n)are unknown parameters to be determined. The functions Ni(x) arecalled the basis functions or approximation functions. Thus, φn is alinear combination of the basis functions Ni(x). The approximationφn of φ must satisfy either all or only some of boundary conditionsof the BVP, depending upon the method of approximation.

3.2. BASIC STEPS IN CLASSICAL METHODS OF APPROXIMATION 71

Determination of N0(x) and Ni(x) (i = 1, 2, . . . , n)

We consider some guidelines that help us in determining N0(x) andNi(x) (i = 1, 2, . . . , n) in (3.1). We classify all specified BCs of theBVP in two types, homogeneous and nonhomogeneous, and considerthe following:

(a) We choose each Ni(x) such that each Ni(x) satisfies requiredhomogeneous boundary conditions as well as homogeneous formsof the required nonhomogeneous boundary conditions. With

this choice of Ni(x) their linear combinationn∑i=1

Ni(x)Ci will

also satisfy these boundary conditions.

(b) With this choice of Ni(x) (i = 1, 2, . . . , n) in (a), we must chooseN0(x) such that it satisfies required nonhomogeneous bound-ary conditions as well as required homogeneous boundary con-ditions.

(c) If we choose Ni(x) (i = 1, 2, . . . , n) as in (a) and N0(x) as in (b),then φn(x) in (3.1) is ensured to satisfy all required boundaryconditions of the BVP.

(d) Ni(x) (i = 1, 2, . . . , n) must constitute a complete and linearlyindependent set so that the rows and columns in the resultingalgebraic systems remain linearly independent.

(e) N0(x) and Ni(x) (i = 1, 2, . . . , n) must also possess desired con-tinuity and differentiability properties so that φn(x) in (3.1) re-mains admissible in the BVP.

(f) Out of all BCs (homogeneous and nonhomogeneous) of the BVP,the required set of BCs are those that remain to be satisfied byφn(x) after constructing the integral form. In other words ifsome boundary conditions are absorbed or used in constructingthe integral form (as in weak form GM), then these must beremoved from the total set of BCs to determine the required setof BCs to be satisfied by φn.

Step 3: Determine the unknown constants Ci by substituting φn in the in-tegral form. When φn is substituted in the integral form and whenwe choose as many test functions as the number of unknowns Ci(i = 1, 2, . . . , n), we obtain a system of n algebraic equations fromwhich Ci’s can be calculated. Knowing the Ci’s, the approximationφn in step 2 gives the solution over Ω. Since the approximation φncontains n basis functions Ni (i = 1, 2, . . . , n), it also is referred toas an n-parameter (Ci’s being the parameters) approximation of φ,hence the reason for the subscript n on φ.


Thus, the classical methods of approximation are rather simple in prin-ciple and the basic steps are straightforward. The difficulty, however, lies inthe determination of Ni (i = 0, 1, . . . , n). When the domain Ω is geomet-rically complicated with involved boundary conditions, determination of Ni

in most cases may not be possible.

Remarks. In the approach described above, the integral form when com-pared with the elements of calculus of variations presented in chapter 2(section 2.3) is only the necessary condition, i.e. variation of some func-tional I(φn) (assumed to exist) set to zero. Extremum principle or sufficientcondition is yet to be considered as without this we can not determine if thesolution φn obtained from the integral form is unique or not. This aspectis considered for various methods of approximation for the three classes ofdifferential operators in later sections by establishing variational consistencyor variational inconsistency of the integral forms by using their definitionspresented in chapter 2 (section 2.3).

3.3 Integral forms using the Fundamental Lemmaof the Calculus of Variations

Let Aφ − f = 0 in Ω be an abstract boundary value problem. Then,the integral form for Aφ − f = 0 can be constructed over Ω using theFundamental Lemma (lemma 2.1). We recall the Fundamental Lemma first.

Lemma. If α(x) is continuous on [a, b] and if

b∫a

α(x)h(x)dx = 0, ∀h(x) ∈ H1(a, b)

then α(x) = 0 ∀x ∈ (a, b).

Since Aφ− f = 0, ∀x ∈ Ω, then∫Ω

(Aφ− f)v dΩ = 0 (3.2)

holds for any v. In (3.2), when compared to Fundamental Lemma, Aφ−f = 0takes the place of α(x) and v replaces h(x).

If A is a differential operator containing highest order derivative of order2m, then for the integrand in (3.2) to be continuous the following must hold[recall that H2m+1(Ω) is the Hilbert space of functions that have continuous

3.3. INTEGRAL FORMS USING THE FUNDAMENTAL LEMMA 73

derivatives up to order 2m and square-integrable, and the 2m+1st derivativeexists but may not be continuous]:

φ ∈ Vn ⊂ H2m+1(Ω) ⊂ H1(Ω) ∀x ∈ Ω

v ∈ V ⊂ H1(Ω) ∀x ∈ Ω(3.3)

where v is called a test function. The choice of test function v must be suchthat (i) v = 0 on Γ if φ = φ0 (specified) on Γ and (ii) v = δφ is a valid choicesince φ = φ0 on Γ implies that v = δφ0 = 0 on Γ. The space Vn of functionsis called the approximation space and V is called the space of test functions.We also note that (3.2) can now be written as∫

Ω

(Aφ− f)v dΩ = (Aφ− f, v)Ω = 0 (3.4)

Thus, the integral form is the scalar product of Aφ−f and the test functionv over Ω. We can also write (3.4) as

B(φ, v) = l(v) (3.5)

where

B(φ, v) = (Aφ, v)

l(v) = (f, v)(3.6)

Both B(·, ·) and l(·) are functionals. B(·, ·) contains both functions φ and vwhereas l(·) only contains the test functions v.

In the following we consider various classical methods of approximationbased on the integral form (3.4) or (3.5) constructed using the fundamentallemma of the calculus of variations.

3.3.1 The Galerkin method

We replace φ by φn in (3.4), an approximation of φ

(Aφn − f, v) = 0 (3.7)

In the Galerkin method, we choose

v = δφn = Nj (j = 1, 2, . . . , n) (3.8)

with v = Nj (j = 1, 2, . . . , n) satisfying the homogeneous form of all spec-ified boundary conditions on Γ, while φ0 satisfies all of the actual specified


boundary conditions of the BVP. Substituting φn from (3.1) and v from (3.8)into (3.7) we obtain(

A(N0(x) +

n∑i=1

CiNi(x))− f,Nj

)= 0, j = 1, 2, . . . , n (3.9)

In this method φn(x) must satisfy all specified BCs of the BVP; hence, therequired boundary conditions to be considered in determining N0(x) andNi(x) (i = 1, 2, . . . , n) are all specified BCs of the BVP. Whether (3.9) can befurther simplified or not depends upon the nature of the differential operator(e.g., linear or nonlinear). We consider different cases in the following.

3.3.1.1 Self-adjoint and non-self-adjoint lineardifferential operators

Using the linearity of the operator, (3.9) can be simplified as( n∑i=1

CiANi(x) +AN0(x), Nj(x))

= (f,Nj(x)), j = 1, 2, . . . , n (3.10)

orn∑i=1

(ANi, Nj)Ci = (f,Nj)− (AN0, Nj), j = 1, 2, . . . , n (3.11)

In matrix form, we can write (3.11) as

[K]C = F (3.12)

In (3.12), [K] is an n× n matrix and C and F are n× 1 vectors; Kij of[K] and Fi of F are given by

Kij = (ANj , Ni), i, j = 1, 2, . . . , n

Fi = (f,Ni)− (AN0, Ni), i = 1, 2, . . . , n(3.13)

Remarks.

(1) Obviously, Kij 6= Kji, that is, the coefficient matrix [K] is not symmet-ric. Hence, [K] is not ensured to be positive-definite for all admissiblechoices of Ni for i = 1, 2, . . . , n.

(2) Whether the integral form in (3.7) is VC or VIC needs to be established.

Theorem 3.1. If Aφ−f = 0 in Ω is a BVP with some boundary conditionsin which A is self-adjoint, then there exists a functional I(φ) given by

I(φ) =1

2(Aφ, φ)− (f, φ), φ ∈ Vn (3.14)


corresponding to Aφ − f = 0 such that (Aφ − f, v) with v = δφ representsthe first variation of I(φ) (i.e., δI(φ)) and if I(φ) is differentiable in itsarguments then δI(φ) = 0 is a necessary condition for an extremum of I(φ).

Proof. Taking the first variation of I(φ)

δI(φ) =1

2(δ(Aφ), φ) +

1

2(Aφ, δφ)− (f, δφ) (3.15)

or

δI(φ) =1

2(A(δφ), φ) +

1

2(Aφ, δφ)− (f, δφ) (3.16)

Since A is self-adjoint(A(δφ), φ

)= (δφ,Aφ) = (Aφ, δφ) (3.17)

Substituting (3.17) into (3.16) we obtain

δI(φ) =1

2(Aφ, δφ) +

1

2(Aφ, δφ)− (f, δφ)

= (Aφ, v)− (f, v) = (Aφ− f, v)(3.18)

If I is differentiable in its arguments, then δI(φ) is continuous and we haveδI = (Aφ− f, v) = 0, a necessary condition for an extremum of I(φ).

Theorem 3.2. If Aφ−f = 0 in Ω is a BVP in which A is a self-adjoint dif-ferential operator, then the integral form resulting from the Galerkin methodwith v = δφ is VIC.

Proof. Let there exist a functional I(φ) such that

δI(φ) = (Aφ− f, v) = 0 with v = δφ, φ, v ∈ Vn (3.19)

Then

δ2I(φ) = δ(δI(φ)) = δ(Aφ− f, v)

= (δ(Aφ− f), v) + (Aφ− f, δv)∀v ∈ V (3.20)

Since A is linear and hence not a function of φ, we have

δ(Aφ) = A(δφ) = Av (3.21)

and δv = 0 (also note that δf = 0 and δA = 0). Using these in (3.20) weobtain

δ2I(φ) = ((δA)φ+A(δφ)− δf, v) + (Aφ− f, δv) (3.22)


Substituting from (3.21) into (3.22)

δ2I(φ) = (Av, v)

≯ 0, minimum

6= 0, saddle point

≮ 0, maximum

v ∈ Vn (3.23)

Hence, δ2I(φ) is not a unique extremum principle. Thus, the integral form(Aφ − f, v) = 0 with v = δφ is VIC when the differential operator is self-adjoint.

When the differential operator is non-self-adjoint it is linear, hence (3.10)- (3.13) must hold. We state two theorems in the following for non-self-adjoint operators and provide their proofs.

Theorem 3.3. If Aφ − f in Ω is a BVP with some boundary conditionsin which A is a non-self-adjoint operator, then the first variation of thefunctional I(φ) defined by

I(φ) =1

2(Aφ, φ)− (f, φ), φ ∈ Vn

does not yield the integral form (Aφ− f, v) = 0 in which v = δφ.

Proof.

δI =1

2δ(Aφ, φ)− δ(f, φ)

=1

2(A(δφ), φ) +

1

2(Aφ, δφ)− (f, δφ)

=1

2(Av, φ) +

1

2(Aφ, v)− (f, v)

(3.24)

Since A is non-self-adjoint, no further simplification is possible in (3.24).Clearly (3.24), that is δI, is not the same as (Aφ− f, v).

Theorem 3.4. If Aφ − f = 0 in Ω is a BVP in which A is a non-self-adjoint operator, then the integral form resulting from the Galerkin method(i.e. (Aφ− f, v) = 0 with v = δφ) is VIC.



Thenδ2I(φ) = δ(δI(φ)) (3.26)

Following the proof of Theorem 3.2,

δ2I(φ) = (Av, v), v = δφ (3.27)


Hence

δ2I(φ) = (Av, v) =

∫Ω

(Av)v dΩ

> 0, minimum

= 0, saddle point

< 0, maximum

∀v ∈ Vn (3.28)

is not ensured. Hence, δ2I(φ) is not a unique extremum principle. Thus,the integral form (Aφ − f, v) = 0 with v = δφ is VIC when the differentialoperator is non-self-adjoint.

3.3.1.2 Non-linear differential operators

In this case also we begin with (Aφ − f, v) = 0 and substitute the ap-proximation φn to obtain

(A(N0(x) +

n∑i=1

CiNi(x))− f,Nj

)= 0, j = 1, 2, . . . , n (3.29)

Since the differential operator A is non-linear, A is a function of φn. We canwrite (3.29) as

(A(N0(x) +

n∑i=1

CiNi(x)), Nj

)= (f,Nj), j = 1, 2, . . . , n (3.30)

(3.30) represents a system of n non-linear algebraic equations in n unknowns,Ci. Specific forms of these equations depends on the specific form of theoperator A.

Theorem 3.5. If Aφ− f = 0 in Ω is a BVP in which A is a non-linear dif-ferential operator, then the integral form resulting from the Galerkin method,that is (Aφ− f, v) = 0 with v = δφ, is VIC.



Then

δ2I(φ) = δ(δI(φ))

= δ(Aφ− f, v)

= ((δA)φ+A(δφ)− δf, v) + (Aφ− f, δv)

(3.32)

since δf = 0

δ2I(φ) = ((δA)φ+Av, v) (3.33)


Obviously

δ2I(φ)

> 0, minimum

= 0, saddle point

< 0, maximum

∀v ∈ Vn (3.34)

does not hold for this case. Hence, δ2I(φ) is not a unique extremum principle.Thus, the integral form (Aφ − f, v) = 0 with v = δφ is VIC when thedifferential operator is non-linear.

3.3.2 The Petrov–Galerkin and weighted-residual methods

In these methods we substitute φn from (3.1) into (3.4), or (3.6) withv = w, a weight function (in weighted-residual method) or test function (inthe Petrov–Galerkin method), to obtain

(Aφn − f, w) = 0 (3.35)

and choosew = Ψj(x), j = 1, 2, . . . , n (3.36)

where w 6= δφn but w = Ψj(x) = 0 on Γ where φ = φ0. Substituting φnfrom (3.1) and w from (3.36) into (3.35) we obtain

(A(N0(x) +n∑i=1

CiNi(x)),Ψj) = (f,Ψj) = 0, j = 1, 2, . . . , n (3.37)

In this method φn(x) must satisfy all BCs of the BVP, hence the requiredBCs to be considered in determining N0(x) and Ni(x) (i = 1, 2, . . . , n) areall BCs of the BVP.


When the differential operator is linear, that is self-adjoint or non-self-adjoint, (3.37) can be written as(

A( n∑i=1

CiNi(x)),Ψj

)= (f,Ψj)− (AN0,Ψj), j = 1, 2, . . . , n (3.38)

or

n∑i=1

(ANi(x),Ψj

)Ci = (f,Ψj)− (AN0,Ψj), j = 1, 2, . . . , n (3.39)

In matrix form we can write (3.39)

[K]C = F (3.40)


Kij of [K] and Fi of F are given by

Kij = (ANj(x),Ψi), i, j = 1, 2, . . . , n

Fi = (f,Ψi)− (AN0,Ψi), i = 1, 2, . . . , n(3.41)

Ψi(x) are referred to as weight functions and hence the name weighted resid-ual due to the fact that (3.35) is a statement of the integral of weighted-residuals with weight functions Ψi. In the Petrov–Galerkin method, Ψi arecalled test functions or weight functions.

Remarks.

(1) Obviously, Kij 6= Kji, that is the coefficient matrix [K], is not sym-metric and, hence, [K] cannot be ensured to be positive-definite for alladmissible choices of Nis and Ψjs.

(2) Whether the integral form (3.39) is variationally consistent or not needsto be established.

Theorem 3.6. If Aφ− f = 0 in Ω is a boundary value problem in which Ais linear, then the first variation of the functional

I(φ) =1

2(Aφ, φ)− (f, φ) (3.42)

set to zero does not correspond to the integral form (Aφ− f, w) = 0 in PGMor WRM.

Proof. Taking the first variation of I(φ) in (3.42) and setting it to zero weobtain

δI(φ) =1

2

(A(δφ), φ

)+

1

2

(Aφ, δφ

)− (f, δφ) (3.43)

Let v = δφ 6= w, then

δI(φ) =1

2(Av, φ) +

1

2(Aφ, v)− (f, δφ) 6= (Aφ− f, w) (3.44)

Theorem 3.7. If Aφ−f = 0 in Ω is a BVP in which A is a linear differentialoperator, then the integral form resulting from the Petrov–Galerkin methodand weighted-residual method, (Aφ− f, w) = 0 with w 6= δφ is VIC.

Proof. Let there exist a functional I(φ) such that δI(φ) = (Aφ− f, w) = 0.Then

δ2I(φ) = δ(δI(φ)) = δ(Aφ− f, w)

= (A(δφ)− δf, w) + (Aφ− f, δw)(3.45)


orδ2I(φ) = (Av,w) because δf = 0 and δw = 0 (3.46)

with v = δφ and w is a weight function. Hence,

δ2I(φ) = (Av,w) =

∫Ω

(Av)w dΩ

> 0, minimum

= 0, saddle point

< 0, maximum

∀v ∈ Vn, ∀w ∈ V

(3.47)is not possible. Hence, δ2I(φ) is not a unique extremum principle. Thus,the integral form (Aφ− f, w) = 0 with w 6= δφ is VIC when the differentialoperator A is linear.


In this case we begin with (Aφ− f, w) = 0 and substitute Ψj for w andφn for φ

(A(N0(x) +n∑i=1

CiNi(x)),Ψj) = (f,Ψj) = 0, j = 1, 2, . . . , n (3.48)

Since the differential operator A is non-linear, A is a function of φn. (3.48)represents a system of non-linear algebraic equations in unknown Ci. Thespecific form of these equations depends upon the specific form of A.

Theorem 3.8. If Aφ − f = 0 in Ω is a BVP in which A is a non-lineardifferential operator, then the integral form (Aφ − f, w) = 0 resulting fromthe Petrov–Galerkin method or weighted-residual method is VIC.


δI(φ) = (Aφ− f, w) (3.49)

where w 6= δφ. Then

δ2I(φ) = δ(δI(φ))

= δ(Aφ− f, w)

=((δA)φ+A(δφ)− δf, w

)+ (Aφ− f, δw)

(3.50)

orδ2I(φ) =

((δA)φ+Av − δf, w

)+ (Aφ− f, δw) (3.51)

since δw = 0 and δf = 0. We have

δ2I(φ) =((δA)v +Av,w

)> 0, minimum

= 0, saddle point

< 0, maximum

∀v ∈ Vn, ∀w ∈ V (3.52)


is not possible. Hence, δ2I(φ) is not a unique extremum principle. Thus,the integral form (Aφ− f, w) = 0 with w 6= δφ is VIC when the differentialoperator is non-linear.

3.3.3 The Galerkin method with weak form

Let Aφ− f = 0 in Ω be a boundary value problem, then in this methodwe begin with (as in GM) integral statement based on fundamental lemma

(Aφ− f, v) = 0 (3.53)

in which v = δφ, and hence, v = 0 on Γ where φ = φ0 (specified). Letthe differential operator contain highest order derivatives of order 2m. Inthis method, we transfer some differentiation form φ to v using integrationby parts. The reasons for doing so are not clear at this stage, but willbe explained shortly in the following sections. In doing so we observe thefollowing.

(1) We have lowered some differentiation on φ but increased by the same onthe test function v.

(2) When a differential operator contains even order derivatives, it is possibleto transfer half of the differentiation from φ to v. In such cases, theintegrand contains terms that have the same orders of derivatives of φand v. This feature has special consequences in relating the resultingintegral form to the elements of calculus of variations (i.e. variationalconsistency).

(3) In the process of transferring differentiation from φ to v by using inte-gration by parts, concomitant consisting of boundary terms, boundaryintegrals or surface integrals (concomitant, henceforth may also be re-ferred to as boundary integrals or boundary terms) results.

(4) Using the concomitant, the dependent variables and their derivatives areclassified into two groups.

(a) primary variables: PVs

(b) secondary variables: SVs

The definition of PVs and SVs result in the classification of the boundaryconditions into two groups as well.

(a) essential boundary conditions (EBCs) - those corresponding to thespecified values of PVs.

(b) natural boundary conditions (NBCs) - those corresponding to thespecified values of SVs.


In the following we consider some definitions that facilitate this processof determining PVs, SVs, EBCs, and NBCs.

Definition 3.1 (Primary Variables (PVs)). Defining dependent vari-ables in the same manner in which the test function (and its derivatives)appear in the concomitant defines primary variables (PVs).

Definition 3.2 (Secondary Variables (SVs)). The coefficient of thetest function (and its derivatives) in the concomitant defines secondaryvariables (SVs).

Definition 3.3 (Essential boundary conditions (EBCs)). Speci-fications of primary variables on some boundaries constitutes essentialboundary conditions (EBCs).

Definition 3.4 (Natural boundary conditions (NBCs)). Specifi-cation of secondary variables on some boundaries constitutes naturalboundary conditions.

At this point we can examine the BCs of the BVP to determine whichones are EBCs and which ones are NBCs. We note that determination ofPVs, SVs, EBCs, and NBCs is strictly using concomitant and not usingthe BCs of the BVP. Thus, in other methods of approximation that donot result in concomitant as there is no integration by parts, there is noconcept of PVs, SVs, EBCs, and NBCs. Only the GM/WF has thesedue to integration by parts.

(5) The concomitant is simplified using boundary conditions of the BVPand their variations.

(6) In step (5) some boundary conditions of the BVP (NBCs) are absorbed(or used). This is important to note due to the fact that in the selectionof Ni, these boundary conditions should not be considered.

The resulting expressions are arranged in the form

B(φ, v) = l(v) (3.54)

In (3.54), the functional B(·, ·) contains those terms that contain both φand v and the functional l(·) constains terms that have v only. The integralform (3.54) is referred to as the weak form of the integral form (3.53). Sincethe starting point in this method is (3.53), we refer to this method morespecifically as “the Galerkin method with weak form.” In the following weconsider three different classes of differential operators: self-adjoint, non-self-adjoint and nonlinear for this method of approximation. We first presentsome important theorems related to the weak form of the BVP for the threeclasses of differential operators.


Theorem 3.9. Let Aφ− f = 0 in Ω be a BVP in which A is a self-adjointdifferential operator and out of all possible weak forms let B(φ, v)− l(v) = 0be a weak form in which B(φ, v) is bilinear and symmetric and l(v) is linear.Then there exists a functional

I(φ) =1

2B(φ, φ)− l(φ)

such that δI = 0 yields the weak form B(φ, v) − l(v) = 0 and δ2I yields aunique extremum principle and, hence, the weak form B(φ, v) − l(v) = 0 isvariationally consistent.

Proof. Let B(φ, v) − l(v) = 0 be a weak form of Aφ − f = 0 in Ω in whichB(φ, v) is bilinear and symmetric and l(v) is linear. Consider the functional

I(φ) =1

2B(φ, φ)− l(φ) (existence of functional I(φ)) (3.55)

Then

δI(φ) =1

2δ(B(φ, φ)− δ(l(φ)) = 0

=1

2(B(δφ, φ) +

1

2(B(φ, δφ)− (l(δφ)) = 0

=1

2(B(v, φ) +

1

2(B(φ, v)− (l(v)) = 0

Since B(·, ·) is symmetric, B(v, φ) = B(φ, v), therefore

δI(φ) = B(φ, v)− l(v) = 0 (necessary condition) (3.56)

and

δ2I(φ) = δ(δI(φ)) = δ(B(φ, v)− l(v))

= B(δφ, v) = B(v, v), v = δφ

Since B(·, ·) is symmetric

δ2I(φ) = B(v, v) > 0 ∀v ∈ Vn (unique extremum principle) (3.57)

Hence, the weak form B(φ, v)− l(v) = 0 is VC.

Remarks.

(1) A solution φ obtained from B(φ, v)− l(v) = 0 minimizes I(φ) due to thefact that δ2I(φ) > 0.

(2) In linear solid mechanics applications, I(φ) represents the total potentialenergy, 1

2B(φ, φ) is the strain energy and l(v) is the potential energy of


loads. Thus, setting the first variation of I(φ) to zero is the well-knownprinciple of minimum total potential energy.

(3) The Galerkin method with weak form for self-adjoint operators in whichB(φ, v) is symmetric is known as the Ritz method (actually, one maycall the weak form Galerkin method as the Ritz method).

Theorem 3.10. Let Aφ − f = 0 in Ω be a BVP in which A is a non-self-adjoint differential operator. Let B(φ, v)− l(v) = 0 be all possible weakforms. Then all such integral forms are variationally inconsistent.

Proof. Let there exist a functional I(φ) such that δI(φ) = 0 yield the weakform B(φ, v) − l(v) = 0. Since A is non-self-adjoint, B(φ, v) is bilinear butnot symmetric (i.e. B(φ, v) 6= B(v, φ)), hence

δ2I(φ) = δ(B(φ, v)− l(v))

= B(δφ, v)

= B(v, v)

> 0

= 0

< 0

∀v ∈ Vn

is not possible because B(·, ·) is not symmetric. Therefore, δ2I(φ) is not aunique extremum principle. Thus, the integral form B(φ, v)− l(v) = 0 withv = δφ is VIC when the differential operator is non-self-adjoint.

Theorem 3.11. Let Aφ− f = 0 in Ω be a BVP in which A is a non-lineardifferential operator and let B(φ, v) − l(v) = be all possible weak forms ofAφ−f = 0 in Ω. Then all such integral forms or weak forms are variationallyinconsistent.

Proof. Let there exist a functional I(φ) such that δI(φ) yields the integralform B(φ, v) − l(v) = 0. Since the differential operator A is non-linear,B(φ, v) is linear in v but not linear in φ and l(v) is linear in v. Therefore,the second variation of I

δ2I(φ) = δ(B(φ, v)− l(v))

= δ(B(φ, v))(3.58)

is a function of φ due to the fact that B(φ, v) is a non-linear function of φ.Thus, δ2I(φ) does not represent a unique extremum principle and, hence,the integral form or weak form B(φ, v)− l(v) = 0 is VIC.


3.3.3.1 Linear differential operators

When the differential A operator is linear, i.e. self-adjoint or non-self-adjoint, with approximation φn of φ, the weak form can be written as

B(φn, v)− l(v) = 0 (3.59)

In (3.59), B(φn, v) is bilinear and l(v) is linear. We substitute approximationφn from (3.9) into (3.59) to obtain

B(N0 +

n∑i=1

CiNi, v)− l(v) = 0 (3.60)

In the Galerkin method with weak form we choose

v = δφn = Nj , j = 1, 2, . . . , n (3.61)

Thus, we obtain

B(N0 +n∑i=1

CiNi, Nj)− l(Nj) = 0, j = 1, 2, . . . , n (3.62)

or

B(

n∑i=1

CiNi, Nj) = l(Nj)−B(N0, Nj), j = 1, 2, . . . , n (3.63)

orn∑i=1

B(Ni, Nj)Ci = l(Nj)−B(N0, Nj), j = 1, 2, . . . , n (3.64)

In matrix form[K]C = F (3.65)

In (3.65), Kij of [K] and Fi of F are given by

Kij = B(Nj , Ni), i, j = 1, 2, . . . , n

Fi = l(Ni)−B(N0, Ni), i = 1, 2, . . . , n(3.66)

Here, [K] is an (n× n) matrix and C, F are (n× 1) vectors.

Remarks.

(1) When the differential operator is self-adjoint, the operator contains onlyeven order derivatives of the dependent variables, hence it is possibleto derive a weak form in which B(·, ·) is bilinear and symmetric andl(·) is linear. Based on theorem 3.9, such integral forms are VC. Thus,therein lies the motivation for the integration by parts and the weak


forms. The integral forms resulting from the Galerkin method are al-ways VIC regardless of the nature of the differential operator. Loweringof differentiation on the dependent variable(s) is a consequence of theweak form, but not the motivation for doing so, due to the fact that ifour approximation is to be admissible in the GDEs (containing deriva-tives of orders higher than in the weak form) in the pointwise sense, thenthe approximation used in the weak form must possess global differen-tiability of higher order than the one required by the weak form. Whenthe differential operator is self adjoint, B(φn, v) = B(v, φn) and henceKij = Kji, that is, B(Nj , Ni) = B(Ni, Nj), i, j = 1, 2, . . . , n

(2) When A is non-self-adjoint, that is, linear but not symmetric, thenB(φn, v) 6= B(v, φn) and hence

B(Nj , Ni) 6= B(Ni, Nj)

thus Kij 6= Kji (i.e. [K] is not symmetric).


When the differential operator A is non-linear it is neither linear norsymmetric. With approximation φn of φ, the weak form can be written as

B(φn, v)− l(v) = 0 (3.67)

B(φn, v) is linear in v but not linear in φn and, hence, is obviously notsymmetric. However, l(v) is linear in v. Substituting for φn from (3.9) into(3.67) and using

v = δφn = Nj , j = 1, 2, . . . , n

we obtain

B(N0 +

n∑i=1

CiNi, Nj

)− l(Nj) = 0, j = 1, 2, . . . , n (3.68)

Since B(·, ·) is not linear in φn, further simplification is not possible withoutknowing specific form of B(·, ·) (i.e., specific form of A). (3.68) represents asystem of non-linear algebraic equations in unknowns Cis.

In GM/WF some boundary conditions may be absorbed (used) in sim-plifying the concomitant resulting as a consequence of integration by parts.Hence, in this method φn only needs to satisfy the remining boundary condi-tions. Thus, in this method the required set of BCs to be used in determiningN0(x) and Ni(x) consist of all BCs minus those that are absorbed in derivingthe weak form.


3.3.4 The least-squares method

In this method, we begin with the construction of the functional I(φ)corresponding to the differential operator.

Let Aφ−f = 0 in Ω be an abstract BVP and let φn be an approximationof φ in Ω. Then

Aφn − f = E in Ω (3.69)

defines the residual function. We proceed as follows.

1. Existence of functional I(φn), residual functional: we construct I(φ) usingE.

I(φn) = (E,E) = (Aφn − f,Aφn − f) (3.70)

2. Necessary condition: If I(φ) is differentiable in φn then δI(φn) = 0 isnecessary condition for an extremum of I(φ).

δI(φn) = (δE,E) + (E, δE) = 0

orδI(φn) = 2(E, δE) = 2g = 0, g = 0 (3.71)

3. Sufficient condition or extremum principle

δ2I(φn) = 2(δE, δE) + 2(E, δ2E)

> 0

= 0

< 0

∀v ∈ Vn must hold (3.72)

for a unique extremum principle.

The necessary condition δI(φn) = 0 is used to find the constants in φn.Variational consistency of this process depends upon whether δ2I yields aunique extremum principle. Specific details of δE, δI, and δ2I as well asVC or VIC are considered in the following for the three classes of differentialoperators. We present theorems and their proofs related to VC and VICof the integral forms in the following. This method is referred to as least-squares method (LSM) or least-squares process (LSP).

Theorem 3.12. Let Aφ− f = 0 in Ω be a boundary value problem in whichA is a self-adjoint operator, let φn be an approximation of φ in Ω and letAφn− f = E be the residual function in Ω. Then the integral form resultingfrom the first variation of the residual functional I(φn) = (E,E) set to zerois VC.

Proof. Since A is linear, E is a linear function of φn and δE is not a functionof φn and hence δ2E = 0

I(φn) = (E,E) = (Aφn − f,Aφn − f) (existence of functional I(φn))


If I(φn) is differentiable in φn, then

δI(φn) = (δE,E) + (E, δE) = 0 = 2(E, δE) = 2g = 0

or

g = 0 with δE = Av is necessary condition

or

g = (Aφn − f,Av) = 0

or

(Aφn, Av) = (f,Av) (3.73)

or

B(φn, v) = l(v)

and

δ2I(φn) = δ(δI(φn)) = 2(δE, δE)

or

δ2I(φn) = 2(δE, δE) = 2(Av,Av) = 2B(v, v) > 0 ∀v ∈ Vn. (3.74)

Hence, δ2I represents a unique extremum principle. Thus, the integral formresulting from δI(φn) = 0, i.e. B(φn, v) = l(v) in (3.73) is VC.

Remarks.

(1) Since δ2I(φn) > 0, a φn from δI(φn) = 0 minimizes I(φn), and theminimum of I(φn) is zero.

(2) When I(φn) → 0, E = Aφn − f → 0 ∀x ∈ Ω, that is, Aφn − f = 0 issatisfied in the pointwise sense in Ω if the integrals are Riemann.

(3) Since the differential operator is linear, B(φn, v) in the integral form(3.73) is bilinear and is also symmetric,

B(φn, v) = B(v, φn) (3.75)

(4) The algebraic system resulting from δI(φn) = 0 yields a symmetric andpositive-definite coefficient matrix.

(5) The property of the symmetry of the operator is not required in theproof or the construction of any of the variations. Only the linearity ofthe operator A is needed.

Theorem 3.13. Let Aφ− f = 0 in Ω be a boundary value problem in whichA is a non-self-adjoint operator, let φn be an approximation of φ in Ω and letAφn− f = E be the residual function in Ω. Then the integral form resulting


from the first variation of the residual functional I(φn) = (E,E) set to zerois VC.

Proof. Since A is linear, E is a linear function of φn and δE is not a functionof φn and hence δ2E = 0.

I(φn) = (E,E) = (Aφn − f,Aφn − f) (existence of functional I(φn))


δI(φn) = (δE,E) + (E, δE) = 2(E, δE) = 2g = 0

org = 0 with δE = Av is necessary condition

or(Aφn, Av) = (f,Av) (3.76)

B(φn, v) = l(v)

and

δ2I(φn) = δ(δI(φn)) = 2(δE, δE) = 2(Av,Av) = 2B(v, v) > 0 ∀v ∈ Vn

Hence, δ2I represents a unique extremum principle. Thus, the integral formresulting from δI(φn) = 0, that is, B(φn, v) = l(v) in (3.76) is VC.

The remarks following theorem 3.12 are precisely applicable here also.

Theorem 3.14. Let Aφ − f = 0 in Ω be a boundary value problem inwhich A is a non-linear operator, let φn be an approximation of φ in Ω andlet Aφn − f = E be the residual function in Ω. Then the integral formresulting from the first variation of the residual functional I(φn) = (E,E)set to zero is VC provided δ2I ∼= (δE, δE) and the system of non-linearalgebraic equations resulting from δI(φn) = 0 are solved iteratively usingNewton-Raphson or Newton’s linear method.

Proof. Since A is non-linear, E is a non-linear function of φn and so δE isalso a function of φn.

I(φn) = (E,E) = (Aφn − f,Aφn − f) (existence of functional I(φn))(3.77)


δI(φn) = (δE,E) + (E, δE) = 2(E, δE) = 2g(φn) = 0

org(φn) = 0 is necessary condition


Since

δE = δ(Aφn − f)

= δA(φn) +A(δφn)

= δA(φn) +Av

(3.78)

we haveg(φn) = (Aφn − f, δA(φn) +Av) = 0 (3.79)

or(Aφn, δA(φn) +Av) = (f, δA(φn) +Av) (3.80)

orB(φn, v) = l(v) (3.81)

Also

δ2I(φn) = 2(δE, δE) + 2(E, δ2E)

> 0

= 0

< 0

∀v ∈ Vn (3.82)

is not possible. Hence, we do not have a unique extremum principle. Atthis stage, the least-squares process is VIC. We rectify this situation in thefollowing.

First we note that based on the necessary condition g(φn) = 0 musthold. We must find a φn iteratively that satisfies g(φn) = 0 where g(φn) is anon-linear function of φn. Let φ0

n be an initial (or assumed) solution. Then

g(φ0n) 6= 0 (3.83)

Let ∆φn be a change in φ0n such that

g(φ0n + ∆φn) = 0 (3.84)

Expanding g(φ0n + ∆φn) in Taylor’s series about φ0

n and retaining only upto linear terms in ∆φn (the Newton–Raphson or Newton’s linear method)

g(φ0n + ∆φn) ∼= g(φ0

n) +∂g(φn)

∂φn

∣∣∣∣φ0n

∆φn = 0 (3.85)

Therefore

∆φn = −

[∂g(φn)

∂φn

∣∣∣∣φ0n

]−1

g(φ0n)

But ∂g(φn)∂φn

= 12δ(δI(φn)) = 1

2δ2I(φn). Hence

∆φn = −1

2[δ2I(φn)]−1

φ0ng(φ0

n) (3.86)


Thus, in order for the coefficient matrix [δ2I(φn)] to be positive-definite, wemust approximate δ2I(φn) in (3.82) by

δ2I(φn) ∼= 2(δE, δE) > 0 (3.87)

where δE is given by (3.78). (3.87) represents a unique extremum principle.The improved value of φn is given by

φn = φ0n + α∗∆φn (3.88)

We choose α∗ such that I(φn) ≤ I(φ0n). (3.88) is referred to as line search.

Thus, the least-squares process based on (3.77), (3.79), (3.86), (3.87) and(3.88) is VC.

Remarks.

(1) Summary of least-squares process for non-linear differential operators ispresented in the following. Let E = Aφn − f in Ω.

I(φn) = (E,E) (existence)

δI(φn) = 2(E, δE) = 2g(φn) = 0 or g(φn) = 0 (necessary condition)

δ2I(φn) ∼= 2(δE, δE) (sufficient condition or extremum principle)

∆φn = −1

2[δ2I]−1

φ0ng(φ0

n)

φn = φ0n + α∗∆φn; α∗ such that I(φn) ≤ I(φ0

n)

Convergence check: Is |gi(φn)| (i = 1, 2, . . . ≤ ∆) less than a smallnumber (error tolerance)? If not converged then repeat (a)–(e).

(2) The scalar α∗ is selected to minimize the error functional I. In allnumerical studies, we generally consider 0 < α∗ ≤ 2. In this range, α∗

is incremented in steps of 0.1 and for each α∗ the error functional I iscalculated. The lowest value of I and a value on either side of I areused to construct a quadratic interpolation from which a value of α∗

is calculated that ensures minimum of I for 0 < α∗ ≤ 2. This valueof α∗ is used in (3.88) to obtain an updated solution φn for the nextiteration. This procedure described above is termed Newton’s methodwith line search. In many cases (but not always) a starting solution φ0

n

with α∗ = 1 may be employed for the first iteration followed by the linesearch for the subsequent iterations. When the updated solution is inthe very close neighborhood of the true solutions, α∗ becomes very closeto 1 and eventually reaches 1 at convergence. Iterations are performeduntil the absolute value of each component of g is below a prescribedtolerance (threshold value for numerically computed zero).


In some cases during the iterative solution procedure it is quite possibleto be at a point on the hypersurface of δI = 0 from where a direction inwhich I is decreasing may not be possible to find. In such a situation,determination of α∗ is not possible (α∗ = 0 is obviously not admissible).In such instances, we let α∗ = 0.005, a very small value. This wouldobviously result in a φn which may cause an increase I(φn) comparedto I(φ0

n). This may be continued for a few iterations with the hopethat from a slightly new location on the hypersurface of δI = 0 it maybe possible to find a direction of decreasing I. For well constructeddiscretizations with adequate p-levels (in finite element processes) thehypersurface of δI = 0 is generally smooth if the theoretical solution isanalytic and, hence, in such cases the situation to set α∗ = 0.005 maynot even arise. Another significant point to note here is that a failurein determining α∗ is an indication that the direction given by ∆φn maynot be correct. The root cause of this of course is the coefficient matrixin (3.86) given by [δ2I]φ0n .

(3) Approximating δ2I amounts to change in slope of the tangent plane tothe hyper surface of δI = 0, but its benefits are enormous. This can beverified by considering a simple case of a non-linear equation f(x) = 0.We remark that [δ2I]φ0n in (3.86) must be calculated using (3.87) toensure that it is positive-definite. Thus, the least-squares process stepsoutlined in remark (1) ensure that the process is VC.


When the differential operator is self-adjoint or non-self-adjoint with ap-proximation φn of φ, the integral form resulting from the least-squares pro-cess can be derived using

E = Aφn − f, I = (E,E), δE = Aδφn = Av

andδI = 2(E, δE) = 2(Aφn − f,Av) = 0 or (Aφn − f, v) = 0

or(Aφn, Av) = (f,Av) Integral form (3.89)

Using

φn = N0(x) +n∑i=1

CiNi(x) and v = δφn = Nj , j = 1, 2, . . . , n

we can write (3.89) as(A(N0(x) +

n∑i=1

CiNi(x)), ANj

)= (f,ANj), j = 1, 2, . . . , n


Since A is linear

n∑i=1

(ANi, ANj)Ci = (fi, ANj)− (AN0(x), ANj), j = 1, 2, . . . , n

or

n∑i=1

B(Ni, Nj)Ci = (fi, ANj)− (AN0(x), ANj), j = 1, 2, . . . , n (3.90)

or[K]C = F (3.91)

where

Kij = B(Nj , Ni) = (ANj , ANi), i, j = 1, 2, . . . , n

Fi = (F,ANi)− (AN0, ANi), i = 1, 2, . . . , n(3.92)

Remarks.

(1) B(·, ·) is bilinear and symmetric, hence, Kij = Kji (i.e. [K] is symmet-ric).

(2) (3.91) is used to calculate the coefficients Ci (i = 1, 2, . . . , n).


Here, also

E = Aφn − f, I = (E,E)

δE = δA(φn) +Aδφn = δA(φn) +Av (3.93)

andδI = 2(E, δE) = 2g(φn) = 0 or g(φn) = 0

org(φn) = (Aφn − f, δA(φn) +Av) = 0 (3.94)

alsoδ2I(φn) ∼= 2(δE, δE)

orδ2I ∼= (δA(φn) +Av, δA(φn) +Av) > 0 ∀φn, v ∈ Vn (3.95)

Since

φn = N0(x) +n∑i=1

CiNi(x)

v = δφn = Nj(x), j = 1, 2, . . . , n

(3.96)


g(φn) = 0 can be written as

g(φn) =(A

(N0 +

n∑i=1

CiNi(x)

)− f, δA(φn) +ANj

)= 0, j = 1, 2, . . . , n

(3.97)and

δ2I = 2(δE, δE) = 2(δA(φn) +ANj , δA(φn) +ANj), j = 1, 2, . . . , n (3.98)

∆φn = −1

2[ δ2I ]−1

φ0ng(φ0

n) (3.99)

φn = φ0n + α∗∆φn (3.100)

where α∗ is determined such that I(φn) ≤ I(φ0n). Convergence of the

Newton–Raphson method is determined based on

|gi(φn)| ≤ ε; i = 1, 2, . . . , n, ε is a preset tolerance (¡¡ 1)

In LSP based on residual functional φn(x) must satisfy all BCs of the BVP,hence the required boundary conditions to be considered in determiningN0(x) and Ni(x) (i = 1, 2, . . . , n) are all BCs of the BVP.

Remarks.

In the LSP based on the residual functional for linear as well as nonlineardifferential operators the necessary condition is given by

g = (E, δE) =

∫Ωx

(Aφn − f) δE dΩ = 0 (3.101)

(a) If we define δE as the test function in (3.101), then (3.101) representsintegral form in PGM. In other words with δE as test function in PGM,the integral form in the PGM becomes same as the integral form in theLSP.

(b) If we define δE as the weight function in (3.101), then (3.101) representsintegral form in WRM. In other words with δE as test function inWRM, the integral form in the WRM becomes same as the integralform in the LSP.

3.3.5 Collocation method

There are other methods of approximation besides those discussed here,such as finite difference methods, finite volume or subdomain method, col-location method, and so on. In the following we discuss only the collocationmethod.

3.4. APPROXIMATION SPACES FOR VARIOUS METHODS OF APPROXIMATION 95

Let Aφ− f = 0 in Ω be a BVP in which the differential operator A canbe self-adjoint, non-self-adjoint or non-linear and let

Aφn − f = E(x, y, z) ∀x, y, z ∈ Ω (3.102)

be the residual function and

φn = N0(x, y, z) +

n∑i=1

CiNi(x, y, z) (3.103)

In this method we choose xj , yj , zj (j = 1, 2, . . . , n) locations in Ω calledcollocation points such that

E(xj , yj , zj) = A(N0(xj , yj , zj)

+

n∑i=1

CiNi(xj , yj , zj))− f = 0, j = 1, 2, . . . , n (3.104)

(3.104) provides n algebraic equations from which Cis are determined.

Remarks.

(1) When the differential operator is linear, (3.104) yields a system of linearsimultaneous algebraic equations.

(2) When the differential operator is non-linear, (3.104) yields a system ofnon-linear algebraic equations.

(3) The coefficient matrix in the algebraic system (3.104) is generally non-symmetric.

(4) The choice of the points (xj , yj , zj) is very critical. The choice controlsaccuracy as well as conditioning of the coefficient matrix in (3.104).

3.4 Approximation Spaces for Various Methods ofApproximation

In this section we briefly discuss the choice of approximation spaces thatcontrol the global differentiability of the approximations. First, we note thatregardless of the choice of method of approximation, if the approximationsare to be admissible in the GDEs in the pointwise sense (dictated by physics)then the global differentiability of the approximation is dictated by the high-est order of the derivatives of the dependent variable(s) in the GDEs. Thus,if the differential operator A in Aφ−f = 0 has highest order derivatives of φof orders 2m (the order of the operator A), then the approximation φn mustbe at least of class C2m(Ω). Thus, if Vn is the approximation space we haveφn ∈ Vn ⊂ H2m+1(Ω). Where 2m+1 is the order of the approximation space


Vn. Secondly, if one wishes to incorporate the higher order global differentia-bility features of the theoretical solution of the BVP in the approximation ofthe solution, then there is obviously need for approximation spaces of ordershigher than 2m. In the following, we discuss minimally conforming spaces.These are the lowest order admissible spaces for a differential operator A oforder 2m.

(a) In the Galerkin method, the Petrov–Galerkin method, and the methodof weighted-residuals, there is no integration by parts and, hence, theintegral forms contain derivatives of up to orders 2m. Thus, for theintegrals to be Riemann, φn ∈ Vn ⊂ Hk(Ω); k ≥ 2m+1, where k = 2m+1is minimally conforming space must hold. On the other hand, if wechoose the integrals in the Lebesgue sense then φn ∈ Vn ⊂ H2m(Ω)is admissible in the integral form. However, such a choice of φn is notadmissible in GDEs in the pointwise sense due to the fact that derivativesof orders 2m of the dependent variables may exhibit discontinuity at afinite number of points in Ω.

(b) In the Galerkin method with weak form, if m∗ < 2m is the highestorders of the derivatives of the dependent variables in the weak formthan φn ∈ Vn ⊂ Hm∗+1(Ω) is admissible in the weak form and theintegrals in the weak form are in the Riemann sense. On the otherhand, if we choose φn ∈ Vn ⊂ Hm∗(Ω), then the integrals in the weakform are in the Lebesgue sense. Such approximations are only admissiblein the GDEs if m∗ = 2m, in which case the admissibility of φn in theGDEs is not strictly in the pointwise sense.

(c) In the least-squares processes, the differentiation is not lowered on thedependent variables, hence φn ∈ Vn ⊂ Hk(Ω), k ≥ 2m+1, with k = 2m+1 being minimally conforming maintains all integrals in the Riemannsense as well as permits admissibility of φn in Aφ−f = 0 in the pointwisesense. When φn ∈ Vn ⊂ H2m(Ω), the integrals in the least-squaresprocess are in the Lebesgue sense and the admissibility of φn in theGDEs is not in the strict pointwise sense.

Some general remarks on approximation methods based on integral formsare in order:

(1) The Galerkin method, the Petrov–Galerkin method and the weighted-residual methods always yield integral forms that are variationally in-consistent.

(2) The Galerkin method with weak form yields VC integral form for self-adjoint operators when the bilinear functional B(·, ·) is symmetric.

(3) The Galerkin method with weak form yields VIC integral forms for non-self-adjoint and non-linear operators.

3.4. INTEGRAL FORMULATIONS OF BVPS 97

(4) Least-squares processes yield integral forms that are VC when the dif-ferential operators are self-adjoint or non-self-adjoint.

(5) Least-squares processes also yield VC integral forms for non-linear dif-ferential operators if

δ2I ∼= 2(δE, δE)

and if the Newton–Raphson method is used for solving the system ofnon-linear algebraic equations resulting from

g(φn) = 0

(6) The minimally conforming order of the approximation space is controlledby the highest orders of the derivatives of the dependent variables in theGDEs if one wishes the approximations φn to be admissible in the GDEsin the strict pointwise sense. The approximation spaces of orders higherthan minimally conforming are beneficial as they allow us to incorporatethe higher order global differentiability features of the theoretical solu-tions in the approximations. This aspect is applicable to all methods ofapproximation.

(7) The Petrov–Galerkin method and least-squares processes:

We remark that in the Petrov–Galerkin method, the integral form isconstructed using (Aφ − f, w) = 0 in which w 6= δφ. In this method ifwe choose w = δ(Aφ) i.e. w = Av when A is linear and w = δ(Aφ) +Av when A is nonlinear, then this method is identical to least-squaresprocesses based on residual functional I = (E,E) where E = Aφn − f ,φn being approximation of φ. With this choice of w the integral form(Aφn − f, w) = 0 becomes variationally consistent (following the detailsfor least-squares process) for all three classes of differential operators.Recently published works [1–4] on unconditionally stable PGM are roundabout means to approach this choice of w considered here and hencethe resulting integral forms in these methods can only approach thoseresulting from LSPs but are not the same. The computational processresulting from such integral forms [1–4] are not the same as those fromLSPs.

3.5 Integral Formulations of BVPs usingthe Classical Methods of Approximations

In this section we develop integral formulations of specific BVPs contain-ing self-adjoint, non-self-adjoint, and non-linear differential operators usingvarious classical methods of approximations. We reaffirm VC and VIC of theresulting integral forms. Details of the choices of N0 and Ni (i = 1, 2, . . . , n)


will also be considered for the specific BVPs and the methods of approxima-tion.

3.5.1 Self-adjoint differential operators

Example 3.1. Consider the 1D generalized diffusion equation

− d

dx

(adφ

dx

)− cφ+ x2 = 0 ∀x ∈ Ω = (0, 1) ⊂ R1 (3.105)

with

φ(0) = φ0 and(adφ

dx

)∣∣∣∣x=1

= q1 (3.106)

where a and c are constants. The differential operator A and f are given by

A = − d

dx

(ad

dx

)− c (3.107)

and

f = −x2 (3.108)

Hence, we can write (3.105) as

Aφ− f = 0

It can be shown that if the differential operator A is linear and A∗ = A, thenA is symmetric provided q0 and q1 are both zero, as these would make theconcomitant zero. In the following we consider various methods of approxi-mation.

The Galerkin method

The integral form using test function v is given by (fundamental lemma)∫Ω

(Aφ− f)v dΩ = (Aφ− f, v) = 0 (3.109)

in which v = δφ and, hence, satisfies the homogeneous form of the boundaryconditions. Substituting for A in (3.109)∫

Ω

(− d

dx

(adφ

dx

)− cφ+ x2

)v dΩ = 0 (3.110)

or

B(φ, v) = l(v) (3.111)


where

B(φ, v) =

∫Ω

(− d

dx

(adφ

dx

)− cφ

)v dΩ (3.112)

l(v) =

∫Ω

−x2v dΩ (3.113)

We note that (3.111) is integral form resulting from the Galerkin method.In this case

(a) B(φ, v) is bilinear but not symmetric, B(φ, v) 6= B(v, φ)

(b) l(v) is linear

Let there exist a functional I(φ) such that δI(φ) = 0 yields the integralform (3.110). Then in order for the integral form (3.110) to be VC, δ2I (i.e.δ(B(φ, v)− l(v))) must yield a unique extremum principle.

δ(B(φ, v)− l(v)) = B(v, v) (3.114)

where

B(v, v) =

∫Ω

(− d

dx

(adv

dx

)− cv

)v dΩ (3.115)

Obviously

B(v, v)

> 0

= 0

< 0

∀v ∈ Vn (3.116)

does not hold. Hence, δ2I(φ) (i.e. δ(B(φ, v)− l(v))) does not yield a uniqueextremum principle. Thus, the Galerkin method for this BVP yields a VICintegral form.

The Petrov–Galerkin method

In this method also, we begin with the integral form based on funda-mental lemma. Let w be a test function or weight function, then we canwrite ∫

Ω

(Aφ− f)w dΩ = (Aφ− f, w) = 0, w 6= δφ (3.117)

or ∫Ω

(− d

dx

(adφ

dx

)− cφ+ x2

)w dΩ = 0 (3.118)


orB(φ,w) = l(w) (3.119)

where

B(φ,w) =

∫Ω

(− d

dx

(adφ

dx

)− cφ

)w dΩ (3.120)

l(w) =

∫Ω

−x2w dΩ (3.121)

and w is a weight function that satisfies the homogeneous form of the bound-ary conditions. Again, we note that here also (3.119) is the integral formresulting from the Petrov–Galerkin method in which:

(a) B(φ,w) is bilinear but not symmetric, B(φ,w) 6= B(w, φ)

(b) l(w) is linear

Let there exist a functional I(φ) such that δI(φ) = 0 yields the integral form(3.118). Then we must show that δ2I yields a unique extremum principle inorder for the integral form (3.118) to be VC:

δ(B(φ,w)− l(w)) = B(v, w) (3.122)

where

B(v, w) =

∫Ω

(− d

dx

(adv

dx

)− cv

)w dΩ (3.123)

Obviously

B(v, w)

> 0

= 0

< 0

∀v ∈ Vn, forallw ∈ V (3.124)

does not hold. Hence, the integral form in (3.118) is VIC. Thus, the Petrov–Galerkin method for this BVP yields a VIC integral form.

The Galerkin method with weak form

Based on fundamental lemma (same as in the case of the Galerkin method),using test function v we can write∫

Ω

(Aφ− f)v dΩ = (Aφ− f, v) = 0 (3.125)

or ∫Ω

(− d

dx

(adφ

dx

)− cφ+ x2

)v dΩ = 0 (3.126)


where v = δφ is a test function.

Step 1: The differential operator has derivatives of even order. We transferone order of differentiation from φ to v using integration by parts∫

Ω

(dvdxadφ

dx− cφv

)dΩ−

[v(adφ

dx

)]1

0

= −1∫

0

x2v dΩ (3.127)

In this case the concomitant 〈Aφ, v〉Γ is

〈Aφ, v〉Γ = −[v(adφ

dx

)]1

0

We note that each term in the integrand on left side of (3.127) contains bothφ and v with the same orders of derivatives. The concomitant is specificallyexpressed as test function v multiplied by the remaining quantities (coef-ficients). This specific form of the term resulting from the integration byparts is essential in what follows.

Step 2: We define PVs, SVs, EBCs and NBCs using concomitant.Using the concomitant we identify the PVs and SVs, as follows.

(a) The dependent variables in the same form in which the test function andits derivatives appear in the concomitant defines primary variables (PVs).Thus, in this specific case φ is the only primary variable.

(b) The coefficient of the test function (and its derivatives) in the concomi-tant defines secondary variables (SVs). In (3.127), adφdx is a secondary vari-able.

(c) Specification of PVs on some boundaries constitutes essential bound-ary conditions, thus φ = φ specified on some boundary Γ1 is an essentialboundary condition.

(d) Specification of the secondary variables on some boundaries constitutesnatural boundary conditions, thus adφdx = q, specified on some boundary Γ2

is a natural boundary condition.

By examining the BCs of the BVP and using the definitions given here,we can classify the BCs of the BVP into EBC and NBC categories.

φ : PV (3.128)

adφ

dx: SV (3.129)

φ(0) = φ0 : EBC, φ0 is specified (3.130)[adφ

dx

]x=1

= q1 : NBC, q1 is specified (3.131)


We note that definitions of PV(s), SV(s), EBC(s) and NBC(s) resultfrom the concomitant and not the definition of BCs in the BVP. A BVPmay or may not have some EBCs and/or NBCs. In this example both EBCand NBC are realized in the definition of the BCs of the BVP.

Step 3: Simplification of the concomitant:

Since

φ(0) = φ0 ⇒ v(0) = δφ(0) = δφ0 = 0 (3.132)(adφ

dx

)∣∣∣∣x=1

= q1 ⇒(adv

dx

)∣∣∣∣x=1

= 0 (3.133)

from (3.127) we can write

∫Ω

(dvdxadφ

dx− cφv

)dΩ−

(vadφ

dx

)∣∣∣∣x=1

+(vadφ

dx

)∣∣∣∣x=0

=

1∫0

−x2v dΩ (3.134)

Substituting from (3.132) and (3.133) into (3.134),

∫Ω

(dvdxadφ

dx− cφv

)dΩ = v(1)q1 −

1∫0

x2v dΩ (3.135)

orB(φ, v) = l(v) (3.136)

where

B(φ, v) =

∫Ω

(dvdxadφ

dx− c φv

)dΩ (3.137)

l(v) = v(1)q1 −∫Ω

x2v dΩ (3.138)

Integral form (3.135) is the weak form associated with the integral form(3.125) with the following attributes:

(a) B(φ, v) is bilinear due to linearity of A(b) B(φ, v) is symmetric, B(φ, v) = B(v, φ), due to the fact that A∗ = Aand due to integration by parts(c) l(v) is linear in v

where φ ∈ Vn ⊂ H2(Ω) and v ∈ Vn ⊂ H2(Ω) are minimally conformingbased on the weak form if the integral in (3.137) is to be in the Riemannsense. The minimally conforming space for the BVP is obviously H3(Ω).


Step 4: Determining VC or VIC of the weak form (3.136):

Since B(·, ·) is bilinear and symmetric and l(·) is linear, the quadratic func-tional (QF) exists and can be constructed using (see Reddy [5])

I(φ) =1

2B(φ, φ)− l(φ) (3.139)

or

I(φ) =1

2

∫Ω

(a(dφdx

)2 − c(φ)2)dΩ− φ(1)q1 +

∫Ω

x2φdΩ (3.140)

(a) This establishes existence of the functional I(φ).(b) Necessary condition: If I(φ) is differentiable in φ, then δI(φ) = 0 is the

necessary condition for an extremum of I(φ) and is given by

δI(φ) =1

2B(v, φ) +

1

2B(φ, v)− l(v) = 0 (3.141)

Since B(φ, v) = B(v, φ), we have

δI(φ) = B(φ, v)− l(v) = 0 (3.142)

which is the weak form (3.136).(c) Extremum principle or sufficient condition

δ2I(φ) = δ(B(φ, v)− l(v)) (3.143)

or

δ2I(φ) = B(v, v) =

∫Ω

(a(dvdx

)2− cv2

)dΩ (3.144)

and for constant c > 0 and a > 0

B(v, v) > 0⇒∫

Ω

(dvdx

)2dΩ∫

Ω v2 dΩ

>c

a, a 6= 0 (3.145)

When (3.145) is satisfied ∀v ∈ Vn, we have a unique extremum principleand the weak form is VC.

Remarks.

(1) I(φ), called the quadratic functional, represents total potential energy.Minimizing of I(φ) (due to δ2I(φ) > 0) using δI(φ) = 0, yields the weakform.

(2) Conditions on c and a given by (3.145) is essential for VC of the weakform.


(3) Spaces:

BVP: Aφ− f = 0 in Ω; φ ∈ Vn ⊂ Hk(Ω); k ≥ 3k = 3 is minimally conforming

WF: B(φ, v) = l(v); φ ∈ Vn ⊂ Hk(Ω); v ∈ Vn ⊂ Hk(Ω)k ≥ 2; k = 2 is minimally conforming

QF: I(φ) = 12B(φ, φ)− l(φ); φ ∈ Vn ⊂ Hk(Ω)

k ≥ 2; k = 2 is minimally conforming

When k = 2 we can not go back from (3.136) or (3.139)to (3.125) which requires k = 3.The minimally conforming choice of k is obviously k = 3.

(4) In this example, we also note that the weak form helps us in achievinga variationally consistent integral form.

(5) The natural boundary condition is absorbed (or naturally absorbed) inthe weak form and, hence, the name natural boundary condition.

The least-squares method

Let φn be an approximation of φ in Ω, then

Aφn − f = E is the residual function (3.146)

(i) We define the residual functional I(φn) as (existence of I(φn))

I(φn) = (E,E) =

∫Ω

[− d

dx

(adφndx

)− cφn + x2

]2dΩ (3.147)

(ii) Necessary condition: If I(φn) is differentiable in φn, then δI(φn) = 0is necessary condition.

δI(φn) = 2g(φn) = 2(E, δE)Ω = 0 or g(φn) = 0 (3.148)

The necessary condition (3.148) gives the desired integral form. Wenote that

E = − d

dx

(adφndx

)− cφn + x2 in Ω (3.149)

δE = − d

dx

(adv

dx

)− cv = Av, v = δφn in Ω (3.150)

Hence, (E, δE) = 0 in (3.148) can be written as

(E, δE) = (Aφn − f, δE) = (Aφn − f,Av) = 0 (3.151)


or

(Aφn, Av) = (Av, f) (3.152)

B(φn, v) = l(v) (3.153)

where

B(φn, v) =(− d

dx

(adφndx

)− cφn,−

d

dx

(adv

dx

)− cv

)(3.154)

l(v) =(− d

dx

(adv

dx

)− cv,−x2

)(3.155)

Integral form (3.153) is the desired integral form resulting from δ(φn) =0. B(·, ·) is bilinear and symmetric and l(·) is linear in v. Due tosymmetry of B(·, ·), B(φn, v) = B(v, φn) holds.

(iii) Sufficient condition or extremum principle:

1

2δ2I(φn) = δ(B(φn, v)− l(v)) = (δE, δE) =

∫Ω

(δE)2 dΩ (3.156)

Hence

1

2δ2I(φn) =

∫Ω

(− d

dx

(adv

dx

)− cv

)2dΩ > 0 ∀v ∈ Vn (3.157)

Thus, δ2I(φn) yields a unique extremum principle. Since the secondvariation δ2I(φn) > 0, a φn from δI(φn) = 0, i.e. from B(φn, v) = l(v)minimizes I(φn) in (3.147). We further note that minimum of I(φn) iszero and when I(φn)→ 0, E → 0 in the pointwise sense, i.e. Aφ−f = 0is satisfied everywhere in Ω provided all integrals are Riemann. Theintegral form resulting from δI(φn) = 0 is variationally consistent.

Example 3.2. Consider the boundary value problem, which arises in bend-ing of beams,

d2

dx2

(bd2φ

dx2

)−Q(x) = 0 in Ω = (0, L) ⊂ R1 (3.158)

with

φ(0) = 0;dφ

dx

∣∣∣∣x=0

= 0,

[bd2φ

dx2

]x=L

= ML,

[d

dx

(bd2φ

dx2

)]x=L

= FL (3.159)

where φ is the transverse displacement and b is a constant. In this case, thedifferential operator A and f are given by

A =d2

dx2

(bd2

dx2

)(3.160)


andf = Q(x) (3.161)


Aφ− f = 0 in Ω

It has been shown previously that

(a) A is linear and A∗ = A

(b) A is symmetric when boundary conditions are homogeneous: ML =FL = 0

We consider various methods of approximation for this BVP.

The Galerkin method

The integral form is given by the following based on fundamental lemma∫Ω

(Aφ− f)v dΩ = (Aφ− f, v) = 0 (3.162)

in which v = δφ is the test function and, hence, satisfies the homogeneousform of the boundary conditions on φ. Substituting for A in (3.162)∫

Ω

[ d2

dx2

(bd2φ

dx2

)−Q(x)

]v dΩ = 0 (3.163)

orB(φ, v) = l(v) (3.164)

where

B(φ, v) =

∫Ω

d2

dx2

(bd2φ

dx2

)v dΩ (3.165)

l(v) =

∫Ω

Qv dΩ (3.166)

In this method (3.164) is the desired integral form in which

(a) B(·, ·) is bilinear but not symmetric as B(φ, v) 6= B(v, φ)

(b) l(·) is linear in v

Let there exist a functional I(φ) such that δI(φ) = 0 yields the integralform (3.163). Then in order for the integral form (3.163) to be VC, we


must show that δ2I(φ) = δ(B(φ, v) − l(v)) must yield a unique extremumprinciple.

δ(B(φ, v)− l(v)) = B(v, v) (3.167)

where

B(v, v) =

∫Ω

d2

dx2

(bd2v

dx2

)v dΩ (3.168)

Clearly

δ2I(φ) = B(v, v)

> 0

= 0

< 0

∀v ∈ Vn (3.169)

does not hold. Hence, the integral form (3.163) does not yield a uniqueextremum principle. Thus, the Galerkin method for this BVP yields a VICintegral form.


Here we begin with integral form based on fundamental lemma using was test function or weight function.∫

Ω

(Aφ− f)w dΩ = (Aφ− f, w) = 0 (3.170)

The test function or the weight function is such that it satisfies the homo-geneous form of the boundary conditions on φ and w 6= δφ. From (3.170),we can write

B(φ,w) = l(w) (3.171)

where

B(φ,w) =

∫Ω

d2

dx2

(bd2φ

dx2

)w dΩ (3.172)

l(w) =

∫Ω

Qw dΩ (3.173)

(a) B(φ,w) is bilinear but not symmetric, i.e. B(φ,w) 6= B(w, φ)

(b) l(w) is linear in w

Let there exist a functional I(φ) such that δI(φ) = 0 yields the integral form(3.171). Then we must show that δ2I(φ) = δ(B(φ,w)− l(w)) yields a uniqueextremum principle in order for the integral form (3.171) to be VC. We have

δ(B(φ,w)− l(w)) = B(v, w) =

∫Ω

d2

dx2

(bd2v

dx2

)w dΩ. (3.174)


Clearly

B(v, w)

> 0

= 0

< 0

∀v ∈ Vn, w ∈ V (3.175)



Based on fundamental lemma we can the following using test function v.∫Ω

(Aφ− f)v dΩ = (Aφ− f, v) = 0 (3.176)

or ∫Ω

[ d2

dx2

(bd2φ

dx2

]−Q

)v dΩ = 0 (3.177)

where v = δφ is a test function. Hence, v must satisfy the homogeneousform of the boundary conditions on φ.

Step 1: We transfer two orders of differentiation from φ to v using integrationby parts∫

Ω

d2v

dx2bd2φ

dx2dΩ +

[v( ddx

(bd2φ

dx2

))]L0

−[dv

dx

(bd2φ

dx2

)]1

0

=

∫Ω

Qv dΩ (3.178)

in which the concomitant 〈Aφ, v〉Γ is given by

〈Aφ, v〉Γ =

[v( ddx

(bd2φ

dx2

))]L0

−[dv

dx

(bd2φ

dx2

)]1

0

Step 2: Determination of PVs, SVs, EBCs and NBCs using concomitant:

Defining φ in the same manner in which the test function and its derivativeappear in the concomitant constitutes primary variables. Thus

φ anddφ

dxare PVs (3.179)

Coefficients of the test function and the coefficients of its derivatives aresecondary variables. Thus

d

dx

(bd2φ

dx2

)and b

d2φ

dx2are SVs (3.180)


Based on (3.179), specification of φ and dφdx on some boundaries constitutes

essential boundary conditions. On the other hand, based on (3.180), specifi-

cations of ddx

(bd

2φdx2

)and bd

2φdx2

on some boundaries constitute natural bound-

ary conditions. Now we can classify the BCs (3.159) of the BVP (3.158) inEBC and NBC categories:

φ(0) = 0 anddφ

dx

∣∣∣∣x=0

= 0 are EBCs[bd2φ

dx2

]x=L

= ML and[ ddx

(bd2φ

dx2

)]x=L

= FL, are NBCs

Again, we note that classification of PVs, SVs, EBCs and NBCs purely re-sults from the concomitant. After we determine these from the concomitant,then we can identify which BCs of the BVP fall into these two catagories aswe have done.

Step 3: Simplification of the concomitant resulting from integration by parts:

From the boundary conditions in (3.159), we have

φ(0) = 0⇒ v(0) = 0

dφ

dx

∣∣∣∣x=0

⇒ dv

dx

∣∣∣∣x=0

= 0[bd2φ

dx2

]x=L

= ML ⇒[bd2v

dx2

]x=L

= 0[d

dx

(bd2φ

dx2

)]x=L

= FL ⇒[d

dx

(bd2v

dx2

)]x=L

= 0

(3.181)

Expanding the boundary terms in (3.178)∫Ω

d2v

dx2bd2φ

dx2dΩ +

[v( ddx

(bd2φ

dx2

))]x=L

−[v( ddx

(bd2φ

dx2

))]x=0

−[dv

dx

(bd2φ

dx2

)]x=L

+

[dv

dx

(bd2φ

dx2

)]x=0

=

∫Ω

Qv dΩ (3.182)

Substituting from (3.181) into (3.182) we obtain∫Ω

d2v

dxbd2φ

dx2dΩ + v(L)FL −

dv

dx

∣∣∣∣x=L

ML =

∫Ω

Qv dΩ (3.183)

Let dvdx

∣∣x=L

= Θ(L). Then (3.184) becomes∫Ω

d2v

dxbd2φ

dx2dΩ =

∫Ω

Qv dΩ− v(L)FL + Θ(L)ML (3.184)


orB(φ, v) = l(v) (3.185)

where

B(φ, v) =

∫Ω

d2v

dxbd2φ

dx2dΩ (3.186)

l(v) =

∫Ω

Qv dΩ− v(L)FL + Θ(L)ML (3.187)

Integral form (3.184) is the weak form resulting from the integral form(3.176) of the BVP.

(a) B(φ, v) is bilinear and also symmetric due to the fact that A∗ = A anddue to integration by parts.(b) l(v) is linear in v

In (3.185), φ ∈ Vn ⊂ Hk(Ω), v ∈ Vn ⊂ Hk(Ω); k ≥ 3 in which k = 3 isminimally conforming as opposed to φ ∈ H5(Ω) and v ∈ H1(Ω) in (3.176).Thus, we see that differentiability requirements have been lowered on φ inthe weak form but increased on v in the weak form compared to the integralform based on fundamental lemma. But, approximations of φ of class C2

are not admissible in the BVP. Thus for this problem we must choose k = 5otherwise we can not go back to (3.176) from (3.185) as (3.176) requiresk = 3.


Here we establish whether the weak form (3.185) is VC or VIC.

(i) Since B(·, ·) is bilinear and symmetric and l(·) is linear, the quadraticfunctional I(φ) is given by

I(φ) =1

2B(φ, φ)− l(φ) (3.188)

or

I(φ) =1

2

∫Ω

b(d2φ

dx2

)2dΩ +

∫Ω

QφdΩ + φ(L)FL −dφ

dx

∣∣∣∣x=L

ML (3.189)

(ii) Necessary condition: If I(φ) is differentiable in φ then δI(φ) = 0 isnecessary condition for an extremum of I(φ).

δI(φ) = δ(1

2B(φ, φ)− l(v)) = 0 (3.190)

=1

2B(v, φ) +

1

2B(φ, v)− l(v) = 0 (3.191)


But B(φ, v) = B(v, φ), hence

δI(φ) = B(φ, v)− l(v) = 0 (3.192)

which is the weak form. That is, δI(φ) = 0 yields the weak form.(iii) Extremum principle or sufficient condition:

δ2I(φ) = δ(δI(φ)) = δ(B(φ, v)− l(v)) (3.193)

Thereforeδ2I(φ) = B(v, v) (3.194)

or

δ2I(φ) =

∫Ω

b(d2v

dx2

)2dΩ > 0 ∀v ∈ V and b > 0 (3.195)

Thus, δ2I(φ) yields a unique extremum principle. Hence, the integralform resulting from the GM/WF is VC for this BVP. Again, we see theimportance of the weak form in obtaining VC integral form when theoperator is linear and when A∗ = A. In this case also I(φ) representstotal potential energy.


Let φn be an approximation of φ in Ω, then

Aφn − f = E (3.196)

is the residual function.

(i) We define the residual functional I(φn) as (existence of I(φn))

I(φn) = (E,E) =

∫Ω

[ d2

dx2

(bd2φ

dx2

)−Q(x)

]2dΩ (3.197)

(ii) Necessary condition: If I(φn) is differentiable in φn, then the necessarycondition is given by δI(φn) = 0.

δI(φn) = 2(E, δE) = 0 or (E, δE) = 0 (3.198)

Where

E =d2

dx2

(bd2φndx2

)−Q(x) (3.199)

δE =d2

dx2

(bd2v

dx2

)= Av, v = δφn (3.200)


Hence, (E, δE) = 0 in (3.198) can be written as

(E, δE) = (Aφn − f, δE) = (Aφn − f,Av) = 0 (3.201)

or(Aφn, Av) = (Av, f) (3.202)

orB(φn, v) = l(v) (3.203)

This is the desired integral form resulting from δI(φn) = 0 where

B(φn, v) =

(d2

dx2

(bd2φndx2

),d2

dx2

(bd2v

dx2

))(3.204)

l(v) =

(Q(x),

d

dx

(bd2v

dx2

))= (f,Av) (3.205)

B(·, ·) is bilinear and symmetric, B(φn, v) = B(v, φn), and l(·) is linearin v.


δ2I(φn) = 2δ(B(φn, v)− l(v)) = 2B(v, v) (3.206)

where

2B(v, v) = 2

(d2

dx2

(bd2v

dx2

),d2

dx2

(bd2v

dx2

))(3.207)

ClearlyB(v, v) > 0 ∀v ∈ Vn (3.208)

Hence, δ2I(φn) yields a unique extremum principle. Therefore theintegral form resulting from δI(φn) = 0 is VC for this BVP and sinceδ2I(φn) > 0, a φn from δI(φn) = 0 minimizes I(φn) in (3.197). Theminimum of I(φn) is zero. Hence, when I(φn)→ 0, E → 0 in Ω in thepointwise sense and, thus, Aφ− f = 0 is satisfied in Ω in the pointwisesense when the integrals are Riemann.

Example 3.3. Consider the boundary value problem that arises in axisym-metric heat transfer

−1

r

d

dr

(krdθ

dr

)− f = 0 ∀ ri ≤ r ≤ ro (3.209)


θ(ri) = θ0

krdθ

dr+ β(θ − θ∞) = 0 at r = ro

(3.210)


The boundary value problem (3.209) describes axisymmetric heat con-duction independent of circumferential coordinate Θ and axial coordinatez in r,Θ, z cylindrical coordinate system, and ri and ro are the inner andouter radii a long cylindrical tube. Boundary conditions (3.210) describefixed temperature θ0 at r = ri and convection heat transfer at r = ro (heatloss in this case), θ is temperature, k is conductivity, β is film coefficient, θ∞is the ambient temperature, and f is the internal heat generation.

The differential operator A in this case is given by

A = −1

r

d

dr

(kr

d

dr

)(3.211)

Thus, we can write (3.209) as

Aθ − f = 0 (3.212)

It can be shown that

(a) A is linear and A∗ = A

(b) A is symmetric when the boundary conditions are homogeneous i.e.when θ0 = 0 at r = ri and kr dθdr = 0 at r = ro

We can consider various methods of approximation for the BVP. SinceA is linear and A∗ = A, GM/WF and LSP are assured to yield VC inte-gral forms whereas GM, PGM, and WRM will yield VIC integral forms;GM, PGM, and WRM methods are straightforward and follow the standardprocedure as shown for examples 3.1 and 3.2, hence here we only considerGM/WF and LSP. We present details in the following.


Based on fundamental lemma we can write the following using test func-tion v. ∫

Ω

(Aθ − f)v dΩ = (Aθ − f, v) = 0 (3.213)

or ∫Ω

(−1

r

d

dr

(krdθ

dr

)− f

)v dΩ = 0 (3.214)

where v = δθ is a test function, hence v must satisfy homogeneous form ofthe boundary conditions on θ. In this BVP, dΩ = 2πr dr, a ring of meanradius r and thickness dr. Hence, we can write (3.214) as

2π

ro∫ri

−1

r

d

dr

(krdθ

dr

)v rdr − 2π

ro∫ri

fv rdr = 0 (3.215)


or

2π

ro∫ri

− d

dr

(krdθ

dr

)v dr − 2π

ro∫ri

fv r dr = 0 (3.216)

Step 1: We transfer one order of differentiation from θ to v2π. Using integra-tion by parts, we can write the following:

2π

ro∫ri

kdv

dr

dθ

drr dr − 2π

[v

(krdθ

dr

)]rori

− 2π

ro∫ri

fv r dr = 0 (3.217)

in which the concomitant 〈Aθ, v〉Γ is given by

〈Aθ, v〉Γ = 2π

[−v(krdθ

dr

)]rori

(3.218)

Hence we can write (3.217) as

2π

ro∫ri

kdv

dr

dθ

drr dr + 〈Aθ, v〉Γ − 2π

ro∫ri

fv r dr = 0 (3.219)

Step 2: Determination of PVs, SVs, EBCs, and NBCs using concomitant:

Defining the dependent variable θ in the same manner in which the testfunction and its derivatives appear in the concomitant constitutes primaryvariables. Coefficients of the test function and the coefficients of the deriva-tives of the test function are secondary variables. Thus,

θ is PV

2πkrdθ

dris SV

(3.220)

Based on (3.218)

θ = θ on some boundary Γ1 is EBC

2πkrdθ

dr= q on some boundary Γ2 is NBC

(3.221)

Now we can classify the BCs (3.210) of the BVP in EBC and NBC categories.Clearly

θ(ri) = θ0 is EBC

krdθ

dr+ β(θ − θ∞) = 0 at r = ro is NBC


We note that classification of PVs, SVs, EBCs, and NBCs purely resultsfrom the concomitant. After we determine these from the concomitant, thenwe can identify which BCs of the BVP fall into these two categories as wehave done.

Step 3: Simplification of the concomitant resulting from integration by parts:

From the boundary conditions (3.210), we have

θ(ri) = θ0 ⇒ v(ri) = 0

krdθ

dr+ β(θ − θ∞) = 0 at r = ro

⇒ krodv

dr

∣∣∣∣ro

+ βv(ro) = 0

(3.222)

Expanding the concomitant in (3.217)

2π

ro∫ri

kdv

dr

dθ

drr dr − v(ro)

[2πkro

dθ

dr

]ro

+ v(ri)

[2πkri

dθ

dr

]ri

− 2π

ro∫ri

fv r dr = 0 (3.223)

Using

v(ri) = 0[2πkro

dθ

dr

]ro

= −2πβ[θ(ro)− θ∞

] (3.224)

in (3.223) we obtain

2π

ro∫ri

kdv

dr

dθ

drr dr + 2πβv(ro)

(θ(ro) − θ∞

)− 2π

ro∫ri

fv r dr = 0 (3.225)

or

2π

ro∫ri

kdv

dr

dθ

drr dr + βv(ro)θ(ro)

− 2π

βv(ro)θ∞ +

ro∫ri

fv r dr

= 0

(3.226)

orB(θ, v)− l(v) = 0 (3.227)


Equation (3.227) is the weak form of the BVP (3.209) with BCs (3.210).

(a) B(θ, v) is bilinear and also symmetric due to the fact that A∗ = A anddue to the integration by parts(b) l(v) is linear in v

In (3.227) θ, v ∈ Vn ⊂ Hk(Ω); k ≥ 2 in which k = 2 is minimally conformingas opposed to θ ∈ Vn ⊂ H3(Ω) and v ∈ H1(Ω) in (3.214). Thus we see thatdifferentiability requirements have been lowered on θ in the weak form butincreased on v compared to the integral form based on fundamental lemma.However, θ of class C1 are not admissible in the BVP.

Step 4: Determining VC or VIC of the weak form:

Here we establish wheter the weak form (3.227) is VC or VIC.

(i) Since B(·, ·) is bilinear and symmetric and l(·) is linear, the quadraticfunctional I(θ) is given by

I(θ) =1

2B(θ, θ)− l(θ) (3.228)

or

1

2

2π

ro∫ri

k

(dθ

dr

)2

r dr + 2πβθ(ro)2

−

2πβθ(ro)θ∞ + 2π

ro∫ri

fθ r dr

= 0 (3.229)

(ii) Necessary condition: If I(θ) is differentiable in θ then δI(θ) = 0 is anecessary condition for an extremum of I(θ).

δI(θ) = δ

(1

2B(θ, θ)− l(θ)

)= 0 (3.230)

=1

2B(v, θ) +

1

2B(θ, v)− l(v) = 0 (3.231)

But B(θ, v) = B(v, θ), hence

δI(θ) = B(θ, v)− l(v) = 0 (3.232)

which is the weak form. That is, δI(θ) = 0 yields the weak form.(iii) Extremum principle or sufficient condition:

δ2I(θ) = δ(δI(θ)) = δ(B(θ, v)− l(v)) (3.233)


Thereforeδ2I(θ) = B(v, v) (3.234)

or

δ2I(θ) = 2π

ro∫ri

k

(dv

dr

)2

r dr + 2πβ(v(ro)

)2> 0 ∀v ∈ V and k, β > 0

(3.235)

Thus δ2I(θ) yields a unique extremum principle. Hence the weak formresulting from GM/WF is VC for this BVP. Again, we see the im-portance of the weak form in obtaining VC integral form when theoperator is linear and when A∗ = A.


Let θn be approximation of θ in Ω = [ri, ro]. Then

Aθn − f = E (3.236)

is the residual function.

(i) We define the residual functional I(θn) as (existence of I(θn))

I(θn) = (E,E) =

∫Ω

E2 dΩ (3.237)

Using (3.209), expanding the differentiation and changing sign through-out, we can write the following for E by replacing θ with θn.

E = k

(d2θndr2

+1

r

dθndr

)+ f (3.238)

In obtaining (3.238) from (3.209), we have assumed that k is constant.

(ii) Necessary condition: If I(θn) is differentiable in θn, then the necessarycondition is given by δI(θn) = 0.

δI(θn) = 2(E, δE) = 0 or (E, δE) = 0 (3.239)

where E is given by (3.238) and δE can be written as

δE = k

(d2v

dr2+

1

r

dv

dr

)= Av, v = δθn (3.240)

Hence, (E, δE) = 0 can be written as

(E, δE) = (Aθn − f, δE) = (Aθn − f,Av) = 0 (3.241)


or(Aθn, Av) = (Av, f) (3.242)

orB(θn, v) = l(v) (3.243)

This is the desired integral form resulting from δI(θn) = 0 where

B(θn, v) =

(k

(d2θndr2

+1

r

dθndr

), k

(d2v

dr2+

1

r

dv

dr

))(3.244)

l(v) =

(f, k

(d2v

dr2+

1

r

dv

dr

))(3.245)

Clearly, B(·, ·) is bilinear and symmetric, B(θn, v) = B(v, θn), and l(v)is linear in v.


δ2I(θn) = 2δ(B(θn, v)− l(v)

)= 2B(v, v) (3.246)

where

2B(v, v) = 2

(k

(d2v

dr2+

1

r

dv

dr

), k

(d2v

dr2+

1

r

dv

dr

))(3.247)

ClearlyB(v, v) > 0 ∀v ∈ Vn (3.248)

Hence δ2I(θn) yields a unique extremum principle. Therefore the in-tegral form resulting from δI(θn) = 0 is VC for this BVP and sinceδ2I(θn) > 0 a θn from δI(θn) = 0 minimizes I(θn) in (3.237). Theminimum of I(θn) is zero. Hence, when I(θn)→ 0, E → 0 in Ω in thepointwise sense when the integrals are Riemann and thus Aθ − f = 0is satisfied in Ω in the pointwise sense.

Example 3.4. Consider the boundary value problem described by the 2DPoisson equation

− ∂

∂x

(∂φ∂x

)− ∂

∂y

(∂φ∂y

)+Q(x, y) = 0 ∀x, y ∈ Ω ⊂ R2 (3.249)

with

φ = φ on Γ1

∂φ

∂xnx +

∂φ

∂yny = q on Γ2

Γ = Γ1 ∪ Γ2 and Ω = Ω ∪ Γ

(3.250)


where φ and q are known. In this case, the differential operator A and f aregiven by

A = − ∂

∂x

( ∂∂x

)− ∂

∂y

( ∂∂y

)(3.251)

and

f = −Q(x, y). (3.252)


(a) A is linear and A∗ = A.

(b) A is symmetric when φ = 0 and q = 0 i.e. when the boundary conditionsare homogeneous.

We consider various methods of approximation.

The Galerkin method

In this method we consider the integral form based on fundamentallemma given by ∫

Ω

(Aφ− f)v dΩ = (Aφ− f, v) = 0 (3.253)

in which v = δφ and, hence, satisfies the homogeneous form of the boundaryconditions on φ. Therefore∫

Ω

[− ∂

∂x

(∂φ∂x

)− ∂

∂y

(∂φ∂y

)+Q(x, y)

]v dΩ = 0 (3.254)

or

B(φ, v) = l(v) (3.255)

where

B(φ, v) = (Aφ, v) (3.256)

l(v) = −∫Ω

Qv dΩ (3.257)

The integral form in this method is given by (3.254) in which

(a) B(φ, v) is bilinear but not symmetric, B(φ, v) 6= B(v, φ)

(b) l(v) is linear

Let there exist a functional I(φ) such that δI(φ) = 0 yields the integral form(3.255). Then in order for the integral form (3.110) to be VC, we must show


that δ2I(φ) = δ(B(φ, v) − l(v))) must yield a unique extremum principle.We have

δ(B(φ, v)− l(v)) = B(v, v), δφ = v (3.258)

where

B(v, v) =

∫Ω

[− ∂

∂x

(∂v∂x

)− ∂

∂y

(∂v∂y

)]v dΩ (3.259)

Obviously

B(v, v)

> 0

= 0

< 0

∀v ∈ Vn (3.260)

does not hold. Hence, δ2I(φ) does not yield a unique extremum principle.Thus, the Galerkin method for this BVP yields a VIC integral form.


In this method also, we begin with the integral form based on fundamen-tal lemma using test function or weight function w.∫

Ω

(Aφ− f)w dΩ = (Aφ− f, w)Ω = 0 (3.261)

or ∫Ω

[− ∂

∂x

(∂φ∂x

)− ∂

∂y

(∂φ∂y

)+Q(x, y)

]w dΩ = 0 (3.262)

The weight function w satisfies the homogeneous form of the boundary con-ditions on φ but w 6= δφ. From (3.262) we can write

B(φ,w) = l(w) (3.263)

where

B(φ,w) =

∫Ω

[− ∂

∂x

(∂φ∂x

)− ∂

∂y

(∂φ∂y

)]w dΩ (3.264)

l(w) =

∫Ω

−Q(x, y)w dΩ (3.265)

The integral form in this method is (3.263) in which:


(b) l(w) is linear


Let there exist a functional I(φ) such that δI(φ) = 0 yields the integral form(3.263). Then we must show that δ2I(φ) yields a unique extremum principlein order for the integral form (3.263) to be VC. We have

δ(B(φ,w)− l(w)) = B(v, w) =

∫Ω

[− ∂

∂x

(∂v∂x

)− ∂

∂y

(∂v∂y

)]w dΩ (3.266)

Clearly

B(v, w)

> 0

= 0

< 0

∀v ∈ Vn, w ∈ V (3.267)



In this method also we begin with the integral form based on fundamentallemma using test function v.∫

Ω

(Aφ− f)v dΩ = (Aφ− f, v) = 0 (3.268)

or ∫Ω

[− ∂

∂x

(∂φ∂x

)− ∂

∂y

(∂φ∂y

)+Q(x, y)

]v dΩ = 0 (3.269)


Step 1: We transfer one order of differentiation from φ to v using integrationby parts. We note that in (3.269), φ ∈ H3 ⊂ H1 and v ∈ H1 are necessaryfor the integral to be in Riemann sense. Integration by parts yields∫

Ω

(∂v∂x

∂φ

∂x+∂v

∂y

∂φ

∂y

)dΩ−

∮Γ

v(∂φ∂xnx +

∂φ

∂yny

)dΓ =

∫Ω

−Qv dΩ (3.270)

The concomitant 〈Aφ, v〉Γ in this case is given by

〈Aφ, v〉Γ = −∮Γ

v(∂φ∂xnx +

∂φ

∂yny

)dΓ

The concomitant is expressed as the test function v multiplied by its coeffi-cient. This form is essential in what follows.



Using the concomitant we define the following.

Definition of primary variable:

The dependent variable in the same form in which the test function appearsin the concomitant is defined as the primary variable (PV). Thus

φ is the PV

Definition of secondary variable:

The coefficient of the test function and the coefficients of the derivativesof the test function in the concomitant is defined as the secondary variable

(SV). In this case the coefficient of the test function v is(∂φ∂xnx + ∂φ

∂yny

)in

the concomitant; hence (∂φ∂xnx +

∂φ

∂yny

)is the SV

Definition of essential boundary conditions:

Specification of the primary variable on a boundary constitutes an essentialboundary condition. Thus

φ = φ specified on some boundary Γ1 is the EBC.

Definition of natural boundary conditions:

Specification of the secondary variable on a boundary constitutes a naturalboundary conditions. Thus

∂φ

∂xnx +

∂φ

∂yny = g specified on some boundary Γ2 is a NBC.

Now we can examine the BCs (3.250) of the BVP (3.249) to determine whichones are EBCs and which ones are NBCs. From (3.250) and the definitionsof EBC and NBC we find that

φ(0) = φ on Γ1 is EBC

and∂φ

∂xnx +

∂φ

∂yny = q on Γ2 is NBC



In this step the concomitant (i.e. (3.270)) is simplified using boundary con-ditions of the problem. Since Γ = Γ1 ∪ Γ2, the boundary integral in (3.270)can be written as the sum of the integrals over Γ1 and Γ2 (we lose the circleas Γ1 and Γ2 are not closed contours) and we can write∫

Ω

(∂v∂x

∂φ

∂x+∂v

∂y

∂φ

∂y

)dΩ−

∫Γ1

v(∂φ∂xnx +

∂φ

∂yny

)dΓ

−∫Γ2

v(∂φ∂xnx +

∂φ

∂yny

)dΓ =

∫Ω

−Qv dΩ (3.271)

From boundary conditions of the BVP

φ = φ⇒ v = δφ = 0 on Γ1 (3.272)

∂φ

∂xnx +

∂φ

∂yny = q ⇒ ∂v

∂xnx +

∂v

∂yny = 0 on Γ2 (3.273)

Hence, the integral over Γ1 vanishes and in the integral over Γ2 we substitutefrom (3.273). Thus, from (3.271) we obtain∫

Ω

(∂v∂x

∂φ

∂x+∂v

∂y

∂φ

∂y

)dΩ =

∫Γ2

vq dΓ−∫Ω

Qv dΩ (3.274)

orB(φ, v) = l(v) (3.275)

where

B(φ, v) =

∫Ω

(∂v∂x

∂φ

∂x+∂v

∂y

∂φ

∂y

)dΩ (3.276)

l(v) =

∫Γ2

vq dΓ−∫Ω

Qv dΩ (3.277)

The integral form (3.274) or (3.275) is the desired integral form and is re-ferred to as the weak form of the BVP. In this case, the differentiation onφ has been lowered or the continuity (or differentiability) requirement hasbeen weakened compared to the integral form (3.253). Hence, the nameweak form.

In (3.274), φ ∈ Vn ⊂ H2(Ω) and v ∈ Vn ⊂ H2(Ω) are minimally admissibleas opposed to φ ∈ Vn ⊂ H3(Ω) and v ∈ Vn ⊂ H1(Ω) in (3.253). Thus, wesee the lowered differentiability requirement on φ but increased on v in theweak form. When φ ∈ Vn ⊂ H2(Ω) it is not admissible in the BVP whichrequires k = 3.


Properties of B(·, ·) and l(·) are

(a) B(·, ·) is bilinear, that is φ and v.

(b) B(φ, v) given by (3.276) is symmetric:

B(v, φ) =

∫Ω

(∂φ∂x

∂v

∂x+∂φ

∂y

∂v

∂y

)dΩ = B(φ, v) (3.278)

Symmetry of B(·, ·) is a direct consequence of the fact that A∗ = A and thatwe have used integration by parts.

(c) l(·) is linear in v.


Since in this case B(·, ·) is bilinear and symmetric and l(·) is linear, thequadratic functional I(φ) exists and is given by

I(φ) =1

2B(φ, φ)− l(φ) (3.279)

or

I(φ) =1

2

∫Ω

[(∂φ∂x

)2+(∂φ∂y

)2]dΩ−

∫Γ2

φq dΓ +

∫Ω

φQdΩ. (3.280)

(i) This establishes existence of the functional I(φ).

(ii) Necessary condition: when I(φ) is differentiable in φ, then the neces-sary condition for an extremum of I(φ) is given by δI(φ) = 0.

δI(φ) = δ[12B(φ, φ)− l(v)

]= 0 (3.281)

=1

2B(v, φ) +

1

2B(φ, v)− l(v) = 0 (3.282)

since B(φ, v) = B(v, φ), we can write

δI(φ) =1

2B(φ, v) +

1

2B(φ, v)− l(v) = 0 (3.283)

or

δI(φ) = B(φ, v)− l(v) = 0 (3.284)

which is the desired integral form and is the weak form. That is,δI(φ) = 0 yields the weak form.

(iii) Extremum principle or sufficient condition:

δ2I(φ) = δ(δI(φ)) = δ(B(φ, v)− l(v)) (3.285)


Therefore

δ2I(φ) = B(v, v) =

∫Ω

[(∂v∂x

)2+(∂v∂y

)2]dΩ > 0 ∀v ∈ V (3.286)

Hence, δ2I(φ) yields a unique extremum principle. Condition (3.286)implies that the a φ from δI(φ) = B(φ, v)− l(v) = 0 minimizes I(φ) in(3.287). Thus, the integral form resulting from the weak form (3.275)is VC.

Remarks.

(1) When the differential operator A in the BVP Aφ − f = 0 in Ω is self-adjoint, then there exists a weak form B(φ, v) = l(v) in which B(φ, v)is bilinear and symmetric and l(v) is linear and the functional I(φ) =12B(φ, φ) − l(v) is always possible. δI(φ) = 0 yields the weak formand δ2I(φ) > 0∀v ∈ V yields a unique extremum principle. A φ fromδI(φ) = 0, i.e. B(φ, v) − l(v) = 0 minimizes I(φ). This weak form isalways VC. The functional I(φ) is called the quadratic functional.

(2) In linear elasticity (linear solid mechanics) when Aφ−f = 0 in Ω are thegoverning differential equations for a stationary process (i.e. a BVP) inwhich A is a self-adjoint differential operator, then 1

2B(φ, φ) is the strainenergy stored by the deforming body (or process). l(φ) is the potentialenergy of loads and I(φ) represents the total potential energy of thedeformed body (or process). Hence, minimization of the functional I(φ)is in fact the principle of minimization of the total potential energy ofthe deformed body.

(3) Thus, in this case we have

(a) Aφ− f = 0 in Ω: BVP(b) weak form: WF(c) functional I(φ) is a quadratic functional: QF

Solutions from all three are in fact equivalent with the proper choice ofspaces.

(4) If we consider the integrals to be in the Riemann sense, then we have

BVP: Aφ− f = 0 in Ω, φ ∈ Vn ⊂ Hk(Ω), k ≥ 3k = 3 is minimally conforming

WF: B(φ, v) = l(v), φ ∈ Vn ⊂ Hk(Ω), v ∈ Vn ⊂ Hk(Ω), k ≥ 2k = 2 is minimally conforming

QF: I(φ) = 12B(φ, φ)− l(v), φ ∈ Vn ⊂ Hk(Ω), k ≥ 2

k = 2 is minimally conforming


With this choice we can go back from the weak form (3.275) or (3.287)to (3.268). This is only possible when k = 3 everywhere.

When φ ∈ Vn ⊂ Hk(Ω) and k ≥ 3, the BVP, QF and WF are allequivalent and the solutions from the WF and QF minimization are alsoadmissible in the BVP in the piontwise sense.

(5) Note that the NBC is absorbed in the weak form and, hence, do notneed to be satisfied by the approximation φn.

(6) This example shows the importance of the weak form. The VIC in-tegral forms in the Galerkin method and the Petrov–Galerkin method(or weighted-residual method) become VC in the Galerkin method withweak form.


Let φn be an approximation of φ in Ω, then Aφn − f = E in Ω is theresidual function.

(i) We define the residual functional I(φn) as

I(φn) = (E,E) =

∫Ω

[− ∂

∂x

(∂φn∂x

)− ∂

∂y

(∂φn∂y

)+Q(x, y)

]2dΩ (3.287)

(ii) Necessary condition: If I(φn) is differentiable in φn, then the necessarycondition for an extremum of I(φ) is given by δI(φn) = 0.

δI(φn) = 2(E, δE) = 0 or (E, δE) = 0 (3.288)


E = − ∂

∂x

(∂φn∂x

)− ∂

∂y

(∂φn∂y

)+Q(x, y) (3.289)

δE = − ∂

∂x

(∂v∂x

)− ∂

∂y

(∂v∂y

)= Av, v = δφn (3.290)

(E, δE) = (Aφn − f,Av) =(− ∂

∂x

(∂φn∂x

)− ∂

∂y

(∂φn∂y

)+Q(x, y),− ∂

∂x

(∂v∂x

)− ∂

∂y

(∂v∂y

))= 0

(3.291)

orB(φn, v) = l(v) (3.292)


This is the integral form resulting from δI(φn) = 0. where

B(φn, v) = (Aφn, Av) =(− ∂

∂x

(∂φn∂x

)− ∂

∂y

(∂φn∂y

),

− ∂

∂x

(∂v∂x

)− ∂

∂y

(∂v∂y

))(3.293)

l(v) = (Q(x), Av) =( ∂∂x

(∂v∂x

)+

∂

∂y

(∂v∂y

), Q)

(3.294)

B(φn, v) is bilinear and symmetric i.e. B(φn, v) = B(v, φn) and l(v) islinear.


δ2I(φn) = δ(2(E, δE)

)= 2(δE, δE)

= 2

∫Ω

(δE)2 dΩ = 2

∫Ω

(− ∂

∂x

(∂v∂x

)− ∂

∂y

(∂v∂y

))2dΩ > 0 ∀v ∈ V

Hence, δ2I(φn) > 0 yields a unique extremum principle. Since δ2I(φn) >0, a φn from B(φn, v)− l(v) = 0 minimizes I(φn) in (3.287). The inte-gral form in (3.292) is obviously VC.

Remarks.

(1) For the integrals to be Riemann

φn ∈ Vn ⊂ Hk(Ω) , k ≥ 3

v ∈ Vn ⊂ Hk(Ω) , k ≥ 3

k = 3 corresponds to the minimally conforming space.

(2) The least-squares process does not require symmetry of the operator,linearity is sufficient.

(3) The minimum of I(φn) is zero. Hence, I(φn) → 0 ⇒ E → 0 in thepointwise sense in Ω, that is Aφ−f = 0 is satisfied by φn in the pointwisesense in Ω provided all integrals are Riemann.

3.5.2 Non-self-adjoint Differential Operators

In this section we consider one dimensional and two dimensional bound-ary value problems containing non-self-adjoint differential operators. Thenon-self-adjoint differential operators are linear but not symmetric.


Example 3.5. Consider the 1D steady-state convection-diffusion equation

dφ

dx− kd

2φ

dx2= 0 ∀x ∈ Ω = (0, 1) ⊂ R1 (3.295)


φ(0) = 1 and φ(1) = 0 (3.296)

where k = 1Pe > 0 is the diffusion coefficient and Pe is the Peclet number.

This BVP is the one dimensional form of the dimensionless energy equationin the absence of viscous dissipation. In this case

A =d

dx− k d

2

dx2(3.297)

f = 0 (3.298)

Hence, (3.295) can be written as

Aφ− f = 0 in Ω (3.299)

It has been shown that the operator A is linear but its adjoint A∗ 6= A,hence A is non-self-adjoint. In the following we consider various methods ofapproximation.

The Galerkin method

In this method we consider the integral form (based on fundamentallemma) using test function v,∫

Ω

(Aφ− f)v dΩ = (Aφ− f, v) = 0 in Ω (3.300)

in which v = δφ and, hence, satisfies the homogeneous form of the boundaryconditions on φ. Substituting for A in (3.300)∫

Ω

(dφdx− kd

2φ

dx2

)v dΩ = 0 (3.301)

orB(φ, v) = l(v) (3.302)

where

B(φ, v) =(dφdx− kd

2φ

dx2, v)

(3.303)

l(v) = 0 (3.304)


We note that (3.302) is the desired integral form in which

(a) B(φ, v) is bilinear but not symmetric i.e. B(φ, v) 6= B(v, φ)

(b) l(v) is linear

Let there exist a functional I(φ) such that δI(φ) = 0 yields the integral form(3.301). Then in order for the integral form (3.301) to be VC, δ2I(φ) =δ(B(φ, v)− l(v)) must yield a unique extremum principle.

δ(B(φ, v)− l(v)) = B(v, v) =(dvdx− k d

2v

dx2, v)

> 0

= 0

< 0

∀v ∈ Vn (3.305)

does not hold. Hence, the variation of the integral form (3.301) does notyield a unique extremum principle. That is, the Galerkin method for thisBVP yields a VIC integral form.


In this method also, we begin with the integral form based on fundamen-tal lemma using a test function or weight function w∫

Ω

(Aφ− f)w dΩ = (Aφ− f, w) = 0 (3.306)

or ∫Ω

(dφdx− kd

2φ

dx2

)w dΩ = 0 (3.307)

or

B(φ,w) = l(w) (3.308)

where

B(φ,w) =

∫Ω

(dφdx− kd

2φ2

dx2

)w dΩ (3.309)

l(w) = 0 (3.310)

and w is a weight function that satisfies the homogeneous form of the bound-ary conditions on φ but w 6= δφ. We note that (3.308) is the desired integralform in which


(b) l(w) is linear


Let there exist a functional I(φ) such that δI(φ) = 0 yields the integral form(3.118). Then we must show that δ2I(φ) yields a unique extremum principlein order for the integral form (3.118) to be VC.

δ(B(φ,w)− l(w)) = B(v, w) =

∫Ω

(dvdx− k d

2v

dx2

)w dΩ (3.311)

Obviously

B(v, w)

> 0

= 0

< 0

∀v ∈ Vn, ∀w ∈ V (3.312)



Based on fundamental lemma and using test function v, we can write∫Ω

(Aφ− f)v dΩ = (Aφ− f, v) = 0 (3.313)

or ∫Ω

(dφdx− kd

2φ

dx2

)v dΩ = 0 (3.314)

or ∫Ω

(dφdxv − vkd

2φ

dx2

)dΩ = 0 (3.315)

where v = δφ is a test function and, hence, must satisfy the homogeneousform of the boundary conditions.

Step 1: There is nothing to be gained by transferring differentiation from φ tov in the first term of (3.315) due to the fact that the non-symmetric natureof this term remains unaltered. In the second term, we transfer one order ofdifferentiation from φ to kv and since k is constant, we can write∫

Ω

(Aφ)v dΩ =

∫Ω

(dφdxv + k

dv

dx

dφ

dx

)dΩ−

[v(kdφ

dx

)]1

0

(3.316)

The concomitant 〈Aφ, φ〉Γ in (3.316) is given by

〈Aφ, φ〉Γ = −[v

((kdφ

dx

))]1

0



Using the definition of primary and second variables and essential and nat-ural boundary conditions, from the concomitant we determine that

φ is PV

kdφ

dxis SV

φ = φ specified on some boundary Γ1 is EBC

kdφ

dx= q specified on some boundary Γ2 is NBC

(3.317)

where φ and q are known. Using these definitions of EBC and NBC, we canidentify the BCs of the BVP (3.295) into EBC and NBC catagories. From(3.296) we find that φ(0) = 1 and φ(1) = 0 both are EBCs. This BVP doesnot have NBCs.

Step 3: Simplification of concomitant:

From boundary conditions of the BVP

φ(0) = 1⇒ v(0) = 0

φ(1) = 0⇒ v(1) = 0(3.318)

Substituting from (3.318) into (3.316), we obtain∫Ω

(Aφ)v dΩ =

∫Ω

(dφdxv + k

dv

dx

dφ

dx

)dΩ (3.319)

orB(φ, v) = l(v) (3.320)

where

B(φ, v) =

∫Ω

(dφdxv + k

dv

dx

dφ

dx

)dΩ (3.321)

l(v) = 0 (3.322)

The weak form (3.313) is the desired integral form in which

(a) B(·, ·) is bilinear(b) B(·, ·) is not symmetric i.e. B(φ, v) 6= B(v, φ)(c) l(·) is linear in v

Since B(·, ·) is not symmetric the quadratic functional I(φ) cannot be con-structed using I(φ) = 1

2B(φ, φ) − l(v). φ ∈ Vn ⊂ H2(Ω), v ∈ Vn ⊂ H2(Ω)


are minimally conforming for the weak form if the integral in (3.321) is tobe Riemann. However the BVP requires φn ∈ Vn ⊂ H3(Ω).


Let there exist a functional I(φ) such that δI(φ) = 0 yields the weak form(3.320). Then, in order to show the variational consistency of the integralform in (3.320), we need to show that δ2I(φ) yields a unique extremumprinciple.

δ2I(φ) = δ(δI(φ)) = δ(B(φ, v)− l(v)) = B(v, v) (3.323)

or

δ2I(φ) =

∫Ω

(dvdxv + k

(dvdx

)2)dΩ (3.324)

We note that∫

Ω k(dvdx

)2dΩ > 0 ∀v ∈ Vn and for k > 0. This is obviously a

consequence of integration by parts in the second term of the integrand in(3.315). Unfortunately, the same does not hold for

∫Ω v

dvdx dΩ, thus

δ2I(φ)

> 0

= 0

< 0

∀v ∈ Vn (3.325)

does not hold. Hence, δ2I(φ) in (3.324) does not yield a unique extremumprinciple. That is, the integral form resulting from the Galerkin methodwith weak form is VIC for this BVP in which the differential operator is nonself-adjoint.


Let φn be an approximation of φ in Ω. Then Aφn − f = E in Ω is theresidual function.


I(φn) = (E,E)Ω =(dφndx− kd

2φ

dx2,dφndx− kd

2φ

dx2

)Ω

=

∫Ω

(dφndx− kd

2φ

dx2

)2dΩ

(3.326)

(ii) Necessary condition: If I(φn) is differentiable in φn, then δI(φn) = 0is necessary condition for an extremum of I(φn).

δI(φn) = 2(E, δE) = 0 or (E, δE) = 0 (3.327)



E =dφndx− kd

2φndx2

(3.328)

δE =dv

dx− k d

2v

dx2= Av, v = δφn (3.329)

Hence(E, δE) = (Aφn − f,Av) = 0 (3.330)

or

(E, δE) =(dφndx− kd

2φndx2

,dv

dx− k d

2v

dx2

)= 0 (3.331)

Therefore, 12δI(φn) = (E, δE) = 0 gives the desired integral form

B(φn, v) = l(v) (3.332)

where

B(φn, v) =(dφndx− kd

2φndx2

,dv

dx− k d

2v

dx2

)(3.333)

l(v) = 0 (3.334)

The desired integral form resulting from this method is (3.332) fromδI(φn) = 0 in which

(a) B(φn, v) is bilinear(b) B(φn, v) is symmetric, B(φn, v) = B(v, φn)(c) l(v) = 0


δ2I(φn) = δ(δI(φn)

)= 2δ

(B(φn, v)− l(v)

)(3.335)

orδ2I(φn) = 2δ

(B(φn, v)

)= 2B(v, v) (3.336)

Thus

δ2I(φn) = 2B(v, v) = 2(dvdx− k d

2

dx2,dv

dx− k d

2

dx2

)(3.337)

Clearlyδ2I(φn) = 2B(v, v) > 0 ∀v ∈ Vn. (3.338)

Hence, δ2I(φn) > 0 yields a unique extremum principle. Thus, theintegral form (3.332) is VC. Since δ2I(φn) > 0, a φn from B(φn, v) −


l(v) = 0 minimizes I(φn) in (3.326). We further note that the minimumof I(φn) is zero. When I(φn)→ 0, then E → 0 in the pointwise sensein Ω. In other words, Aφ − f = 0 in Ω is satisfied in the pointwisesense provided all integrals are Riemann. We note that even thoughthe differential operator is non-self-adjoint, the integral form resultingfrom the least-squares process is VC.

Example 3.6. Consider the 2D steady-state convection-diffusion equation

ux∂φ

∂x+ uy

∂φ

∂y− k∂

2φ

∂x2− k∂

2φ

∂y2= 0 ∀(x, y) ∈ Ω ⊂ R2 (3.339)

with

φ(0) = φ on Γ1

k∂φ

∂xnx + k

∂φ

∂yny = q on Γ2

(3.340)

where Γ = Γ1∪Γ2 is such that Ω = Ω∪Γ and ux and uy are known velocities

in the x and y direction, k > 0 is the diffusion coefficient, and φ and q areknown data. In this case, the differential operator A and f are given by

A = ux∂

∂x+ uy

∂

∂y− k ∂

2

∂x2− k ∂

2

∂y2(3.341)

and

f = 0. (3.342)

Thus, we can write

Aφ− f = 0 in Ω (3.343)


(a) A is linear

(b) The adjoint A∗ 6= A for this BVP

We consider various methods of approximation in the following.

The Galerkin method

In this method we consider the integral form based on fundamentallemma using a test function v.∫

Ω

(Aφ− f)v dΩ = (Aφ− f, v) = 0 (3.344)


or ∫Ω

(ux∂φ

∂x+ uy

∂φ

∂y− k∂

2φ

∂x2− k∂

2φ

∂y2

)v dΩ = 0 (3.345)

in which v = δφ and, hence, satisfies the homogeneous form of the boundaryconditions on φ. Integral (3.345) can be written as

B(φ, v) = l(v) (3.346)

where

B(φ, v) =

(ux∂φ

∂x+ uy

∂φ

∂y− k∂

2φ

∂x2− k∂

2φ

∂y2, v

)(3.347)

l(v) = 0 (3.348)

The desired integral form in this method is (3.346) in which

(a) B(φ, v) is bilinear

(b) B(φ, v) is not symmetric, B(φ, v) 6= B(v, φ)

(c) l(v) is linear

Let there exist a functional I(φ) such that δI(φ) = 0 yields the integral form(3.345). Then in order for the integral form (3.345) to be VC, we must showthat δ2I(φ) = δ(B(φ, v)− l(v))) yields a unique extremum principle.

δ2I(φ) = B(v, v) =

(ux∂v

∂x+ uy

∂v

∂y− k ∂

2v

∂x2− k∂

2v

∂y2, v

)(3.349)

Obviously

B(v, v)

> 0

= 0

< 0

∀v ∈ Vn (3.350)



In this method also, we begin with the integral form based on fundamen-tal lemma using a test function or weight function w.∫

Ω

(Aφ− f)w dΩ = (Aφ− f, w) = 0 (3.351)


or ∫Ω

(ux∂φ

∂x+ uy

∂φ

∂y− k∂

2φ

∂x2− k∂

2φ

∂y2

)w dΩ = 0 (3.352)

where w is a weight function that satisfies the homogeneous form of theboundary conditions on φ but w 6= δφ; (3.352) can be written as

B(φ,w) = l(w) (3.353)

where

B(φ,w) =

(ux∂φ

∂x+ uy

∂φ

∂y− k∂

2φ

∂x2− k∂

2φ

∂y2, w

)(3.354)

l(w) = 0 (3.355)

The desired integral form in this method is (3.353) in which

(a) B(φ,w) is bilinear

(b) B(φ,w) is not symmetric i.e. B(φ,w) 6= B(w, φ)

(c) l(w) is linear


δ2I(φ) = δ(B(φ,w)− l(w)) = B(v, w) 6= B(w, v); v = δφ (3.356)

or

δ2I(φ) =

(ux∂v

∂x+ uy

∂v

∂y− k ∂

2v

∂x2− k∂

2v

∂y2, w

)(3.357)

Obviously

δ2I(φ)

> 0

= 0

< 0

∀v ∈ Vn, w ∈ V (3.358)



Based on fundamental lemma we can write the following using test func-tion v. ∫

Ω

(Aφ− f)v dΩ = (Aφ− f, v) = 0 (3.359)


or (ux∂φ

∂x+ uy

∂φ

∂y− k∂

2φ

∂x2− k∂

2φ

∂y2, v

)= 0 (3.360)

or ∫Ω

(ux∂φ

∂xv + uy

∂φ

∂yv − k∂

2φ

∂x2v − k∂

2φ

∂y2v

)dΩ = 0 (3.361)


Step 1: There is nothing to be gained by transferring differentiation from φto v in the first two terms of (3.361) since the non-symmetric nature of theseterms would remain unchanged. In the third and fourth terms in (3.361) wetransfer one order of differentiation from φ to kv and since k is constant wecan write ∫

Ω

(ux∂φ

∂xv + uy

∂φ

∂yv + k

∂v

∂x

∂φ

∂x+ k

∂v

∂y

∂φ

∂y

)dΩ

−∮

Γv(k∂φ

∂xnx + k

∂φ

∂yny

)dΓ = 0 (3.362)

The concomitant 〈Aφ, v〉Γ in (3.362) is given by

〈Aφ, v〉Γ = −∮

Γv(k∂φ

∂xnx + k

∂φ

∂yny

)dΓ

We note that each term in the resulting integral over Ω due to integrationby parts is symmetric in φ and v. This is the main reason for transferringone order of differentiation from φ to kv in the third and fourth terms in(3.361).


Using the definitions of primary and secondary variables and essential andnatural boundary conditions, we have

φ is PV(k∂φ

∂xnx + k

∂φ

∂yny

)is SV

φ = φ specified on some boundary Γ1 is EBC(k∂φ

∂xnx + k

∂φ

∂yny

)= q specified on some boundary Γ2 is NBC

(3.363)


Thus, the boundary conditions in (3.340) contain both essential and naturalboundary conditions.


k∂φ

∂xnx + k

∂φ


Step 3: Simplification of the concomitant using BCs of the BVP:

First we split the integral over Γ into Γ1 and Γ2 in the concomitant in (3.362)

∫Ω

(ux∂φ

∂xv + uy

∂φ

∂yv + k

∂v

∂x

∂φ

∂x+ k

∂v

∂y

∂φ

∂y

)dΩ

−∫

Γ1

v(k∂φ

∂xnx + k

∂φ

∂yny

)dΓ−

∫Γ2

v(k∂φ

∂xnx + k

∂φ

∂yny

)dΓ = 0 (3.364)

From the boundary conditions of the BVP

φ = φ0 ⇒ v = 0 on Γ1

k∂φ

∂xnx + k

∂φ

∂yny = q ⇒ k

∂v

∂xnx + k

∂v

∂yny = 0 on Γ2

(3.365)


(ux∂φ

∂xv + uy

∂φ

∂yv + k

∂v

∂x

∂φ

∂x+ k

∂v

∂y

∂φ

∂y

)dΩ =

∫Γ2

vq dΓ (3.366)

orB(φ, v) = l(v) (3.367)

where

B(φ, v) =

∫Ω

(ux∂φ

∂xv + uy

∂φ

∂yv + k

∂v

∂x

∂φ

∂x+ k

∂v

∂y

∂φ

∂y

)dΩ (3.368)

l(v) =

∫Γ2

vq dΓ (3.369)

The integral form (3.367) is the weak form of the BVP and is the desiredintegral form in which

(a) B(·, ·) is bilinear in φ and v(b) B(·, ·) is not symmetric, B(φ, v) 6= B(v, φ)(c) l(·) is linear in v


In (3.367), φ ∈ Vn ⊂ H2(Ω) and v ∈ Vn ⊂ H2(Ω) are minimally conformingif the integrals in (3.366) are to be Riemann, however for the BVP φ ∈ Vn ⊂H3(Ω) is essential.


Let there exist a functional I(φ) such that δI(φ) = 0 yields the integral form(3.366). Then in order to show the variational consistency of the integralform (3.366) we need to show that δ2I(φ) yields a unique extremum principle.

δ2I(φ) = δ(B(φ, v)− l(v)

)= B(v, v) (3.370)

where

B(v, v) =

∫Ω

(ux∂v

∂yv + uy

∂v

∂yv + k

(∂v∂x

)2+ k(∂v∂y

)2)dΩ (3.371)

We note that∫

Ω k((

∂v∂x

)2+(∂v∂y

)2)dΩ > 0 ∀v ∈ Vn and k > 0. This is

obviously a consequence of integration by parts and the main reason fortransferring one order of differentiation from φ to kv in the third and fourthterms in (3.361). Clearly

δ2I(φ) = B(v, v)

> 0

= 0

< 0

∀v ∈ Vn (3.372)

Hence, δ2I(φ) in (3.370) does not yield a unique extremum principle. Thus,the integral form resulting from the Galerkin method with weak form is VICfor this BVP in which the differential operator is non-self-adjoint.




I(φn) = (E,E) = (Aφn, Aφn) (3.373)

or

I(φn) =

(ux∂φn∂y

+ uy∂φn∂y− k∂

2φn∂x2

− k∂2φn∂y2

,

ux∂φn∂y


2φn∂x2

− k∂2φn∂y2

)dΩ (3.374)


(ii) Necessary condition: If I(φn) is differentiable in φn, then δI(φn) = 0is necessary condition for an extremum of I(φn).

δI(φn) = 2(E, δE) = 0 or (E, δE) = 0 (3.375)


E = ux∂φn∂y


2φn∂x2

− k∂2φn∂y2

= Aφn (3.376)

δE = ux∂v

∂y+ uy

∂v

∂y− k ∂

2v

∂x2− k∂

2v

∂y2= Av, v = δφn (3.377)


(E, δE) = (Aφn, Av) = 0 (3.378)

orB(φn, v) = l(v) (3.379)

This is the desired integral form resulting from δI(φn) = 0 in which

B(φn, v) =

(ux∂φn∂y


2φn∂x2

− k∂2φn∂y2

,

ux∂v

∂y+ uy

∂v

∂y− k ∂

2v

∂x2− k∂

2v

∂y2

)(3.380)

l(v) =

(f, ux

∂v

∂y+ uy

∂v

∂y+ k

∂2v

∂x2+ k

∂2v

∂y2

)= 0, since f = 0 (3.381)

B(·, ·) is bilinear and symmetric, B(φn, v) = B(v, φ) and l(v) is linearin v.


δ2I(φn) = δ(δI(φn)

)= 2B(v, v) (3.382)

where

B(v, v) =

(ux∂v

∂y+ uy

∂v

∂y− k ∂

2v

∂x2− k∂

2v

∂y2,

ux∂v

∂y+ uy

∂v

∂y− k ∂

2v

∂x2− k∂

2v

∂y2

)> 0 ∀v ∈ Vn holds

(3.383)


Hence, δ2I(φn) > 0 yields a unique extremum principle and so theintegral form (3.379) is VC. Since δ2I(φn) > 0, a φn from δI(φn) = 0minimizes I(φn) in (3.374). Further, we note that the minimum ofI(φn) is zero and so when I(φn)→ 0, E → 0 in the pointwise sense inΩ. In other words, Aφ− f = 0 in Ω is satisfied in the pointwise senseprovided all integrals are Riemann.

3.5.3 Non-linear Differential Operators

In this section we consider two examples of model problems with non-linear differential operators.

Example 3.7. Consider the 1D steady-state Burgers equation

φdφ

dx− kd

2φ

dx2= 0 ∀x ∈ Ω = (0, 1) ⊂ R1 (3.384)


φ(0) = 1 and φ(1) = 0 (3.385)

where k = 1Re > 0 is the diffusion coefficient and Re is the Reynolds number.

This BVP is the one dimensional form of the dimensionless energy equationin the absence of viscous dissipation. In this case

A = φd

dx− k d

2

dx2(3.386)

f = 0 (3.387)


Aφ− f = 0 in Ω (3.388)

It is straightforward to show that the operator A is not a linear differentialoperator and, hence, is not symmetric. In the following we consider variousmethods of approximation.

The Galerkin method

In this method we consider the integral form based on fundamentallemma given by ∫

Ω

(Aφ− f)v dΩ = (Aφ− f, v)Ω = 0 (3.389)


in which the test function v = δφ and, hence, satisfies the homogeneous formof the boundary conditions on φ. Substituting for A in (3.389)∫

Ω

(φdφ

dx− kd

2φ

dx2

)v dΩ = 0 (3.390)

orB(φ, v) = l(v) (3.391)

where

B(φ, v) =(φdφ

dx− kd

2φ

dx2, v)

(3.392)

l(v) = 0 (3.393)

We note that (3.391) is the desired integral form in which

(a) B(φ, v) is linear in v but not linear in φ since the differential operator isnon-linear

(b) l(v) is linear

(c) B(φ, v) is obviously not symmetric, i.e. B(φ, v) 6= B(v, φ)

Let there exist a functional I(φ) such that δI(φ) = 0 yields the integral form(3.390). Then in order for the integral form (3.390) to be VC, δ2I(φ) =δ(B(φ, v)− l(v)) must yield a unique extremum principle.

δ(B(φ, v)− l(v)) = δ(B(φ, v))

=(vdφ

dx+ φ

dv

dx− k d

2v

dx2, v)

> 0

= 0

< 0

∀φ ∈ Vn; ∀v ∈ Vn (3.394)

does not hold. Hence, the integral form (3.390) does not yield a uniqueextremum principle. That is, the Galerkin method for this BVP in whichthe differential operator is non-linear yields a VIC integral form.


In this method also, we begin with the integral form based on fundamen-tal lemma using a test function or a weight function w.∫

Ω

(Aφ− f)w dΩ = (Aφ− f, w)Ω = 0 (3.395)

or ∫Ω

(φdφ

dx− kd

2φ

dx2

)w dΩ = 0 (3.396)


orB(φ,w) = l(w) (3.397)

where

B(φ,w) =

∫Ω

(φdφ

dx− kd

2φ2

dx2

)w dΩ (3.398)

l(w) = 0 (3.399)

and w is a weight function that satisfies the homogeneous form of the bound-ary conditions on φ and w 6= δφ. We note that (3.397) is the desired integralform in which

(a) B(φ,w) in linear in w but not in φ

(b) l(w) is linear

(c) B(φ,w) is not symmetric, B(φ,w) 6= B(w, φ)


δ2I(φ) = δ(B(φ,w)− l(w)) = δ(B(φ,w))

=

∫Ω

(vdφ

dx+ φ

dv

dx− k d

2v

dx2

)w dΩ (3.400)

Obviously

δ2I(φ)

> 0

= 0

< 0

∀φ ∈ Vn, ∀w ∈ V (3.401)

does not hold. Hence, the integral form in (3.396) is VIC. Thus, the Petrov–Galerkin method for this BVP yields a VIC integral form when the operatoris non-linear.


Based on fundamental Lemma∫Ω

(Aφ− f)v dΩ = (Aφ− f, v) = 0 (3.402)

or ∫Ω

(φdφ

dx− kd

2φ

dx2

)v dΩ = 0 (3.403)


or ∫Ω

(φdφ

dxv − vkd

2φ

dx2

)dΩ = 0 (3.404)

where v = δφ is a test function and, hence must satisfy the homogeneousform of the boundary conditions on φ.

Step 1: There is nothing to be gained by transferring differentiation from φto φv in the first term of (3.404) due to the fact that the non-symmetricnature of this term remains unaltered. In the second term, we transfer oneorder of differentiation from φ to kv and since k is constant and f = 0, wecan write∫

Ω

(Aφ− f)v dΩ =

∫Ω

(φdφ

dxv + k

dv

dx

dφ

dx

)dΩ−

[v(kdφ

dx

)]1

0

= 0 (3.405)

In this case the concomitant 〈Aφ, v〉Γ is given by

〈Aφ, v〉Γ = −[v(kdφ

dx

)]1

0


Using the definition of primary and second variables and essential and nat-ural boundary conditions, we have

φ is PV

kdφ

dxis SV

φ = φ specified on some boundary Γ1 is EBC

kdφ

dx= q specified on some boundary Γ2 is NBC

(3.406)

Using (3.406), we can identify the BCs of the BVP (3.385) in EBC and/orNBC catagories. Both boundary conditions in (3.385) are essential. Thereare no natural boundary conditions in this BVP.


From boundary conditions we have

φ(0) = 1⇒ v(0) = 0

φ(1) = 0⇒ v(1) = 0(3.407)


Substituting from (3.407) into (3.405) after expanding the concomitant, weobtain ∫

Ω

(Aφ− f)v dΩ =

∫Ω

(φdφ

dxv + k

dv

dx

dφ

dx

)dΩ = 0 (3.408)

orB(φ, v) = l(v) (3.409)

where

B(φ, v) =

∫Ω

(φdφ

dxv + k

dv

dx

dφ

dx

)dΩ (3.410)

l(v) = 0 (3.411)

Integral form (3.409) is the desired weak form resulting from the integralform (3.402). We note the following properties of bilinear and linear forms.

(a) B(·, ·) is linear in v but not linear in φ(b) B(·, ·) is not symmetric, B(φ, v) 6= B(v, φ)

These properties are a consequence of the fact that the differential operatoris non-linear• l(·) is linear in v

φ ∈ Vn ⊂ H2(Ω), v ∈ Vn ⊂ H2(Ω) are minimally conforming if the integralin (3.410) is to be Riemann, but for the BVP H3(Ω) is minimally conformingspace.


Let there exist a functional I(φ) such that δI(φ) = 0 yields the weak form(3.409). Then, in order to show the variational consistency of the integralform in (3.409), we need to show that δ2I(φ) yields a unique extremumprinciple.

δ2I(φ) = δ(δI(φ)) = δ(B(φ, v)− l(v)) = δ(B(φ, v)) (3.412)

or

δ2I(φ) =

∫Ω

(v2dφ

dx+ φ

dv

dxv + k

(dvdx

)2)dΩ

> 0

= 0

< 0

∀φ ∈ Vn, ∀v ∈ Vn

(3.413)does not hold. Hence, δ2I(φ) in (3.413) does not yield a unique extremumprinciple. That is, the integral form resulting from the Galerkin methodwith weak form is VIC for this BVP in which the differential operator isnon-linear.





I(φn) = (E,E) =(φndφndx− kd

2φndx2

, φndφndx− kd

2φndx2

)(3.414)

(ii) Necessary condition: If I(φn) is differentiable in φn then δI(φn) = 0 isnecessary condition for an extremum of I(φn).

δI(φn) = 2(E, δE)Ω = 2g(φn) = 0 or g(φn) = 0 (3.415)


E = φndφndx− kd

2φndx2

(3.416)

δE = vdφndx

+ φndv

dx− k d

2v

dx2, v = δφn (3.417)

Hence, g(φn) = 0 can be written as

g(φn) = (E, δE)

=(φndφndx− kd

2φndx2

, vdφndx

+ φndv

dx− k d

2v

dx2

)= 0

(3.418)

or

B(φn, v) = l(v) (3.419)

where

B(φn, v) = (E, δE) (3.420)

l(v) = 0 (3.421)

Therefore

B(φn, v)− l(v) = B(φn, v) = g(φn) = 0 (3.422)

is the desired integral form resulting from δI(φn) = 0 in which

(a) B(φn, v) is linear in v but not linear in φn(b) B(φn, v) is not symmetric symmetric, B(φn, v) 6= B(v, φn)

(c) l(v) = 0



δ2I(φn) ∼= 2(δE, δE) (3.423)

Thus

δ2I(φn) = 2(vdφndx

+ φndv

dx− k d

2v

dx2, vdφndx

+ φndv

dx− k d

2v

dx2

)(3.424)

Clearlyδ2I(φn) > 0 ∀φn ∈ Vn, v ∈ Vn (3.425)

Hence, δ2I(φn) > 0 yields a unique extremum principle. Thus, the inte-gral form (3.419) is VC for this BVP in which the differential operatoris non-linear. The solution procedure for finding a φn that satisfiesg(φn) = 0 follows the usual procedure, i.e. Newton’s linear methodwith line search.

Example 3.8. Consider the 2D steady-state Burgers equation

φ∂φ

∂x+ φ

∂φ

∂y− k∂

2φ

∂x2− k∂

2φ

∂y2= f(x, y) in Ω ⊂ R2 (3.426)

with

φ(0) = φ on Γ1

k∂φ

∂xnx + k

∂φ

∂yny = q on Γ2

(3.427)

where Γ = Γ1 ∪ Γ2 is such that Ω = Ω ∪ Γ, k is the diffusion coefficient andφ and q are known data. In this case, the differential operator A and f aregiven by

A = φ∂

∂x+ φ

∂

∂y− k ∂

2

∂x2− k ∂

2

∂y2, f = f(x, y) (3.428)

Thus, we can writeAφ− f = 0 in Ω (3.429)

It is straight forward to show that A is non-linear and, hence, A is notsymmetric either. We now consider various methods of approximation.

The Galerkin method

In this method we consider the integral form based on fundamentallemma using a test function v∫

Ω

(Aφ− f)v dΩ = (Aφ− f, v) = 0 (3.430)


in which v = δφ and, hence satisfies the homogeneous form of the boundaryconditions on φ or(

φ∂φ

∂x+ φ

∂φ

∂y− k∂

2φ

∂x2− k∂

2φ

∂y2− f, v

)= 0 (3.431)

or

B(φ, v) = l(v) (3.432)

where

B(φ, v) =

(φ∂φ

∂x+ φ

∂φ

∂y− k∂

2φ

∂x2− k∂

2φ

∂y2, v

)(3.433)

l(v) = (f, v) (3.434)

(a) B(φ, v) is linear in v but not in φ


(c) l(v) is linear

In this case (3.432) is the desired integral form. Let there exist a functionalI(φ) such that δI(φ) = 0 yields the integral form (3.431). Then in order forthe integral form (3.431) to be VC, we must show that δ2I(φ) = δ(B(φ, v)−l(v))) must yield a unique extremum principle.

δ2I(φ) =

(v∂φ

∂x+ φ

∂v

∂x+ v

∂φ

∂y+ φ

∂v

∂y− k ∂

2v

∂x2− k∂

2v

∂y2, v

)(3.435)

Obviously

B(v, v)

> 0

= 0

< 0

∀φ ∈ Vn, v ∈ Vn (3.436)



In this method also, we begin with the integral form based on fundamen-tal lemma using test function or weight function w∫

Ω

(Aφ− f)w dΩ = (Aφ− f, w) = 0 (3.437)


where w is a weight function that satisfies the homogeneous form of theboundary conditions on φ and is such that w 6= δφ. Substituting for A from(3.428) (

φ∂φ

∂x+ φ

∂φ

∂y− k∂

2φ

∂x2− k∂

2φ

∂y2− f, w

)= 0 (3.438)

orB(φ,w) = l(w) (3.439)

where

B(φ,w) =

(φ∂φ

∂x+ φ

∂φ

∂y− k∂

2φ

∂x2− k∂

2φ

∂y2, w

)(3.440)

l(w) = (f, v) (3.441)

(a) B(φ,w) is linear in v but not in φ

(b) B(φ,w) is not symmetric, B(φ,w) 6= B(w, φ)

(c) l(w) is linear in v

The integral form in (3.439) is the desired integral form in this case. Letthere exist a functional I(φ) such that δI(φ) = 0 yields the integral form(3.438). Then we must show that δ2I(φ) yields a unique extremum principlein order for the integral form (3.438) to be VC.

δ2I(φ) = δ(B(φ,w)− l(w)) = δ(B(φ,w)) (3.442)

or

δ2I(φ) =

(v∂φ

∂x+ φ

∂v

∂x+ v

∂φ

∂y+ φ

∂v

∂y− k ∂

2v

∂x2− k∂

2v

∂y2, w

)(3.443)

Clearly

δ2I(φ)

> 0

= 0

< 0

∀φ, v ∈ Vn, w ∈ V (3.444)



Based on fundamental lemma, we can write the following using test func-tion v ∫

Ω

(Aφ− f)v dΩ = (Aφ− f, v) = 0 (3.445)


where v = δφ is a test function. Hence, v must satisfy the homogeneousform of the boundary conditions. Substituting for A from (3.428)∫

Ω

[(φ∂φ

∂x+ φ

∂φ

∂y

)v − k

(∂2φ

∂x2+∂2φ

∂y2

)v

]dΩ = (f, v) (3.446)

Step 1: There is nothing to be gained by transferring differentiation from φto v in the first two terms of (3.446) since the non-symmetric nature of theseterms would remain unchanged. In the last two terms we transfer one orderof differentiation from φ to kv and since k is constant we can write∫

Ω

((φ∂φ

∂x+ φ

∂φ

∂y

)v + k

(∂v∂x

∂φ

∂x+∂v

∂y

∂φ

∂y

))dΩ

−∮Γ

v(k∂φ

∂xnx + k

∂φ

∂yny

)dΓ = (f, v) (3.447)

The concomitant 〈Aφ, v〉Γ in (3.447) is given by

〈Aφ, v〉Γ = −∮Γ

v(k∂φ

∂xnx + k

∂φ

∂yny

)dΓ

Step 2: Determination of PVs, SVs, EBCs and NBCs using the concomitant:

Using the definitions of primary and secondary variables and essential andnatural boundary conditions, we have the following from the concomitant.

φ is PV(k∂φ

∂xnx + k

∂φ

∂yny

)is SV

φ = φ specified on some boundary Γ1 is EBC(k∂φ

∂xnx + k

∂φ

∂yny

)= q specified on some boundary Γ2 is NBC

(3.448)

Using (3.448) we can identify BCs (3.427) of the BVP (3.426) in EBC and/orNBC categories.


k∂φ

∂xnx + k

∂φ


Thus, the boundary conditions in (3.427) contain both essential and naturalboundary conditions.



Since Γ = Γ1∪Γ2, we can split the integral over Γ into integrals over Γ1 andΓ2 in (3.447)∫

Ω

((φ∂φ

∂x+ φ

∂φ

∂y

)v + k

(∂v∂x

∂φ

∂x+∂v

∂y

∂φ

∂y

))dΩnotag (3.449)

−∫Γ1

v(k∂φ

∂xnx + k

∂φ

∂yny

)dΓ−

∫Γ2

v(k∂φ

∂xnx + k

∂φ

∂yny

)dΓ = (f, v)

From boundary conditions (3.427) we have

φ = φ0 ⇒ v = 0 on Γ1

k∂φ

∂xnx + k

∂φ

∂yny = q ⇒ k

∂v

∂xnx + k

∂v

∂yny = 0 on Γ2

(3.450)


[(φ∂φ

∂x+ φ

∂v

∂y

)v + k

(∂v∂x

∂φ

∂x+∂v

∂y

∂φ

∂y

)]dΩ =

∫Γ2

vq dΓ + (f, v) (3.451)

or

B(φ, v) = l(v) (3.452)

where

B(φ, v) =

∫Ω

[(φ∂φ

∂x+ φ

∂φ

∂y

)v + k

(∂v∂x

∂φ

∂x+∂v

∂y

∂φ

∂y

)]dΩ (3.453)

l(v) =

∫Γ2

vq dΓ + (f, v) (3.454)

The integral form (3.452) is the desired weak form resulting from (3.445).

(a) B(φ, v) is linear in v but not in φ


(c) l(v) is linear in v

These properties are due to the fact that the differential operator A is non-linear. In (3.452), φ ∈ Vn ⊂ H2(Ω) and v ∈ Vn ⊂ H2(Ω) are minimally ifthe integrals in (3.452) are to be Riemann.



Let there exist a functional I(φ) such that δI(φ) = 0 yields the integral form(3.452). Then in order to show the variational consistency of the integralform (3.452) we need to show that δ2I(φ) yields a unique extremum principle.

δ2I(φ) = δ(B(φ, v)− l(v)) = δ(B(φ, v)) (3.455)

or

δ2I(φ) =

∫Ω

(v∂φ

∂x+ φ

∂v

∂x+ v

∂φ

∂y+ φ

∂v

∂y+ k(∂v∂x

∂v

∂x+∂v

∂y

∂v

∂y

))dΩ (3.456)

Clearly,

δ2I(φ)

> 0

= 0

< 0

∀φ ∈ Vn, ∀v ∈ Vn (3.457)

is not possible. Hence, δ2I(φ) in (3.456) does not yield a unique extremumprinciple. Thus, the integral form resulting from the Galerkin method withweak form is VIC for this BVP in which the differential operator is non-linear.


Let φn be an approximation of φ in Ω. Then Aφn − f = E in Ω is theresidual function where

E = φn∂φn∂x

+ φn∂φn∂y− k∂

2φn∂x2

− k∂2φn∂y2

(3.458)


I(φn) = (E,E) (3.459)

(ii) Necessary condition: If I(φn) is differentiable in φn, then δ(I(φn)) = 0is necessary condition for an extremum of I(φn):

δI(φn) = 2(E, δE) = 2g(φn) = 0 or g(φn) = 0 (3.460)

where

δE = v∂φn∂x

+φn∂v

∂x+v

∂φn∂y

+φn∂v

∂y−k ∂

2v

∂x2−k∂

2v

∂y2, v = δφn (3.461)

3.6. NUMERICAL EXAMPLES 153

The necessary condition gives the desired integral form. We can alsowrite

1

2δI(φn) =

(φn∂φn∂x


2φn∂x2

− k∂2φn∂y2

− f,

v∂φn∂x

+ φn∂v

∂x+ v

∂φn∂y

+ φn∂v

∂y− k ∂

2v

∂x2− k∂

2v

∂y2

)= g(φn) = 0 (3.462)

or

B(φn, v) = l(v) (3.463)

where

B(φn, v) =

(φn∂φn∂x


2φn∂x2

− k∂2φ

∂y2,

v∂φn∂x

+ φn∂v

∂x+ v

∂φn∂y

+ φn∂v

∂y− k ∂

2v

∂x2− k∂

2v

∂y2

)(3.464)

l(v) =

(f, v

∂φn∂x

+ φn∂v

∂x+ v

∂φn∂y

+ φn∂v

∂y− k ∂

2v

∂x2− k∂

2v

∂y2

)(3.465)


δ2I(φn) ∼= 2(δE, δE) = 2

∫Ω

(δE)2 dΩ > 0 ∀v ∈ Vn, φn ∈ Vn (3.466)

Hence, δ2I(φn) > 0 yields a unique extremum principle and so theintegral form (3.463) is VC. We find a φn using Newton’s linear methodwith line search that satisfies g(φn) = 0.

3.6 Numerical Examples

In this section we present numerical solutions for a variety of 1D and 2Dboundary value problems described by self-adjoint, non-self-adjoint and non-linear differential operators using the Galerkin method (GM), the Petrov–Galerkin method (PGM), the weighted-residual method (WRM), the Galerkinmethod with weak form (GM/WF), and the least-squares processes (LSP).First we consider some general guidelines (also discussed in earlier sections)that are helpful in considering various methods of approximations.


In all methods of approximation, we always approximate the theoreticalsolution φ by φn, an n-parameter approximation of φ given by

φn = N0(x) +n∑i=1

CiNi(x) (3.467)

where x could be x, (x, y), or (x, y, z) depending upon whether the BVP is1D, 2D, or 3D, respectively; N0(x) and Ni(x) are known functions. Guide-lines for determining N0(x) and Ni(x) for various methods of approximationhave already been discussed.

Convergence of the approximation φn to the theoretical solution φEstablishing the convergence of φn to φ and the rate at which φn con-

verges is an important and essential aspect of the computations. However,the convergence of φn to φ implies that we must show that the error betweenφ and φn in some norm diminishes at some rate as n is increased.

In the simple cases, it may be possible to determine the theoretical so-lution φ, but in general this may not be possible. We follow an alternateapproach described in the following to assess error in the computed solutionφn.

Let φn be an n-parameter solution of a BVP. Then using the error func-tion E = Aφ − f ∀x ∈ Ω we define the residual functional and I = (E,E),the square of the L2-norm of the residual E over the domain of definition Ω.We note that the residual functional I is quadratic in E and, hence, is alwayspositive and its minimum is zero. When φn → φ, the E → 0 and I → 0.Thus, the proximity of I to zero is an absolute measure of the accuracy ofφn. This approach does not require the theoretical solution φ but clearlyestablishes the accuracy of φn. How quickly or slowly φn approaches φ as nis increased in different methods of approximation can be easily establishedby studying the behavior of I or

√I versus n. We follow this approach in

the numerical studies to determine the convergence rates and the superiorityof one method of approximation over others.

Example 3.9. Consider the 1D diffusion equation

−d2φ

dx2− φ+ x2 = 0 ∀x ∈ Ω = (0, 1) ⊂ R1 (3.468)

withφ(0) = 0 and φ(1) = 0 (3.469)

The Galerkin method

Following the details presented earlier, for this case we have the integralform

B(φ, v) = l(v) (3.470)


where

B(φ, v) =(d2φ

dx2+ φ, v

)(3.471)

l(v) = (x2, v) (3.472)

and v = δφ. Let

φn = N0(x) +n∑x=1

CiNi(x) (3.473)

where φn is the approximation of φ and v = δφn = Nj(x) (j = 1, 2, . . . , n).In this case the differential operator is self-adjoint. Substituting φn in placeof φ and Nj(x) in place of v in (3.471) and (3.472) and regrouping the termswe obtain the following from (3.470),

n∑i=1

(d2Ni

dx2+Ni, Nj

)Ci = (x2, Nj)−

(d2N0

dx2+N0, Nj

), j = 1, 2, . . . , n

(3.474)Equations (3.474) can be written in matrix form as

[K]C = F (3.475)

in which Kij of [K] and Fi of F are given by

Kij =(d2Nj

dx2+Nj , Ni

), i = 1, 2, . . . , n, j = 1, 2, . . . , n (3.476)

Fi = (x2, Ni)−(d2N0

dx2+N0, Ni

), i = 1, 2, . . . , n (3.477)

Obviously, Kij 6= Kji, i.e. [K] is not symmetric. This is a consequence ofthe VIC integral.

We have N0(x) = 0 due to the fact that both boundary conditions (3.469)are homogeneous. We choose Ni(x) such that each Ni(x) satisfies the homo-geneous form of all boundary conditions of the problem. Therefore

Ni(0) = Ni(1) = 0, i = 1, 2, . . . , n must hold (3.478)

In choosing Ni(x), we may proceed as follows. Let N1(x) = a0+a1x in whicha0 and a1 are evaluated using N1(0) = 0 and N1(1) = 0 which gives a0 = 0and a1 = 0. So this choice is of no consequence. Let N1(x) = a0+a1x+a2x

2.Then using N1(0) = 0 and N1(1) = 0 we have a0 = 0 and a1 +a2 = 0. Thus,we can choose either a1 = −a2 or a2 = −a1, which yields

N1(x) = a2x(x− 1) or N1(x) = a1x(1− x) (3.479)


From (3.479) we note that the function in one choice i.e. N1(x) is the negativeof the function in other choice. Furthermore a1, a2, or −a1,−a2 can all beabsorbed in the constant C1 in the approximation φn. Thus, we can chooseeither of the two forms in (3.479). Consider

N1(x) = x(1− x) (3.480)

Next we consider N2(x). First, we note that a general expression for N2(x)must contain an x3 term (based on (3.480)) and can only have three arbitraryconstants due to the fact that: (1) we have only two boundary conditions,i.e. N2(0) = 0 and N2(1) = 0, to evaluate the constants and (2) the determi-nation of N2(x) within an arbitrary constant is satisfactory due to the factthat the constant gets absorbed in C2 in the approximation φn. Thus, wehave two choices:

N2(x) = a0 + a1x+ a3x3 (3.481)

orN2(x) = a0 + a2x

2 + a3x3 (3.482)

Using (3.481) with N2(0) = 0 and N2(1) = 0 we obtain

a0 = 0

a1 = −a3 ⇒ N2(x) = a3x(x2 − 1)

a3 = −a1 ⇒ N2(x) = a1x(1− x2)

(3.483)

or using (3.482) with N2(0) = 0 and N2(1) = 0 we obtain

a0 = 0

a2 = −a3 ⇒ N2(x) = a3x(x2 − 1)

a3 = −a2 ⇒ N2(x) = a2x(1− x2)

(3.484)

Since the polynomial terms in the two choices of N2(x) in (3.483) are thenegative of each other and the sign and the constants a3 or a1 can be ab-sorbed in C2 in (3.473), either of the two choices are satisfactory. We choosethe following for N2(x):

N2(x) = x(1− x2) (3.485)

orN2(x) = x2(1− x) (3.486)

The constants a1 and a2 in (3.483) and (3.486) are absorbed in C2 in theapproximation φn. Thus, we see that for a choice of N1(x) in (3.480) thereare two choices of N2(x) ((3.485) and (3.486)). N1(x) is a polynomial ofup to degree two in x whereas both choices of N2(x) are polynomials of upto degree three in x. A continuation of this procedure used for establishing


N1(x) and N2(x) to determine Ni(x), i = 3, . . . , n is obviously difficult dueto the fact that an expression for Ni(x) must contain terms a0 and aix

i butcan only have one more term of degree lower than xi. This is due to thefact that we only have two conditions, namely Ni(0) = 0 and Ni(1) = 0, toevaluate Ni(x) within an arbitrary constant. Thus, there are many choicespossible. This simple exercise points out the difficulty in establishing Ni(x)in the classical methods of approximation. At this stage, we abandon thisapproach but pursue a slightly different line of thinking. We note that N1(x)and N2(x) are algebraic polynomials of up to highest degrees of two andthree. Thus, by induction, we expect Ni(x) to be an algebraic polynomialof highest degree i in x. Hence, using induction we can write

Choice 1: Ni(x) = x(1− xi), i = 1, 2, . . . , n (3.487)

or

Choice 2: Ni(x) = xi(1− x), i = 1, 2, . . . , n (3.488)

with N0(x) = 0. Thus, in this case there are at least two possible choices ofNi(x) (i = 1, 2, . . . , n). Other possible choices may also be admissible. Forexample, Ni(x) = x(1 − x)i is also admissible, though at this stage we donot have a way to derive these directly as we have done in case of (3.487)and (3.488).

One parameter approximation: For this case choices 1 and 2 are the same,thus we can use

N0(x) = 0

N1(x) = x(1− x), v = N1(x)

The computed [K], F, and C are

[K] = [−0.3], F = 0.05, C = −0.166667. (3.489)

Hence

φn = (−0.166667)x(1− x)

I = (E,E) = (Aφn − f,Aφn − f) = 0.8796296× 10−1

Two parameter approximation: Here, we choose

N0(x) = 0

N1(x) = x(1− x)

N2(x) = x(1− x2) : choice 1

N2(x) = x2(1− x) : choice 2


with v = Nj(x) (j = 1, 2).

Choice 1:

[K] =

[−0.30 −0.4500−0.45 −0.7238

], F =

0.05000.0833

and

C =

0.08943−0.17073

Hence

φn = (0.08943)x(1− x) + (−0.17073)x(1− x2)

I = (E,E) = (Aφn − f,Aφn − f) = 0.4644502× 10−2

Choice 2:

[K] =

[−0.30 −0.1500−0.15 −0.1238

], F =

0.05000.0333

and

C =

−0.08130−0.17073

Hence

φn = (−0.081301)x(1− x) + (−0.17073)x2(1− x)

I = (E,E) = (Aφn − f,Aφn − f) = 0.4644501× 10−2

1e-25

1e-20

1e-15

1e-10

1e-05

1e+00

1 2 3 4 5 6 7 8 9

Resi

du

al

fun

cti

on

al I

Number of terms n

GM, GM/WF, PGMLSP

Figure 3.1: Example 3.9: Residual functional I versus the number of terms n


-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0 0.2 0.4 0.6 0.8 1

So

luti

on

φn

Distance x

GM, PGM, GM/WF

n = 1n = 2n =3-6

Figure 3.2: Example 3.9: Solution φn versus distance x

Remarks.

(1) Residual functional I decreases by one order of magnitude when n isincreased from one to two, clearly confirming the much improved ap-proximation φn for n = 2.

(2) For both choices of N2(x), identical values of I confirm identical φn ∀x ∈Ω = (0, 1). Hence, either choice 1 or choice 2 can be used for highervalues of n. We consider choice 1 and compute solutions for values of nup to 9.

(3) With progressively increasing n, we expect to obtain progressively re-duced I values which would indicate progressively improved approxima-tion φn.

(4) Figure 3.1 shows a plot of residual functional I versus the number ofterms n. I reduces from O(10−1) to O(10−16) for n = 1 to n = 9. Almostconstant slope of the I versus n for n up to 8 indicates no change in theconvergence rate of I with increasing n.

(5) Figure 3.2 shows plots of φn versus x for n = 1, . . . , 6. For n > 3, thesolution φn agrees quite well with the theoretical solution.

(6) From (3.476) we note that the coefficient matrix [K] is not symmetric.However, for both of these choices of Ni(x), calculated [K] is symmetricas evident for n = 2 (choices 1 and 2).



In this method we have

B(φ,w) = l(w) (3.490)

in which

B(φ,w) =(d2φ

dx2+ φ,w

)l(w) = (x2, w)

(3.491)

where w 6= δφ. Here, w must be zero where φ is specified and, hence, we canchoose w = Nj(x) (j = 1, 2, . . . , n). We approximate φ by φn given by

φn = N0(x) +

n∑i=1

CiNi(x) (3.492)

Clearly, φn must satisfy all BCs of the BVP. Thus, the N0(x) and Ni(x)established in the Galerkin method are applicable here as well. Secondly, thechoice of w = Nj(x) (j = 1, 2, . . . , n) is admissible as well. With the choiceof PGM and WRM are the same as the Galerkin method. For the choiceof w = Nj , different than Nj(x), the PGM or WRM will obviously yielddifferent φn compared to GM. However, the superiority of one choice overthe other can be established by examining I = (E,E) = (Aφn− f,Aφn− f)resulting from the different choices. Results shown in Figs. 3.1 and 3.2 arefor w = Nj(x) (j = 1, 2, . . . , n), same as in case of the Galerkin method.



B(φ, v) = l(v) (3.493)

in which

B(φ, v) =

∫Ω

(dφdx

dv

dx− φv

)dΩ (3.494)

l(v) = −(x2, v) (3.495)

where v = δφ. We choose

φn = N0(x) +

N∑i=1

CiNi(x), v = Nj(x), j = 1, 2, . . . , n (3.496)



n∑i=1

1∫0

(dNi

dx

dNj

dx−NiNj

)Ci dx

= −1∫

0

x2Nj dx−1∫

0

(dN0

dx

dNj

dx−N0Nj

)Ci dx, j = 1, 2, . . . , n (3.497)

Equation (3.497) can be written in the matrix form as

[K]C = F (3.498)


Kij =

1∫0

(dNj

dx

dNi

dx−NjNi

)dx, i, j = 1, 2, . . . , n (3.499)

Fi = −1∫

0

x2Ni dx−1∫

0

(dN0

dx

dNi

dx−N0Ni

)dx, i = 1, 2, . . . , n (3.500)

Clearly, Kij = Kji and, therefore, [K] is symmetric (a consequence of VCintegral form).

The functions N0(x) and Ni(x) determined in the Galerkin method areapplicable here also as both BCs are EBCs with v = Nj(x) (j = 1, 2, . . . , n).

One parameter approximation:

N0(x) = 0

N1(x) = x(1− x), v = N1(x)

The computed [K], F and C are

[K] = [0.3], F = −0.05, C = −0.166667

Hence, the approximation φn becomes

φn = (−0.166667)x(1− x)

I = (E,E) = (Aφn − f,Aφn − f) = 0.8796296× 10−1


N0(x) = 0

N1(x) = x(1− x)

N2(x) = x(1− x2) : choice 1

N2(x) = x2(1− x) : choice 2


with v = Nj(x) (j = 1, 2).

Choice 1:

[K] =

[0.30 0.45000.45 0.7328

], F =

−0.0500−0.0833

and

C =

0.08943−0.17073

Hence

φn = (0.089431)x(1− x) + (−0.17073)x(1− x2)

I = (E,E) = (Aφn − f,Aφn − f) = 0.4644502× 10−2

Choice 2:

[K] =

[0.30 0.15000.15 0.1238

], F =

−0.0500−0.0333

and

C =

−0.08130−0.17073

Hence

φn = (−0.0813)x(1− x) + (−0.17073)x2(1− x)

I = (E,E) = (Aφn − f,Aφn − f) = 0.4644501× 10−2

Remarks.


(2) For both choices of N2(x), identical values of I confirm identical φn ∀x ∈Ω = (0, 1) for both choices (can be seen in the graphs in Figs. 3.2 and3.1). Hence, for higher values of n we can use either choice. We considerchoice 1 and compute solutions for up to n = 9.

(3) With progressively increasing n, we obtain progressively reduced I valuesindicating progressively improved approximation φn.

(4) A comparison of [K] and f in this method with the Galerkin methodclearly confirms differences. The functional I for n = 1 and n = 2matches exactly with the Galerkin method for n = 1 and n = 2, con-firming that φn in both methods of approximation for n = 1 and n = 2are identical. This is also confirmed by identical values of Ci in thismethod and the Galerkin method for n = 1 and n = 2.

(5) Plots of the solution φn versus x and the residual functional I versus nare shown in Figs. 3.2 and 3.1.




B(φ, v) = l(v) (3.501)

with

B(φ, v) = (Aφ,Av) =(d2φ

dx2+ φ,

d2v

dx2+ v)

(3.502)

l(v) = (f, v) = (x2, v) (3.503)

with v = δφ. We choose

φn = N0(x) +

N∑i=1

CiNi(x), v = Nj(x), j = 1, 2, . . . , n (3.504)

Substituting φn from (3.504) into (3.501), we obtain

n∑i=1

(d2Ni

dx2+Ni,

d2Nj

dx2+Nj

)Ci dx

=(−d

2N0

dx2−N0 + x2,

d2Nj

dx2+Nj

), j = 1, 2, . . . , n (3.505)

Equation (3.505) can be written in matrix form as

[K] = C = F (3.506)

in which Kijof [K] and Fi of F are given by

Kij =(d2Nj

dx2+Nj ,

d2Ni

dx2+Ni

), i, j = 1, 2, . . . , n (3.507)

Fi =(−d

2N0

dx2−N0 + x2,

d2Ni

dx2+Ni

), i = 1, 2, . . . , n (3.508)

Clearly, Kij = Kji, therefore, [K] is symmetric (a consequence of the VCintegral form in the least-squares method).

The functions N0(x) and Ni(x) determined in the Galerkin method areapplicable here also with v = Nj(x) (j = 1, 2, . . . , n).


N0(x) = 0

N1(x) = x(1− x), v = N1(x)


[K] = [3.3667], F = −0.61667, C = −0.18317



φn = (−0.18317)x(1− x)

I = (E,E) = (Aφn − f,Aφn − f) = 0.8704621× 10−1


N0(x) = 0

N1(x) = x(1− x)

N2(x) = x(1− x2), choice 1

N2(x) = x2(1− x), choice 2

with v = Nj(x) (j = 1, 2).

Choice 1:

[K] =

[3.3667 5.0505.0500 10.476

], F =

−0.6167−1.4167

and

C =

0.07104−0.16947

Hence

φn = (0.017038)x(1− x) + (−0.16947)x(1− x2)

I = (E,E) = (Aφn − f,Aφn − f) = 0.37231298× 10−2

Choice 2:

[K] =

[3.3667 1.68331.6833 3.7429

], F =

−0.6167−0.8000

and

C =

−0.09843−0.16947

Hence

φn = (−0.098433)x(1− x) + (−0.16947)x2(1− x)

I = (E,E) = (Aφn − f,Aφn − f) = 0.37231298× 10−2

Remarks.


(2) For both choices of N2(x), identical values of I confirm identical φn ∀x ∈Ω = (0, 1) for both choices. For higher values of n it suffices to considerchoice 1.


(3) With progressively increasing n, we obtain progressively reduced I valuesindicating progressively improved approximation φn.

(4) For both n = 1 and n = 2, we note that least-squares method yieldslowest values of I compared to all other methods of approximation. Thisis of course no surprise due to the fact that least-squares processes arebased on minimization of I, whereas other methods of approximationare not. The significance of this aspect of least-squares processes is thatif our objective is to satisfy the GDEs as accurately as we desire (byincreasing n) in Ω, then a φn from least-squares method gives us thebest solution compared to all other methods of approximation.

(5) A plot of the residual functional I versus n is shown in Fig. 3.1. Weconfirm that least-squares method yields the lowest value of I for eachn compared to all other methods of approximation.

(6) Graphs of solution φn versus x for different values of n are shown inFig. 3.3. Beyond n = 3, the approximation φn agrees well with thetheoretical solution φ.

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0 0.2 0.4 0.6 0.8 1

Solu

tion φn

Distance x

LSM

n = 1n = 2n =3-6

Figure 3.3: Example 3.9, LSM: Solution φn versus distance x

Example 3.10. Consider the 1D diffusion equation

−d2φ

dx2− φ+ x2 = 0 ∀x ∈ Ω = (0, 1) ⊂ R1 (3.509)

with

φ(0) = 0 anddφ

dx

∣∣∣∣x=1

= 1 (3.510)


This boundary value problem is exactly the same as in example 17 exceptthat at x = 1 we have a Neumann boundary condition as opposed to Dirichletboundary condition in example 17. The main purpose of this example isto illustrate that a simple change in boundary conditions has a significantinfluence on the choice of functions N0(x) and Ni(x) (i = 1, 2, . . . , n).

The Galerkin method

The integral form remains the same as in the previous example.

B(φ, v) = l(v) (3.511)

where

B(φ, v) =(d2φ

dx2+ φ, v

)(3.512)

l(v) = (x2, v) (3.513)

Let

φn = N0(x) +

n∑x=1

CiNi(x) (3.514)

and

v = δφn = Nj(x), 1, . . . , n (3.515)

Substituting (3.514) and (3.515) into (3.511) and rearranging terms, we ob-tain

n∑i=1

(d2Ni

dx2+Ni, Nj

)Ci = (x2, Nj)−

(d2N0

dx2+N0, Nj

), j = 1, 2, . . . , n

(3.516)Equations (3.516) can be written in matrix form as

[K]C = F (3.517)


Kij =(d2Nj

dx2+Nj , Ni

); i, j = 1, 2, . . . , n (3.518)

Fi = (x2, Ni)−(d2N0

dx2+N0, Ni

), i = 1, 2, . . . , n (3.519)

Obviously, Kij 6= Kji, i.e. [K] is not symmetric. This is a consequence ofthe VIC integral.


The function N0(x) must satisfy all of the boundary conditions of the

BVP, N0(0) = 0, and dN0dx

∣∣∣x=1

= 1 as φn(0) = 0 and dφndx

∣∣∣x=1

= 1 must hold.

Thus, the choice ofN0(x) = x (3.520)

is admissible. Next, each Ni(x) must satisfy the homogeneous form of allboundary conditions. Therefore

Ni(0) = 0; i = 1, 2, . . . , n

dNi

dx

∣∣∣∣x=1

= 0, i = 1, 2, . . . , n(3.521)

Following the procedure described in the previous example, we find that ifwe choose

N1(x) = a0 + a1x (3.522)

then both a0 and a1 are zero due to the fact that the two boundary conditionsto be used to evaluate a0 and a1 in (3.521) are homogeneous. If we choose

N1(x) = a0 + a1x+ a2x2 (3.523)

then following the previous example and by using (3.521) we obtain

N1(x) = x(2− x) (3.524)

For N2(x) we have two possible choices

N2(x) = a0 + a1x+ a3x3 (3.525)

which gives

N2(x) = x(1− 1

3x2) (3.526)

andN2(x) = a0 + a2x

2 + a3x3 (3.527)

which gives

N2(x) = x2(1− 2

3x) (3.528)

Using (3.524), (3.526), (3.528) and (3.520) we can write

N0(x) = x (3.529)

N1(x) = x(2− x) (3.530)

Ni(x) = x(

1− i− 1

i+ 1xi), i = 2, 3, . . . , n choice I (3.531)

Ni(x) = xi(

1− i

i+ 1x), i = 2, 3, . . . , n choice II (3.532)


We can use either of the expressions for Ni(x). We choose the first choice ofNi(x) given by (3.531) in the numerical studies. We note that as n → ∞,(3.531) and (3.532) both yields algebraic polynomials of degree infinity.

One parameter approximation: Using

N0(x) = x

N1(x) = x(2− x), v = N1(x)


[K] = [−0.80], F = 0.11667, C = −0.14583

Hence

φn = (−0.14583)x(2− x)

I = (E,E) = (Aφn − f,Aφn − f) = 0.9837963× 10−2


N0(x) = x

N1(x) = x(2− x)

N2(x) = x(1− 1

3x2), choice I (3.531)

with v = Nj(x) (j = 1, 2). The computed [K], F and C are

[K] =

[−0.8000 0.18890.1889 −0.9206× 10−1

], F =

0.11667

−0.27778× 10−1

and

C =

−0.14468

0.48769× 10−2

Hence

φn = x+ (−0.14468)x(2− x) + (0.48769× 10−2)x(1− 1

3x2)

I = (E,E) = (Aφn − f,Aφn − f) = 0.9272824× 10−2

Figure 3.4 shows plots of the solution φn versus x for different values ofn. It appears that for all values of n the computed solution φn is in goodagreement with the theoretical solution beyond n = 5. This can also beconfirmed by the graph of I versus n shown in Fig. 3.5. I is of the order of10−10 for n = 5 conforms extremely good accuracy of φn.


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 0.2 0.4 0.6 0.8 1

So

luti

on

φn

Distance x

GM

n = 1-6

Figure 3.4: Example 3.10, GM: Solution φn versus distance x

1e-25

1e-20

1e-15

1e-10

1e-05

1e+00

1 2 3 4 5 6 7 8 9

Resi

du

al

fun

cti

on

al I

Number of terms n

GMGM/WF

PGMLSP

Figure 3.5: Example 3.10: Residual functional I versus number of terms n


The choice of N0(x) and Ni(x) remain the same as in the Galerkinmethod. However, we may choose v = w = Nj (j = 1, 2, . . . , n) is theweight function, independent of the basis functions. Based on fundamentalLemma, w must be zero where φ is specified, hence the only requirement in


the choice of w = Nj(x) (j = 1, . . . , n) is that

Nj(0) = 0, j = 1, 2, . . . , n (3.533)

Thus

Nj(x) = xj , j = 1, 2, . . . , n (3.534)

is an admissible choice. The details of the integral forms are as follows.

B(φ,w) = l(w) (3.535)

in which

B(φ,w) =(d2φ

dx2+ φ,w

)(3.536)

l(w) = (x2, w) (3.537)

w 6= δφ = Nj(x) = xj , j = 1, 2, . . . , n (3.538)

φn = N0(x) +n∑i=1

CiNi(x) (3.539)

Substituting (3.538) and (3.539) into (3.536) and (3.537) we obtain

n∑i=1

(d2Ni

dx2+Ni, Nj

)Ci = (x2, Nj)−

(d2N0

dx2+N0, Nj

), j = 1, 2, . . . , n

(3.540)Equations (3.540) can be written in matrix form

[K] = C = F (3.541)


Kij =(d2Nj

dx2+Nj , Ni

); i, j = 1, 2, . . . , n (3.542)

Fi = (x2, Ni)−(d2N0

dx2+N0, Ni

), i = 1, 2, . . . , n (3.543)


N0(x) = x

N1(x) = x(2− x), N1(x) = x


[K] = [0.58333], F = −0.083333, C = −0.14286



φn = x+ (−0.14286)x(2− x)

I = (E,E) = (Aφn − f,Aφn − f) = 0.95238095× 10−2


N0(x) = x

N1(x) = x(2− x)

N2(x) = x(1− 1

3x2), choice I (3.531)

with w = Nj(x) (j = 1, 2), N1(x) = x, and N2(x) = x2. The computed [K],F, and C are

[K] =

[0.58333 −0.216670.36667 −0.24444

], F =

−0.8333× 10−1

−0.0500

and

C =

−0.15103

−0.21994× 10−1

Hence

φn = (−0.15103)x(2− x) + (−0.21994× 10−1)x(1− 1

3x2)

I = (E,E) = (Aφn − f,Aφn − f) = 0.1324869× 10−1

Plots of φn versus x for different values of n are shown in Fig. 3.6. Here also,all values of n produce reasonable approximations. Beyond n = 5, values ofI of the order of 10−10 or lower (Fig. 3.5) confirm extremely good accuracyof φn.



B(φ, v) = l(v) (3.544)

in which

B(φ, v) =

∫Ω

(dφ

dx

dv

dx+ φv) dΩ (3.545)

l(v) = −(x2, v) + v|1 (3.546)


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 0.2 0.4 0.6 0.8 1

Solu

tion φn

Distance x

PGM

n = 1-6

Figure 3.6: Example 3.10, PGM: Solution φn versus distance x

where v = δφ. We choose

φn = N0(x) +

N∑i=1

CiNi(x), v = Nj(x); j = 1, 2, . . . , n (3.547)


n∑i=1

1∫0

(dNi

dx

dNj

dx−NiNj

)Ci dx = −(x2, Nj)+Nj(1), j = 1, 2, . . . , n (3.548)


[K]C = F (3.549)


Kij =

1∫0

(dNj

dx

dNi

dx−NjNi

)dx (3.550)

Fi = −1∫

0

x2Ni dx+Ni(1) (3.551)



In this method the boundary condition dφdx

∣∣∣x=1

= 1 is absorbed in the

weak form , hence we only have φ(0) = 0 left to be satisfied by φn. Sinceφ(0) = 0 is homogeneous, we have N0(x) = 0 and the choice Ni(x) = xi

(i = 1, 2, . . . , n) is admissible. Thus, in this method we have

N0(x) = 0 (3.552)

Ni(x) = xi, i = 1, 2, . . . , n (3.553)


N0(x) = 0

Ni(x) = xi, v = Nj(x), i, j = 1, 2, . . . , n


[K] = [0.66667], F = 0.75000, C = 1.1250


φn = (1.1250)x

I = (E,E) = (Aφn − f,Aφn − f) = 0.59375× 10−1


N0(x) = 0

N1(x) = x

N2(x) = x2

with v = Nj(x) (j = 1, 2). The computed [K], F, and C are

[K] =

[0.6667 0.75000.7500 1.1333

], F =

0.750000.80000

and

C =

1.2950−0.15108

Hence

φn = (1.2950)x+ (−0.15108)x2

I = (E,E) = (Aφn − f,Aφn − f) = 0.1055846× 10−1

Figure 3.7 shows plots of φn versus x for different values of n. For n > 2, thecomputed solutions are quite accurate. This can be confirmed by I versus ngraph shown in Fig. 3.5.


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 0.2 0.4 0.6 0.8 1

Solu

tion φn

Distance x

GM/WF

n = 1n =2-6

Figure 3.7: Example 3.10, GM/WF: Solution φn versus distance x



B(φ, v) = l(v) (3.554)

with

B(φ, v) = (Aφ,Av) =(d2φ

dx2+ φ,

d2v

dx2+ v)

(3.555)

l(v) = (f, v) = (x2, v) (3.556)


φn = N0(x) +

N∑i=1

CiNi(x), v = Nj(x), j = 1, 2, . . . , n (3.557)


n∑i=1

(d2Ni

dx2+Ni,

d2Nj

dx2+Nj

)Ci dx

=(−d

2N0

dx2−N0 + x2,

d2Nj

dx2+Nj

), j = 1, 2, . . . , n (3.558)


[K] = C = F (3.559)


in which Kij of [K] and Fi of f are given by

Kij =(d2Nj

dx2+Nj ,

d2Ni

dx2+Ni

), i, j = 1, 2, . . . , n (3.560)

Fi =(−d

2N0

dx2−N0 + x2,

d2Ni

dx2+Ni

), i = 1, 2, . . . , n (3.561)


The choice of N0(x) and Ni(x) (i = 1, 2, . . . , n) in this method is obvi-ously the same as in the Galerkin method due to same required BCs. Thus,we can choose

N0(x) = x

N1(x) = x(2− x)

Ni(x) = x(1− i− 1

i+ 1xi), i = 2, . . . , n choice I (see GM)

(3.562)


N0(x) = x

N1(x) = x(2− x), v = N1(x)


[K] = [1.8667], F = −0.21667, C = −0.11607.


φn = x+ (−0.11607)x(2− x)

I = (E,E) = (Aφn − f,Aφn − f) = 0.818145238× 10−2


N0(x) = x

N1(x) = x(2− x)

N2(x) = x(1− 1

3x2)


[K] =

[1.8667 0.52220.5222 1.1079

], F =

−0.21667−0.02778


and

C =

−0.12562

0.34140× 10−1

Hence

φn = x+ (−0.12565)x(2− x) + (0.34140× 10−1)x(1− 1

3x2)

I = (E,E) = (Aφn − f,Aφn − f) = 0.70634539× 10−2

Graphs of φn versus x for different values of n are shown in Fig. 3.8.I versus n is shown in Fig. 3.5. We note that I is lowest for least-squaresmethod for all values of n. This is of course no surprise as least-squaresprocesses are based on minimization of I, hence will produce the lowest Icompared to all other methods of approximation.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 0.2 0.4 0.6 0.8 1

Solu

tion φn

Distance x

LSM (LSP)

n = 1n = 2n =3-6


Example 3.11. Consider the 1D beam equation

d2

dx2

(EI

d2φ

dx2

)− q = 0, ∀x ∈ Ω = (0, L) ⊂ R1 (3.563)

with

φ(0) = 0,dφ

dx

∣∣∣∣x=0

= 0 (3.564)(EI

d2φ

dx2

)∣∣∣∣x=L

= ML,d

dx

(EI

d2φ

dx2

)∣∣∣∣x=L

= FL (3.565)



The operator is self-adjoint provided ML = 0 and FL = 0. In the fol-lowing, we consider an approximate solution φn using the Galerkin methodwith weak form. As shown earlier, the integral form (weak form) in this caseis variationally consistent. The weak form can be written as

B(φ, v) = l(v) (3.566)

in which

B(φ, v) =

L∫0

EId2v

dx2

d2φ

dx2dΩ (3.567)

l(v) =

L∫0

vq dx− v(L)FL +(dvdx

)∣∣∣∣x=L

ML (3.568)

Let φn be an approximation of φ given by

φn = N0(x) +N∑i=1

CiNi(x), v = Nj(x), j = 1, 2, . . . , n (3.569)

Substituting (3.569) into (3.566), (3.567) and (3.568) yields

n∑i=1

( L∫0

EId2Ni

dx2

d2Nj

dx2dx

)Ci = −

L∫0

EId2Nj

dx2

d2N0

dx2dx

+

L∫0

Nj(x)q dx−Nj(L)FL +(dNj

dx

)∣∣∣∣x=L

ML, j = 1, 2, . . . , n (3.570)


[K]C = F (3.571)


Kij =

L∫0

EId2Nj

dx2

d2Ni

dx2dx, i, j = 1, 2, . . . , n (3.572)

Fi = −L∫

0

EId2Ni

dx2

d2N0

dx2dx+

L∫0

Niq dx−Ni(L)FL +(dNj

dx

)∣∣∣∣x=L

ML

(3.573)



First, we note that boundary conditions (3.564) are essential boundaryconditions whereas (3.565) are natural boundary conditions. Hence, theboundary conditions in (3.565) are absorbed in the weak form (3.566) anddo not need to be considered in determining N0(x) and Ni(x). Secondly, theessential boundary conditions in (3.564) are homogeneous, hence

N0(x) = 0 (3.574)

and Ni(x) must be chosen such that

Ni(0) = 0 (3.575)(dNi

dx

)∣∣∣∣x=0

= 0 (3.576)

Therefore

Ni(x) = xi+1, i = 1, 2, . . . , n (3.577)

are admissible. For numerical studies, we choose L = 1, EI = 1, q = 1.0,FL = −1.0, and ML = −1.0.


N0(x) = 0

N1(x) = x2, v = N1(x)


[K] = [4.0], F = −0.66667, C = −0.16667


φn = (−0.16667)x2

I = (E,E) = (Aφn − f,Aφn − f), meaningless in this case


N0(x) = 0, N1(x) = x2, N2(x) = x3


[K] =

[4.0 6.06.0 12.0

], F =

−0.66667

−0.17500× 101


and

C =

0.20833−0.25000

Hence

φn = (0.20833)x2 + (−0.25000)x3

I = (E,E) = (Aφn − f,Aφn − f), meaningless in this case

Three parameter approximation: Here, we choose

N0(x) = 0, N1(x) = x2, N2(x) = x3, N3(x) = x4

with v = Nj(x) (j = 1, 2, 3). The computed [K], F, and C are

[K] =

4.0 6.0 8.06.0 12.0 18.08.0 18.0 28.8

, F =

−0.66667

−0.17500× 101

−0.28000× 101

and

C =

0.25000−0.33333

0.41667× 10−1

Hence

φn = (0.2500)x2 + (−0.33333)x3 + (0.416667× 10−1)x4

I = (E,E) = (Aφn − f,Aφn − f) = 0.2839899× 10−26

Remarks.

(1) For n = 1, 2, φn is not admissible in the BVP and, hence, computationof I is meaningless.

(2) When n = 3, I = O(10−26) indicating that n = 3 corresponds to the the-oretical solution of this BVP (this can be verified by integrating (3.563)and using (3.564) and (3.565) to find constants of integration).

(3) Graphs of φn versus x for n = 1, 2, 3 are shown in Fig. 3.9.

Example 3.12. Consider the 1D convection-diffusion equation

dφ

dx− 1

Pe

d2φ

dx2= 0 ∀x ∈ Ω = (0, 1) ⊂ R1 (3.578)

withφ(0) = 1 and φ(1) = 0 (3.579)

The operator is non-self-adjoint, hence all methods of approximation ex-cept least-squares processes yield VIC integral forms. In the following, weconsider the Galerkin method with weak form and least-squares method.


-0.18

-0.16

-0.14

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0 0.2 0.4 0.6 0.8 1

Solu

tion φn

Distance x

GM/WF

n = 1n = 2n = 3

Figure 3.9: Example 3.11: Solution φn versus distance x


The weak form is given by

B(φ, v) = l(v) (3.580)

in which

B(φ, v) =

1∫0

(dφ

dxv +

1

Pe

dφ

dx

dv

dx

)dΩ (3.581)

l(v) = 0 (3.582)

Let φn be an approximation of φ given by

φn = N0(x) +

N∑i=1

CiNi(x), v = Nj(x), j = 1, 2, . . . , n (3.583)


n∑i=1

( 1∫0

(dNi

dxNj +

1

Pe

dNi

dx

dNj

dx

)dx

)Ci

= −1∫

0

(dN0

dxNj +

1

Pe

dN0

dx

dNj

dx

)dx, j = 1, 2, . . . , n (3.584)



[K]C = F (3.585)


Kij =

1∫0

(dNj

dxNi +

1

Pe

dNj

dx

dNi

dx

)dx, i = 1, 2, . . . , n, j = 1, 2, . . . , n

(3.586)

Fi = −1∫

0

(dN0

dxNi +

1

Pe

dN0

dx

dNi

dx

)dx (3.587)

Clearly, Kij 6= Kji and, therefore, [K] is non-symmetric (a consequence ofVIC integral form). In the numerical studies we choose Pe = 10.

Since both boundary conditions are essential, φn(x) must satisfy both ofthem. Let N0(x) be such that N0(0) = 1 and N0(1) = 0, hence the choice

N0(x) = 1− x (3.588)

is admissible. We choose Ni(x) such that

Ni(0) = 0 and Ni(1) = 0 (3.589)

Therefore, we can choose

Ni(x) = xi(1− x), i = 1, 2, . . . , n (3.590)


N0(x) = 1− xN1(x) = x(1− x), v = N1(x)


[K] = [0.33333× 10−1], F = 0.16667, C = 5.0


φn = (1− x) + (5.0)x(1− x)

I = (E,E) = (Aφn − f,Aφn − f) = 0.833333× 101



N0(x) = 1− xN1(x) = x(1− x)

N2(x) = x2(1− x)

with v = Nj(x); j = 1, 2. The computed [K], F, and C are

[K] =

[0.033333 0.0666670.033333 0.080000

], F =

0.166670.25000

and

C =

−0.75000× 101

0.62500× 101

Hence

φn = (1− x) + (−0.75× 101)x(1− x) + (0.625× 101)x2(1− x)

I = (E,E) = (Aφn − f,Aφn − f) = 0.234375× 101

The three, five, ten, and higher term approximate solutions are calculatedas well. Figure 3.10 shows a plot of the solution φn versus x for differentvalues of n. For n > 10, the computed solution is quite accurate. This canbe confirmed by the I versus n graph shown in Fig. 3.11.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

0 0.2 0.4 0.6 0.8 1

So

luti

on

φn

Distance x

GM/WF

n = 1n = 2n = 3n =5,10

Figure 3.10: Example 3.12, GM/WF: Solution φn versus distance x


1e-07

1e-06

1e-05

1e-04

1e-03

1e-02

1e-01

1e+00

1e+01

1 2 3 4 5 6 7 8 9 10

Resi

du

al

fun

cti

on

al I

Number of terms n

GM/WF

Figure 3.11: Example 3.12, GM/WF: Residual functional I versus n


The integral form is given by

B(φ, v) = l(v) (3.591)

with

B(φ, v) =(dφdx− 1

Pe

d2φ

dx2,dv

dx− 1

Pe

d2v

dx2

)(3.592)

l(v) = 0 (3.593)


φn = N0(x) +

N∑i=1

CiNi(x), v = Nj(x), j = 1, 2, . . . , n (3.594)


n∑i=1

(dNi

dx− 1

Pe

d2Ni

dx2,dNj

dx− 1

Pe

d2Nj

dx2

)Ci =

(−dN0

dx+

1

Pe

d2N0

dx2,dNj

dx− 1

Pe

d2Nj

dx2

), j = 1, 2, . . . , n (3.595)

Equations (3.595) can be written in matrix form as

[K]C = F (3.596)



Kij =(dNj

dx− 1

Pe

d2Nj

dx2,dNi

dx− 1

Pe

d2Ni

dx2

), i = 1, 2, . . . , n; j = 1, 2, . . . , n

(3.597)

Fi =(−dN0

dx+

1

Pe

d2N0

dx2,dNi

dx− 1

Pe

d2Ni

dx2

), i = 1, 2, . . . , n (3.598)


The choice of N0(x) and Ni(x) (i = 1, 2, . . . , n) in the Galerkin methodholds here as well due to same required BCs. Therefore

N0(x) = 1− xNi(x) = xi(1− x), i = 1, 2, . . . , n

(3.599)


N0(x) = 1− xN1(x) = x(1− x), v = N1(x)


[K] = [0.37333], F = 0.20000, C = 0.53571.


φn = (1− x) + (0.53571)x(1− x)

I = (E,E) = (Aφn − f,Aφn − f) = 0.89285714


N0(x) = 1− xN1(x) = x(1− x)

N2(x) = x2(1− x)

with v = Nj(x) (j = 1, 2). The computed [K], F, and C are

[K] =

[0.37333 0.460000.46000 0.62000

], F =

0.200000.30000

and

C =

−0.70470

1.0067

3.7. SUMMARY 185

Hence

φn = (1− x) + (−0.70470)x(1− x) + (1.0067)x2(1− x)

I = (E,E) = (Aφn − f,Aφn − f) = 0.83892617

The three, five, ten and higher term approximations are computed forthis case as well. Graphs of φn versus x for different values of n are shown inFig. 3.12; I versus n is shown in Fig. 3.13. Once again, we note that I in least-squares method is always lower compared to the Galerkin method with weakform. For Pe = 10, the theoretical solution is sufficiently diffused and, hence,smooth. Even then, up to ten terms are required for reasonable accuracy. AsPe increases, a sharp front develops near x = 1. Computation of solutionsfor such cases using these classical methods will become prohibitive.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 0.2 0.4 0.6 0.8 1

So

luti

on

φn

Distance x

LSM (LSP)

n = 1n = 2n = 3n = 5n =10


3.7 Summary

In this chapter, we have considered classical integral methods of approx-imation: the Galerkin method, the Petrov–Galerkin method, the weighted-residual method, the Galerkin method with weak form, and the least-squaresprocess for boundary value problems containing self-adjoint, non-self-adjoint,and non-linear differential operators. For each method of approximation,formulation details are presented for each of the three types of differentialoperators including the determination of variational consistency or inconsis-tency of the resulting integral forms. It has been shown and proven that


1e-07

1e-06

1e-05

1e-04

1e-03

1e-02

1e-01

1e+00

1e+01

1 2 3 4 5 6 7 8 9 10

Resi

du

al

fun

cti

on

al I

Number of terms n

GM/WFLSP

Figure 3.13: Example 3.12, LSM: Error functional I versus n

(i) the integral forms resulting from the Galerkin method, and the Petrov–Galerkin method are always VIC, (ii) the integral form resulting from theGalerkin method with weak form is only variationally consistent when thedifferential operator is self-adjoint and when B(·, ·) in the weak form is sym-metric, (iii) the least-squares processes always yields variationally consistentintegral forms for self-adjoint and non-self-adjoint operators and (iv) least-squares processes also yield VC integral forms for non-linear operators ifin δ2I, δ2E is neglected and if the resulting non-linear algebraic equationsfrom δI = 0 are solved using Newton’s linear method (or Newton–Raphsonmethod).

It has been shown that VC integral forms yield computational processesin which the coefficient matrices are symmetric and positive-definite, hencehave real and positive eigenvalues and real bases i.e. real eigenvectors.Hence, the computations remain unconditionally stable and non-degenerate.The VIC integral forms, on the other hand, yield non-symmetric coefficientmatrices which are not always ensured to be positive-definite. Such coeffi-cient matrices may have partial or completely complex bases. Unconditionalstability of computations in such cases cannot always be ensured. One mustshow on a problem-by-problem basis whether the computations remain sta-ble. The LBB condition, Lax-Milgram theorem and Banach theorem aremeans of accomplishing this. A significant point to note here is that whenthe integral forms are VC, the LBB condition, Lax-Milgram theorem andBanach theorem are automatically satisfied.

3.7. SUMMARY 187

For each method of approximation, approximation spaces are discussedin conjunction with Riemann and Lebesgue measures. In section 3.5.1, avariety of boundary value problems are considered to present the details ofvarious methods of approximation for each of the three types of differentialoperators. In section 3.6, specific numerical studies are presented to illus-trate various details of approximation φn, computations of constants Ci inφn as well as the residual functional I as a measure of the accuracy of thecomputed solution. Form the material presented in section 3.6, it is ratherclear that a major difficulty in the classical methods of approximation isthe determination of N0(x) and Ni(x). Since their determination requiressatisfying some or all BCs of the BVP (depending upon the method of ap-proximation), when the domain of definition Ω is two or three dimensionaland when the boundary conditions are not simple, the determination ofN0(x) and Ni(x) is extremely difficult if not impossible. This is perhaps themain reason that these methods can rarely be applied to boundary valuesproblems of practical interest.

We note that in the methods of approximation discussed in this chapterthe domain of definition Ω of the boundary value problem is not discretizedand the approximation φn of φ is global over Ω and hence must satisfy appro-priate boundary conditions of the BVP. We emphasize that the methods ofapproximation despite their lack of usefulness in practical applications, formthe mathematical foundations of the finite element processes due to the factthat all fundamental principles of the finite element method are derived fromthese methods except two important aspects: (i) in finite element processesΩ is discretized and (ii) for a subdomain of the discretization we have localapproximations that are based on interpolation theory and are independentof the boundary conditions of the boundary value problem, thus avoiding allof the difficulties associated with the determination of N0(x) and Ni(x) inthe classical methods of approximation.

Problems

Consider boundary value problems 3.1 to 3.7. For each BVP construct and/or show thefollowing [(a) and (b)] using non-discretized domain of definitions of the boundary valueproblems.

(a) Construct an integral form using the Galerkin method. Show whether the integralform is VC or VIC. Discuss continuity requirements on the functions for the integralsin the integral form to be (i) in Riemann sense (ii) in Lebesgue sense. Define theassociated spaces.

(b) Construct an integral form using the Galerkin method with weak form.

(i) Identify PVs, SVs, EBCs, and NBCs.(ii) Simplify the expressions resulting from integration by parts using BCs and expressthe final results in the following form:

B(·, ·) = l(·)


(iii) Determine the nature of the functionals B(·, ·) and l(·), that is bilinearity, sym-metry, linearity, etc.(iv) Discuss minimally conforming spaces for:

(1) the mathematical model(2) the weak form(3) the BVP, i.e. the mathematical model, integral form, weak form, etc. for equiva-

lency between them.

3.1 Consider one dimensional heat conduction equation.

− d

dx

(adT

dx

)+ f = 0, 0 < x < 1 = Ω ⊂ R1

T (0) = 0 ; adT

dx+ h(T − T∞) = q at x = 1

where a = a(x), f = f(x) are known functions and h, T∞ and q are constants.

3.2 Consider a beam on elastic foundation.

d2

dx2

(bd2w

dx2

)+ kw + f = 0, 0 < x < L = Ω ⊂ R1

w = 0 at x = 0 and x = L

bd2w

dx2= 0 at x = 0 and x = L

b = b(x) and f = f(x) are known functions and k is a constant.

3.3 Consider longitudinal deformation of a bar with end spring, an eigenvalue problem.

− d

dx

(adu

dx

)+ λu = 0, 0 < x < L = Ω ⊂ R1

u(0) = 0(adu

dx+ ku

)∣∣∣∣x=L

= 0

a = a(x) is a known function and λ and k are constants.

3.4 Consider a second order BVP

− ∂

∂x

(a11

∂u

∂x+ a12

∂u

∂y

)− ∂

∂y

(a21

∂u

∂x+ a22

∂u

∂y

)+ f = 0 ∀x, y ∈ Ω ⊂ R2

u = u0 on Γ1(a11

∂u

∂x+ a12

∂u

∂y

)nx +

(a21

∂u

∂x+ a22

∂u

∂y

)ny = t0 on Γ2

where

aij = aji, i = 1, 2; j = 1, 2

f = f(x, y) ∀x, y ∈ Ω

Γ = Γ1 ∪ Γ2

u0 and t0 are known functions on Γ1 and Γ2 respectively and nx, ny are the directioncosines of a unit exterior normal to the boundary Γ2.

3.5 Consider one dimensional steady state convection diffusion equation.

dφ

dx− 1

Pe

d2φ

dx2= 0, 0 < x < 1 = Ω ⊂ R1

φ(0) = 1, φ(1) = 0

3.7. SUMMARY 189

Pe > 0 is known data.

3.6 Consider one dimensional steady state convection Burgers equation, a nonlinear ordi-nary differential equation.

φdφ

dx− 1

Re

d2φ

dx2= 0, 0 < x < 1 = Ω ⊂ R1

φ(0) = 1, φ(1) = 0

Re > 0 and is known.

3.7 Consider the following non-linear ordinary differential equation.

− d

dx

(udu

dx

)+ f = 0, 0 < x < 1 = Ω ⊂ R1

du

dx

∣∣∣∣x=0

= 0, u(1) =√

2

f = f(x) is a known function.


− d

dx

((1 + x)

du

dx

)= 0, 0 < x < 1 = Ω ⊂ R1

u(0) = 0, u(1) = 1

Construct an integral form of the BVP using the Galerkin method with weak form. Ap-proximate u by un, an n-parameter approximation using

un = N0(x) +

n∑i=1

CiNi(x)

(a) Provide details fo the integral form. Express it in the form B(·, ·) = l(·).(b) Establish VC or VIC of the integral form.(c) Establish general expressions for the coefficient matrix and the right hand side vector.(d) Specialize your results for n = 2 and compute coefficients C1 and C2 using

N0(x) = x

Ni(x) = xi(x− 1), i = 1, 2

3.9 Consider the following BVP, bending of simply supported beam subjected to uniformloading.

d2

dx2

(EI

d2w

dx2

)− f = 0 ; 0 < x < L = Ω ⊂ R1

w = EId2w

dx2= 0 at x = 0 and x = L

(a) Construct an integral form of the BVP using the Galerkin method with weak form.(b) Establish VC or VIC of the weak form.(c) Approximate w by wn

wn = N0(x) +

n∑i=1

CiNi(x)

(d) Establish general expressions for the coefficient matrix and right hand side vector.(e) Specialize your solution for n = 2, a two parameter approximation using

N0(x) = 0, Ni(x) = xi(L− x), i = 1, 2


Compute coefficients C1 and C2.


− 2ud2u

dx2+(dudx

)2= 4, 0 < x < 1 = Ω ⊂ R1

u(0) = 1, u(1) = 0

(a) Approximate u by un

un = N0(x) +

n∑i=1

CiNi(x)

(b) Find coefficient C1 (i.e. for n = 1) using

(i) the Galerkin method(ii) the least-squares method(iii) the Petrov–Galerkin method (or the weighted-residual method) with w1 = 1. Use

N0(x) = (1− x), N1(x) = x(1− x)

in all cases.

3.11 Consider the following eigenvalue problem.

− d

dx

((1 + x)

du

dx

)= λu, 0 < x < 1 = Ω ⊂ R1

u(0) = u(1) = 0

(a) Construct an integral form of the BVP using the Galerkin method with weak form.(b) Establish VC or VIC of the weak form.(c) Approximate u by

un = N0(x) +

n∑i=1

CiNi(x)

(d) Establish general expressions for the coefficient matrix and right hand side vector.(e) Specialize your solution for n = 2 and find eigenvalues λ1 and λ2 and the correspondingnormalized eigenvectors. Use

N0(x) = 0, Nj(x) = xj(x− 1), j = 1, 2, . . . , n

3.12 Consider the following BVP

−∂2u

∂x2− ∂2u

∂y2= 1 ∀x, y ∈ Ω ⊂ R2

The domain of definition Ω is a unit square (shown in the figure). u = 0 on Γ, closedboundary of the unit square.

(a) Construct an integral form of the BVP using the Galerkin method with weak form.(b) Establish VC or VIC of the weak form.(c) Approximate u by un

un = N0(x, y) +

n∑i=1

CiNi(x, y)

(d) Establish general expressions for the coefficient matrix and right hand side vector.(e) Specialize your results for n = 1 using

N0(x, y) = 0 and N1(x, y) = sinπx sinπy


y

1

1

x

and calculate the coefficient C1.

3.13 Consider the following BVP, an eigenvalue problem.

− d2u

dx2− λu = 0, 0 < x < 1 = Ω ⊂ R1

u(0) = u(1) = 0

Use collocation method with collocation points at

x =1

4, x =

1

2, x =

3

4

and calculate first three eigenvalues of the problem using n = 3 and

N0(x) = 0, Ni(x) = xi(x− 1), i = 1, 2, 3

[5–20]

References for additional reading[1] L. Demkowicz and J. Gopalakrishnan. A class of discontinuous petrovgalerkin meth-

ods. part I: The transport equation. Comp. Meth. in Applied Mech. and Eng.,199:1558–1572, 2010.

[2] L. Demkowicz and J. Gopalakrishnan. A class of discontinuous petrovgalerkin meth-ods. II. optimal test functions. Num. Meth. for Partial Diff. Equations, 27:70–105,2011.

[3] L. Demkowicz, J. Gopalakrishnan, and A. H. Niemi. A class of discontinuous petro-vgalerkin methods. part III: Adaptivity. Applied Num. Math., 62:396–427, 2011.

[4] J. Zitelli, I. Muga, L. Demkowicz, J. Gopalakrishnan, D. Pardo, and V.M. Calo. Aclass of discontinuous petrovgalerkin methods. part IV: The optimal test norm andtime-harmonic wave propagation in 1d. Applied Num. Math., 230:2406–2432, 2011.






[9] G. Mikhlin. Numerical Performance of Variational Methods. Woltes-Noordhoff,Svoningen, 1971.








[17] J. T. Oden and J. N. Reddy. Variational Methods in Theoretical Mechanics. JohnWiley, New York, 2nd edition, 2002.

[18] J. N. Reddy. Energy Principles and Variational Methods in Applied Mechanics. JohnWiley, New York, (3rd in print) 2nd edition, 2002.

[19] C. Lanczos. The variational principles of mechanics. University of Toronto Press,Toronto, 4th edition, 1970.

[20] G. Strang and G. Fix. An Analysis of the Finite Element Method. Prentice Hall, NewJersey, 1973.

4

The Finite Element Method

4.1 Introduction

The classical methods of approximation of the solutions of BVPs de-scribed in Chapter 3 are hampered primarily due to the fact that determi-nation of the basis functions used in approximating φ of Aφ − f = 0 in Ωfor the non-discretized domain Ω becomes increasingly difficult for practicalproblems of interest. The basis functions, apart from satisfying continuity,linear independence, completeness, and appropriate boundary conditions de-pending upon the method, there exists no systematic method for establish-ing them. Yet, the accuracy of the approximation is highly dependent uponthem. Thus, classical methods of approximation are not regarded as prac-tical and competitive for approximating the solutions of BVPs of practicalinterest.

We remark on some important features that are essential for a computa-tional method to be effective in practical applications.

(1) The method must have a sound mathematical foundation which mustresult in a transparent computational infrastructure without further as-sumptions or approximations.

(2) The method should be independent of the geometric description, that is,the domain of definition of the BVP, the nature of physical or transportproperties, boundary conditions and the nature of external disturbances.

(3) The method must address the following three classes of differential op-erators that account for the totality of all BVPs.

(a) self-adjoint(b) non-self-adjoint(c) non-linear

without additional assumptions or approximations than those used inobtaining the mathematical descriptions of the corresponding physicalprocesses. The resulting computational infrastructure must be mathe-matically consistent and should require no additional ad hoc remedies.

(4) The method should be convergent and unconditionally stable regardlessof the choice of the degree or the order of the basis functions as long assuch choices are admissible.

193

194 THE FINITE ELEMENT METHOD

(5) The method must permit higher degree and higher order of approxima-tions so that coarser discretizations may remain practical with the de-sired degree of global smoothness that is dictated by the desired physicsin the computational process.

(6) Lastly, the method must lend itself as a systematic procedure that canbe automated for use on digital computers with inherent and built-inadaptivity.

The finite element method is indeed one such method that permits us tosatisfy all of the above requirements. In applying the finite element methodfor approximating solutions of BVPs, one must adhere to the following basicsteps regardless of the type of operator and the method of approximationused in constructing the integral form.

4.2 Basic steps in the finite element method

We recall that in classical methods of approximation, one approximatesthe solutions of the BVP Aφ− f = 0 over the domain of definition Ω in thecontinuum sense, i.e. in such methods there is no concept of discretizationand desired regularity is incorporated in the process. The finite elementmethod derives all of its basic mathematical principles from classical meth-ods of approximation except that the domain of definition of Ω of the BVPis discretized into subdomains and the principles of classical methods ofapproximation are applied to the discretization and thereby to the subdo-mains. Thus, the finite element method can be thought of as piecewise (apiece being a subdomain or a finite element) application of classical meth-ods of approximation. This is the fundamental difference between classicalmethods of approximation and the finite element method. This concept ofdiscretization and dealing with subdomains allows the method to incorpo-rate flexibility in all aspects which overcomes all of the shortcomings of theclassical methods of approximation. In the following, we provide a system-atic description and details of all of the basic steps involved in the finiteelement method for approximating solutions of the BVPs. These basic stepsare identical regardless of the type of differential operator and the specificphysical processes from which they arise.

4.2.1 Discretization

The domain Ω = Ω∪Γ of the BVP Aφ−f = 0 is subdivided into smallersubdomains. Each subdomain is of finite size and is considered connected tothe neighboring subdomains along its boundaries. A subdomain is referred toas a finite element. Thus, a finite element is a subdomain of finite dimensions

4.2. BASIC STEPS IN THE FINITE ELEMENT METHOD 195

of the original domain Ω. Symbolically,

Ω =

M⋃e=1

Ωe (4.1)

in which Ω and Ω are open and closed domains of the BVP and Ωe and Ωe

are open and closed domains of a typical finite element e. This process ofsubdividing a domain Ω into subdomains Ωe is called discretization. Now,one could think of the domain Ω as the assembly of the subdomains Ωe.The collection of finite elements Ωe is called a finite element mesh. If all theelements of the mesh are of the same size and shape, then the mesh is calleduniform. Otherwise, we refer to them as non-uniform (or graded) meshes.Graded meshes contain elements of various sizes (but often the same shape)in order to generate a bias in the computational process. For each elementΩe we define some points on the boundaries of the element, which are callednodes. The choice of the nodal points is not arbitrary. The node points helpus in (a) defining the mathematical description of the geometrical shape ofthe element as well as (b) the behaviors of dependent variables over the ele-ment. With these nodes on the boundaries of elements, one could now thinkof an element Ωe as being connected to the neighboring elements only at thenodes provided the connection at the nodes automatically ensures a contin-uous connection along all of its boundaries with the neighboring elements.The connection of an element to its neighbors is in the mathematical senseof continuity and differentiability of the dependent variables and, hence isdependent upon the inter-element behaviors of the dependent variables.

In the finite element mesh, the nodes and the elements are uniquely num-bered. The choice of numbers for the nodes is irrelevant as long as they areunique. One also establishes an origin of a unique coordinate system (atany convenient location) in which the position coordinates of the nodes areuniquely identified. Two typical examples of discretizations are shown inFigs. 4.1 and 4.2. We consider Ω = [0, 1], a line segment (see Fig. 4.1). Fig-ure 4.1 (b) shows a four element discretization of Ω using two-node elements,the process of disconnecting and reconnecting the elements at the boundariesof the neighboring elements and a typical element e with its domain Ωe withlocal node numbers 1 and 2 isolated from the discretization. In Fig. 4.2, weconsider a 2D domain Ω of a BVP, discretized using four four-node quadri-lateral elements, the process of disconnecting and reconnecting the elementsand a typical element e with its domain Ωe and the local node numbers1, 2, 3, 4 isolated from the discretization. Use of local node numbers for anelement Ωe is helpful in deriving the mathematical details. Correspondencebetween the element numbers, their local node numbers and the nodes ofthe whole discretization can be easily established.


1

1

2 3

1 32

1

1

2 3 4

1 32 4

e

e

xΩ = [0, 1]

y

y

a typical inter-element boundary

7

1

y

a 3-node typical element e

Ωe = [xe, xe+2]

xe

(local node numbers 1, 2 and 3)x

(c)

xe+1 xe+2

y

a typical inter-element boundary

4 4 53322

21 3 4 5

x

y

a 2-node typical element e

Ωe = [xe, xe+1]

xe1xe

(local node numbers 1 and 2)x

(b)

x

2 3 764 5

32 3 4 5 5 6

1 2

1 32

A three-element uniform discretization of domain Ω, the assembly ofsubdomains, and a typical three-node element e with its local nodenumbers and its domain Ωe

A four-element uniform discretization of domain Ω, the assembly of sub-domains, and a typical two-node element e with its local node numbersand its domain Ωe

(a) One dimensional domain Ω

Figure 4.1: 1D domain Ω of the BVP and its discretization


1

21

4 5

O

b

a

y

x

(a) A 2D domain Ω of a BVP

e

21

4 3

x

y

a typical four-node quadri-lateral element e with localnode numbers

Discretization of 2D domain Ω, an assembly of elements, and a typical element e withits local node numbers and domain Ωe.

(b)

y

x

3

94

6

7

5

8

3

9

44

78

1

4

2

65

2

8

5

5 6

32321

Individual elementsMesh

Figure 4.2: 2D domain Ω of the BVP and its discretization


The end result of discretization is that we have a finite element mesh withM elements and N nodes clearly identified. The position coordinates of thenodes are known and a typical representative element e with its domain Ωe

of the discretization clearly identified. The values of dependent variablesand/or their derivatives at the nodes of the entire discretization, referred toas the degrees of freedom (dofs) are the quantities of interest.

4.2.2 Construction of integral forms over an element

In the study of the classical methods of approximation, for the bound-ary value problem Aφ − f = 0 in Ω, we construct an integral form overthe domain Ω = Ω

⋃Γ, Γ being closed boundary of Ω. This was done one

of the two ways: (1) starting with the fundamental lemma, which lead toGM, PGM, WRM, and GM/WF methods of approximation, or (2) usingresidual functional in which the first variation of the residual functional isset to zero to obtain the integral form. This method is called least-squaresmethod or process (LSP). A correspondence between these integral formsand the calculus of variations leads to the definitions of variationally consis-tent or variationally inconsistent integral forms. The variationally consistentintegral forms yield algebraic systems in which the coefficient matrices aresymmetric and positive definite and, hence, the unconditional stability of thecomputational processes is ensured. The variationally inconsistent integralforms, on the other hand, yield algebraic systems in which the coefficientmatrices are non-symmetric and, hence, unconditional stability of the com-putational processes is not always ensured. All of the concepts, principlesand stated findings and conclusions in connection with classical methods ofapproximation presented in chapter 3 precisely hold for the finite elementprocesses as well.

In the following sections we consider details of the finite element processesthat are directly derived from classical methods of approximation: GM,PGM, WRM, GM/WF and LSP. In each case, details are presented for atypical element e with domain Ωe of the discretization ΩT of the domain ofdefinition Ω of the BVP Aφ− f = 0. In all cases we consider the following.

Let ΩT =⋃Me=1 Ωe be the discretization of Ω in which Ωe = Ωe ∪ Γe

is the domain of an element e with closed boundary Γe. Let φh be theapproximation of φ over ΩT and φeh be the approximation of φ over Ωe, then

φh =⋃Me=1 φ

eh. Let φn be the n-term approximation of φ over Ω in the

classical methods of approximation.


4.2.2.1 Integral forms for GM, PGM, and WRM

Using the approximation φn over Ω for the BVP Aφ − f = 0, we canwrite the integral statement∫

Ω

(Aφn − f)v dΩ = (Aφn − f, v)Ω = 0, Ω = Ω ∪ Γ (4.2)

based on the fundamental lemma, in which v is the test function such thatv = 0 on Γ∗ if φ = φ0 on Γ∗ of Γ. In GM method v = δφn, hence, v = 0on Γ∗ is satisfied. In PGM v = ψ 6= δφn, but ψ = 0 on Γ∗ if φ = φ0 on Γ∗

must hold. In WRM v = w 6= δφn (weight function), but w = 0 on Γ∗ musthold if φ = φ0 on Γ∗. Using (4.2) and approximation φh of φ over ΩT , wecan write the integral form over discretization ΩT as∫

ΩT

(Aφh − f)v dΩ = (Aφh − f, v)ΩT = 0 (4.3)

Since the integral in (4.3) is a functional and since ΩT =⋃e Ωe, and φh =⋃

e φeh using (4.3) we can write

(Aφh − f, v)ΩT =∑e

∫Ωe

(Aφeh − f)v dΩ =∑e

(Aφeh − f, v)Ωe = 0 (4.4)

or ∑e

(Aφeh, v)Ωe =∑e

(f, v)Ωe (4.5)

or ∑e

Be(φeh, v) =∑e

le(v) (4.6)

In (4.6), Be(φeh, v) and le(v) are functionals for an element e with domainΩe.

We remark that if the approximation φh over ΩT is such that the in-tegrand in (4.3) is continuous everywhere in ΩT , then (4.4) holds in theRiemann sense. Otherwise, it holds in the Lebesgue sense in which casediscontinuity in the behavior of the integrand over the inter-element bound-aries (sets of measure zero) is neglected. This, of course, depends upon theglobal differentiability of φh which obviously depends upon the global differ-entiability achievable from φeh due to the fact that φh =

⋃e φ

eh. Here, φeh is

called the local approximation as it is local to the element e with domain Ωe

without regard to the other elements in ΩT and boundary conditions of theBVP. The local approximation φeh is established using interpolation theory.Once we know φeh, details of Be(φeh, v) and le(v) for an element e can be


established. We note that in GM v = δφeh, in PGM v = ψ 6= δφeh and inWRM v = w 6= φeh. The choices of ψ and w in PGM and WRM is notarbitrary and must conform to the same restrictions locally over Γe as in thecase of classical PGM and WRM over Γ∗ of Γ the closed boundary of Ω. Thedetails of Be(φeh, v) and le(v) are presented in the following sections for GM,PGM and WRM. The desired integrals in these methods for an element eare (Aφeh, v)Ωe = Be(φeh, v) and le(v) = (f, v)Ωe .

Remarks.

(1) In these methods Be(φeh, v) is always non-symmetric, i.e. Be(φeh, v) 6=Be(v, φeh). This is of course due to the variational inconsistency of theintegral forms as shown in the classical methods of approximation. Theconsequence of this is that the element matrix resulting from Be(φeh, v)is not symmetric.

(2) When the differential operator A is linear, Be(φeh, v) results in linearsimultaneous algebraic relations for the element e in the nodal degreesof freedom for the element due to the fact that Be(φeh, v) is bilinear.

(3) On the other hand if the differential operator A is non-linear, thenBe(φeh, v) is a non-linear function of φeh (but linear in v) and as a con-sequence the resulting algebraic relations for an element e are also non-linear in the nodal degrees of freedom for the element e. The elementmatrix is not symmetric as well.

(4) The variational inconsistency of the integral forms over ΩT cannot berestored by any mathematically justifiable means.

(5) le(v) is independent of φeh and is linear in v regardless of the nature ofthe differential operator.

4.2.2.2 Integral form for GM/WF

In this method we also begin with (4.2) and arrive at (4.3) and then(4.5) or (4.6) for the discretization ΩT . We consider (Aφeh − f, v)Ωe over anelement e with domain Ωe in which v = δφeh,

(Aφeh − f, v)Ωe = (Aφeh, v)Ωe − (f, v)Ωe (4.7)

We transfer some differentiation from φeh to v in (Aφeh, v)Ωe using integrationby parts to obtain

(Aφeh, v) = Be(φeh, v)− l˜e(v) (4.8)

in which l˜e(v) is concomitant 〈Aφeh, v〉Γe , consisting of boundary terms or

boundary integrals resulting as a consequence of integration by parts. Themotivation for integration by parts is to see if Be(φeh, v) = Be(v, φeh), i.e. if


symmetric Be(·, ·) is possible as it would lead to symmetric element matrices(see section 4.2.4). Substituting from (4.8) into (4.7) we obtain

(Aφeh − f, v)Ωe = Be(φeh, v)− l˜e(v)− (f, v)Ωe (4.9)

or

(Aφeh − f, v)Ωe = Be(φeh, v)− le(v) (4.10)

in which

le(v) = l˜e(v) + (f, v)Ωe (4.11)

Using φeh and v = δφeh details of Be(·, ·) and le(·) can be easily established(see section 4.2.4). (4.10) is the desired integral form based on GM/WF forthe eth element with domain Ωe.

Remarks.

(1) When the differential operator A in Aφ − f = 0 is self-adjoint, i.e.contains only even order derivatives of the dependent variable(s), then itis possible to transfer half of the orders of differentiation from φeh to v andobtain Be(φeh, v) that is symmetric and, hence Be(φeh, v) = Be(v, φeh).This is obviously due to variational consistency of the integral form inthe classical GM/WF for self-adjoint operators. The consequence of thisis that the element matrix resulting from Be(φeh, v) is symmetric. Dueto linearity of the differential operator A, the algebraic relations for anelement e are linear in the nodal degrees of freedom for the element.

(2) When the differential operator A is non-self-adjoint, Be(φeh, v) alwaysremains non-symmetric due to variational inconsistency of the integralform in the classical GM/WF for non-self-adjoint operators. The ele-ment matrix resulting from Be(φeh, v) is non-symmetric but the algebraicrelations for the element e are linear in the nodal degrees of freedom forthe element due to the fact that Be(φeh, v) is bilinear.

(3) If the differential operator A is non-linear, then Be(φeh, v) is a non-linearfunction of φeh (but linear in v) and, hence the algebraic relations for theelement e are non-linear in nodal degrees of freedom for the element andthe element matrix is non-symmetric. The variational inconsistency ofthe integral form in this method is clearly due to variationally inconsis-tent integral form in classical GM/WF.

(4) When the differential operator A is non-self-adjoint or non-linear vari-ational inconsistency of the integral forms cannot be restored by anymathematically justifiable means.

(5) le(v) is independent of φeh and is linear in v regardless of the nature ofthe differential operator.


(6) Using the concomitant we can identify PVs, SVs, EBCs, and NBCs inthe same manner as in classical methods of approximation. However,the BCs of the BVP cannot be used to simplify the concomitant as theintegral form is over the domain Ωe of an element e of the discretization

ΩT =⋃e

Ωe whereas the BCs can only be used for the whole domain (i.e.

ΩT ) as in classical methods of approximation. This is a fundamentaldifference in GM/WF over Ω and Ωe.

4.2.2.3 Integral form based on residual functional

Using the approximation φn over Ω for the BVP Aφ − f = 0 we candefine the residual function E over Ω

E = Aφn − f (4.12)

The residual functional I(φn) can be constructed using E

I(φn) = (E,E)Ω (4.13)

using (4.12) and (4.13) and approximation φh of φ over ΩT , we can write thefollowing for the residual function E(φh) and the residual functional I(φh)over the discretization ΩT

E = Aφh − f holds everywhere in ΩT (4.14)

andI(φh) = (E,E)ΩT (4.15)

or

I(φh) =

M∑e=1

Ie(φeh) =∑e

(Ee, Ee)Ω (4.16)

in which Ie(φeh) is the residual functional for an element e and Ee is theresidual function over Ωe. Ee(φeh) and Ie(φeh) are given by Ee = Aφeh − fand Ie = (Ee, Ee)Ωe .

In least-squares process we take the first variation of the residual func-tional I(φh) and set it to zero provided I(φh) is differentiable in φh (basedon calculus of variations).

δI(φh) =

M∑e=1

δIe(φeh) =∑e

((δEe, Ee)Ωe + (Ee, δEe)Ωe

)(4.17)

But (δEe, Ee)Ωe = (Ee, δEe)Ωe , hence

δI(φh) = 2∑e

(Ee, δEe) = 2∑e

ge = 2g = 0 or g = 0 (4.18)


Using (Ee, δEe)Ωe , we can write

(Ee, δEe)Ωe = Be(φeh, v)− le(v) (4.19)

This is the desired integral form for an element e with domain Ωe based onthe residual functional or least-squares method.

Remarks.

(1) When the differential operator is linear, i.e. self-adjoint or non-self-adjoint, Be(φeh, v) is linear in φeh as well as v, that is, bilinear and is alsosymmetric (Be(φeh, v) = Be(v, φeh)). This is due to variational consis-tency of the integral form in classical LSP for self-adjoint and non-self-adjoint differential operators. The element coefficient matrix resultingfrom Be(φeh, v) is symmetric and the algebraic relations for the elementare linear in the nodal degrees of freedom for the element.

(2) When the differential operator A is non-linear, then Be(φeh, v) (or g)is a non-linear function of φeh but is linear in v. The element matrixis non-symmetric and its coefficients are functions of φeh. However, asshown in classical least-squares method of approximation if we find a φhsatisfying g = 0 using Newton’s linear method and neglect (Ee, δ2Ee)in δ2Ie(φen), then the finite element process based on LSP becomes VCand the symmetry of the element matrices and the assemble coefficientmatrix in the algebraic system is restored as well.

(3) le(v) is independent of φeh and is always linear in v regardless of thenature of the differential operator.

4.2.3 The local approximation φeh of φ over an element

In order to obtain the element matrices (i.e. the algebraic relations) fromBe(φeh, v) and vectors from le(v) we need local approximations φeh of φ overΩe. The local approximations φeh of φ over Ωe can be constructed usinginterpolation theory presented in chapter 8. The domain Ωe of each elementis mapped into the natural coordinate space (chapter 8) in a two-unit length,two-unit square or a two-unit cube with the origin of the natural coordinatesystem at the center by utilizing mappings such as

x = x(ξ) =∑

Ni˜ (ξ)xi in R1 (4.20)

x = x(ξ, η) =∑

Ni˜ (ξ, η)xi

y = y(ξ, η) =∑

Ni˜ (ξ, η)yi

in R2 (4.21)


x = x(ξ, η, ζ) =∑

Ni˜ (ξ, η, ζ)xi

y = y(ξ, η, ζ) =∑

Ni˜ (ξ, η, ζ)yi

z = y(ξ, η, ζ) =∑

Ni˜ (ξ, η, ζ)zi

in R3 (4.22)

In (4.20)–(4.22), Ni˜ (·) are the local approximation functions or shape func-

tions in the natural coordinate space and xi, yi, zi are nodal coordinates ofthe element nodes in the physical coordinate space. The sum in (4.20)–(4.22)is over suitably chosen nodes (see chapter 8). Using (4.20)–(4.22), relation-ships for mapping of lengths, areas and volumes in the (ξ, η, ζ) and (x, y, z)coordinate systems can be derived. Also, the derivatives of the dependentvariable(s) φ in the two coordinate systems can be related.

Based on the interpolation theory presented in chapter 8, we can alsowrite the following for the local approximation φeh of φ over Ωe.

φeh(ξ) =

n∑i=1

Ni(ξ)δei = [N(ξ)]δe in R1 (4.23)

φeh(ξ, η) =

n∑i=1

Ni(ξ, η)δei = [N(ξ, η)]δe in R2 (4.24)

φeh(ξ, η, ζ) =

n∑i=1

Ni(ξ, η, ζ)δei = [N(ξ, η, ζ)]δe in R3 (4.25)

when [N(·)] is the basis or approximation function matrix in the natural co-ordinate space and δe are the nodal degrees of freedom containing values ofφ and/or its derivatives at the nodes of the element. The choice of Ni(·) andthe resulting δe depend upon the degree of local approximation p (p-level)and the order k of the approximation space defining global differentiabilityof order (k− 1) of φh =

⋃eφeh. Thus, k controls the global smoothness of φh

over ΩT through φeh. Equations (4.23)–(4.25) completely define the behaviorof each dependent variable over a typical element e with domain Ωe of thediscretization ΩT .

4.2.4 Element matrices and vectors resulting from theintegral form and the local approximation

In this section we utilize the integral forms derived in section 4.2.2 usingvarious methods of approximation and the local approximation defined insection 4.2.3 to derive the details of the element matrices and vectors, i.e thealgebraic relations for an element e with domain Ωe.


4.2.4.1 Galerkin method, Petrov–Galerkin method,and weighted residual method

In the integral forms resulting from Galerkin method (GM), Petrov–Galerkin method (PGM), and weighted residual method (WRM) for an el-ement e with domain Ωe, we have (see section 4.2.2)

Be(φeh, v)Ωe = (Aφeh, v)Ωe (4.26)

andle(v) = (f, v)Ωe (4.27)

Let the local approximation be defined by

φeh =

n∑i=1

Niδei (4.28)

Then

v = δφeh = Nj , j = 1, 2, . . . , n GM

v = w = Nj , j = 1, 2, . . . , n PGM or WRM(4.29)

In the following we consider two cases: when the differential operator A is(1) linear (i.e. self-adjoint or non-self-adjoint) and (2) non-linear.

(1) Linear differential operators

In this case Be(φeh, v) is bilinear, that is linear in φeh as well as v. Wesubstitute from (4.28) and (4.29) in (4.26) and (4.27) and consider GM,i.e. v = Nj ; j = 1, 2, . . . , n.

Be(φeh, v) =(A( n∑i=1

Niδei

), Nj

), j = 1, 2, . . . , n (4.30)

le(v) = (f,Nj)Ωe , j = 1, 2, . . . , n (4.31)

Since the operator A is linear, (4.26) can be written as

Be(φeh, v) =( n∑i=1

(ANi)δei , Nj

)Ωe

=n∑i=1

(ANi, Nj)Ωe δei , j = 1, 2, . . . , n (4.32)

Equations (4.32) and (4.31) can be written using matrix and vectornotation as

Be(φeh, v) = [Ke]δe (4.33)

le(v) = F e (4.34)


in which Keij of [Ke] and F ei of F e are given by

Keij = Be(Nj , Ni) = (ANj , Ni)Ωe , i, j = 1, 2, . . . , n (4.35)

F ei = le(Ni) = (f,ANi)Ωe , i = 1, 2, . . . , n (4.36)

Equations (4.33)–(4.36) provide details of the element matrix and vectorfor an element e of the discretization. We note that Ke

ij 6= Keji, i.e. the

element matrix [Ke] is not symmetric, a consequence of the VIC integralform in the classical method of approximation based on GM. In the caseof PGM and WRM, v = Nj and, hence, the details in (4.30)–(4.36)remain the same except that Nj are replaced by Nj .

(2) Non-linear differential operators

We begin with (4.26) and (4.27). Since the differential operator A isnon-linear, Be(φeh, v) is a non-linear function of φeh but linear in v. Wesubstitute (4.28) and (4.29) into (4.26) and (4.27) and consider GM.

Be(φeh, v) =(A( n∑i=1

Niδei

), Nj

)Ωe, j = 1, 2, . . . , n (4.37)

le(v) = (f,Nj)Ωe , j = 1, 2, . . . , n (4.38)

Since A is a non-linear differential operator, (4.37) is a system of n non-linear algebraic relations in δei . We can write (4.37) and (4.38) in thematrix and vector form as

Be(φeh, v) = [Ke]δe (4.39)

le(v) = F e (4.40)

The coefficients Keij of [Ke] are a function of δe and Ke

ij 6= Keji; that

is, [Ke] is not symmetric, a consequence of the VIC integral form inGM. The explicit expression for Ke

ij can be obtained once we know the

specific form of the operator A. In the case of PGM and WRM, v = Nj

but the remaining details remain the same as in GM except that Nj arereplaced by Nj .

4.2.4.2 Galerkin method with weak form

In the Galerkin method with weak form (GM/WF), we have v = δφehand we obtain a weak form using (Aφeh, v)Ωe

(Aφeh, v)Ωe = Be(φeh, v)− l˜e(v) (4.41)

Therefore

(Aφeh − f, v)Ωe = Be(φeh, v)− l˜e(v)− (f, v)Ωe = Be(φeh, v)− le(v) (4.42)


where

le(v) = (f, v)Ωe + l˜e(v) (4.43)

The right side of (4.41) is a consequence of transferring some differentiationfrom φeh to v using integration by parts. Be(·, ·) is the integral over Ωe

whereas l˜e(v) are the boundary terms or boundary integrals (concomitant).

We consider local approximations φeh

φeh =

n∑i=1

Niδei (4.44)

and

v = δφeh = Nj , j = 1, 2, . . . , n (4.45)

We consider three cases: when the differential operator A is self-adjoint,non-self-adjoint and non-linear.

(1) Self-adjoint differential operators

In this case, A is linear and symmetric and, hence, contains only evenderivative terms. Therefore, we can transfer half of the differentiationfrom φeh to v in obtaining (4.41). In doing so, we find that

Be(φeh, v) = Be(v, φeh) (4.46)

That is, Be(·, ·) is symmetric. Furthermore, since the differential op-erator A is linear, Be(·, ·) is bilinear, i.e. linear in φeh as well as v.Substituting (4.44) and (4.45) into (4.41) and (4.42), we obtain

(Aφeh, v)Ωe = Be( n∑i=1

Niδei , Nj

)− l˜e(Nj), j = 1, 2, . . . , n (4.47)

(f, v)Ωe = (f,Nj)Ωe , j = 1, 2, . . . , n (4.48)

Since Be(·, ·) is bilinear and le(v) is linear, the right sides of (4.46) and(4.47) can be written in the following form

(Aφeh, v)Ωe = [Ke]δe − P e (4.49)

(f, v)Ωe = F e (4.50)


Keij = Be(Nj , Ni) = Be(Ni, Nj), i, j = 1, 2, . . . , n (4.51)

F ei = (f,Ni)Ωe , i = 1, 2, . . . , n (4.52)


Therefore

(Aφen − f, v)Ωe = [Ke]δe − P e − F e

The vector P e is due to l˜e(Nj) (j = 1, 2, . . . , n) and is called a vector

of secondary variables. Obviously Keij = Ke

ji; that is, [Ke] is symmetric,a direct consequence of the VC integral form possible only due to thefact that A∗ = A. The explicit forms of Ke

ij and P e depend upon thedifferential operator A.

(2) Non-self-adjoint differential operators

In this case the differential operator is linear but not symmetric. We maytransfer some differentiation from φeh to v in (Aφeh, v)Ωe . It is importantto consider the possible situations in which it is advantageous to doso. First, when the differential operator is non-self-adjoint, it definitelycontains some terms with odd derivatives but may (or may not) alsocontain some terms with even derivatives. When A contains some termswith even derivative terms, then in these terms we can transfer half of thedifferentiation from φeh to v in (Aφeh, v)Ωe using integration by parts. Bydoing so, we ensure that the contribution of these resulting terms to theelement matrix [Ke] becomes symmetric. This helps in restoring somestability of the resulting computational process. In the remaining termswith odd derivatives of the dependent variable, we may also transfersome differentiation from φeh to v in (Aφeh, v)Ωe to lower differentiabilityrequirements in the resulting integral. However, for such terms theircontribution to [Ke] remains non-symmetric. The end result is that wehave (similar to (4.41) and (4.42))


Therefore

(Aφeh− f, v)Ωe = Be(φeh, v)− l˜e(v)− (f, v)Ωe = Be(φeh, v)− le(v) (4.54)

in which l˜e(v) is concomitant which only occurs if some differentiation

is transferred from the dependent variable(s) to v. The local approxi-mation can be written as

φeh =n∑i=1

Niδei (4.55)

and

v = δφeh = Nj , j = 1, 2, . . . , n (4.56)


Since the operator A is linear (but not symmetric), Be(·, ·) is bilinearbut not symmetric due to A being non-symmetric. Substituting from(4.55) and (4.56) into (4.53) and (4.54), we obtain


Niδi, Nj

)− l˜e(Nj), j = 1, 2, . . . , n (4.57)

(f, v)Ωe = (f,Nj)Ωe , j = 1, 2, . . . , n (4.58)

Since Be(φeh, v) is bilinear and le(v) is linear. From (4.57) and (4.58) wecan write

(Aφeh, v)Ωe = [Ke]δe − P e (4.59)

(f, v)Ωe = F e (4.60)


Keij = Be(Nj , Ni) 6= Be(Ni, Nj), i, j = 1, 2, . . . , n (4.61)

F ei = (f,Ni)Ωe , i = 1, 2, . . . , n (4.62)

Therefore

(Aφeh − f, v)Ωe = [Ke]δe − P e − F e

The vector P e is due to l˜e(Nj) (j = 1, 2, . . . , n) and is called vector

of secondary variables. Obviously, Keij 6= Ke

ji; that is, [Ke] is not sym-metric, a consequence of the VIC integral form due to the fact that thedifferential operator A is not symmetric. The explicit forms of Ke

ij andP e depend upon the differential operator A.


Here the differential operator is neither linear nor symmetric. In suchcases it is possible that the differential operator A contains terms witheven and odd derivatives in addition to non-linear terms. The rulesfor transferring differentiation from φeh to v in (Aφeh, v)Ωe for the termscontaining even and odd derivatives of the dependent variables remainthe same as discussed in (2) when the differential operator A is non-self-adjoint. For the non-linear terms, there is generally no incentive toperform integration by parts. The final integral form in this case alsoresults in


Therefore

(Aφeh− f, v)Ωe = Be(φeh, v)− l˜e(v)− (f, v)Ωe = Be(φeh, v)− le(v) (4.64)


The local approximation

φeh =

n∑i−1

Niδei (4.65)

v = δφeh = Nj , j = 1, 2, . . . , n (4.66)

In this case, Be(φeh, v) is non-linear in φeh but linear in v and le(v) islinear in v. Substituting from (4.65) and (4.66) into (4.63) and (4.64)


Niδi, Nj

)− l˜e(Nj), j = 1, 2, . . . , n (4.67)

(f, v)Ωe = (f,Nj)Ωe , j = 1, 2, . . . , n (4.68)

(4.67) and (4.68) can also be written in the matrix and vector form

(Aφeh, v)Ωe = [Ke]δe − P e (4.69)

(f, v)Ωe = F e (4.70)

Therefore

(Aφeh − f, v)Ωe = [Ke]δe − P e − F e

The coefficients of [Ke] are functions of δe due to the fact that thedifferential operator A is non-linear. Explicit forms of Ke

ij of [Ke] canbe obtained when A is defined. In this case also P e is a vector of sec-ondary variables. Furthermore, Ke

ij 6= Keji, i.e. [Ke] is not symmetric, a

consequence of the VIC integral form due to the fact that the differentialoperator A is non-linear.

4.2.4.3 Least-squares process based on residualfunctional

In the least-squares process based on residual functional (LSP), we con-sider Ee = Aφeh − f , δEe and (Ee, δEe)Ωe for an element e with domainΩe. We consider two cases: when the differential operator A is linear (i.e.self-adjoint and non-self-adjoint) and when the differential operator A isnon-linear.

(1) Linear differential operators

In this case, for an element e with domain Ωe we have

Ee = Aφeh − f (4.71)

δEe = Av (4.72)


Hence, for the necessary condition we have

ge = (Ee, δEe)Ωe = (Aφeh − f,Av)Ωe

= (Aφeh, Av)Ωe − (f,Av)Ωe = Be(φeh, v)− le(v) (4.73)

Be(·, ·) is bilinear and symmetric and le(v) is linear. Consider

φeh =n∑i=1

Niδei (4.74)

andδφh = v = Nj , j = 1, 2, . . . , n (4.75)

Substituting from (4.74) and (4.75) into (4.73)

ge =(A( n∑i=1

Niδei

), ANj

)Ωe− (f,ANj)Ωe , j = 1, 2, . . . , n (4.76)

Since A is linear, using (4.76) we can write

ge =

n∑i=1

(ANi , ANj

)Ωeδei − (f,ANj)Ωe , j = 1, 2, . . . , n (4.77)

or in matrix and vector notation

ge = [Ke]δe − F e (4.78)


Keij = (ANj , ANi)Ωe , i, j = 1, 2, . . . , n (4.79)

F ei = (f,ANi), i = 1, 2, . . . , n (4.80)

Clearly, Keij = Ke

ji; that is, [Ke] is symmetric, a consequence of the VCintegral form in LSP for linear differential operator.


Consider BVPs in which the differential operator is non-linear. In thiscase also, ge is given by

ge = (Ee, δEe)Ωe (4.81)

where

Ee = Aφeh − f (4.82)

δEe = Av + δA(φeh) (4.83)


holds for an element e with domain Ωe. We note that since A in non-linear, δA 6= 0. Substituting from (4.82) and (4.83) in (4.81)

ge =(Aφeh − f,Av + δA(φeh)

)(4.84)

Consider

φeh =n∑i=1

Niδei (4.85)

δφen = v = Nj , j = 1, 2, . . . , n. (4.86)

Substituting from (4.85) and (4.86) into (4.84) and noting that A is afunction of φeh,

ge =(A( n∑i=1

Niδi)− f, Av + δA

( n∑i=1

Niδei

))(4.87)

We note that A is a function of φeh and, hence, δA 6= 0. Thus, geis a non-linear function of δei . (4.87) can be written in matrix andvector notation, but we do not do so. Instead as shown in the classicalmethods, we find a

δ =⋃e

δe (4.88)

that satisfies

g =M∑i=1

ge = 0 (4.89)

using iterative methods such as Newton’s linear method with line search.Details follow the classical least-squares method discussed earlier but arerepeated for finite element processes based on LSP in section 4.2.8.

4.2.5 Assembly of element equations: GM, PGM, WRM,GM/WF and LSP when A is linear

Recall that for the discretization ΩT =⋃e Ωe, we have in all finite element

processes based on various methods of approximation except LSP for non-linear differential operator the following.GM, PGM, WRM:

M∑e=1

(Aφeh − f, v)Ωe =M∑e=1

Be(φeh, v)−M∑e=1

le(v) = 0

=

M∑e=1

[Ke]δe −M∑e=1

F e = 0 (4.90)


GM/WF:

M∑e=1

(Aφeh − f, v)Ωe =M∑e=1

Be(φeh, v)−M∑e=1

l˜e(v)−M∑e=1

le(v) = 0

=M∑e=1

[Ke]δe −M∑e=1

P e −M∑e=1

F e = 0 (4.91)

LSP: When A is linear

2

M∑e=1

(Ee, δEe)Ωe = 2

M∑e=1

(Aφeh − f,Av)Ωe = 2( M∑e=1

Be(φeh, v)− le(v))

= 0

orM∑e=1

(Aφeh, Av)Ωe −M∑e=1

(f,Av)Ωe = 0

orM∑e=1

[Ke]δe −M∑e=1

F e = 0 (4.92)

By examining (4.90)–(4.92), we note that (4.91) can be used when thefinite element processes are derived using GM, PGM, WRM, GM/WF andLSP (for linear operators) except that P e only exists in the case GM/WF.Thus, we consider the following to hold for all finite element processes exceptthose derived using LSP for non-linear differential operators. Consider

M∑e=1

[Ke]δe =M∑e=1

P e+M∑e=1

F e (4.93)

in which [Ke] is the element matrix, δe are the nodal degrees of freedom foran element e and P e, F e are vectors of secondary variables and the loadvector due to the non-homogeneous part in the mathematical description ofthe BVP. From (4.93), for ΩT =

⋃e Ωe we can write

[K]δ = P+ F (4.94)

in which

δ =⋃e

δe, P =

M∑e=1

P e ; F =

M∑e=1

F e, [K] =

M∑e=1

[Ke]

(4.95)where [K] is the result of the sum, assembly or addition of the elementmatrices [Ke] and likewise P and F are the sum, assembly or additions


of P e and F e. δ are the nodal degrees of freedom for ΩT . (4.94)holds for the discretization ΩT . The rules for the assembly of the elementequations are derived using the following.

(1) The functionals such as B(·, ·) and l(·) are the sum of the functionalsBe(·, ·) and le(·), l˜e(·) for the individual elements.

(2) At the mating boundaries of the elements, δe at the common nodesmust have unique values. This is also expressed by

δ =⋃e

δe

These conditions are known as inter-element continuity conditions onthe nodal variables and must be considered for each element of the dis-cretization ΩT .

More details on how to assemble the element equations will be presented inchapters 5, 6 and 7 in connection with specific model problems and numericalstudies.

4.2.6 Consideration of boundary conditions in theassembled equations

4.2.6.1 GM, PGM, WRM, and LSP based on theresidual functional

First we note that in GM, PGM, WRM, and LSP based on residualfunctional there is no integration by parts hence the concepts of PVs, SVs,EBCs, and NBCs do not exist. Hence in the element equations and thereforethe assembled equations there is no concept of secondary variables P e andP. The assembled equations in this case are of the form

[K]δ = F (4.96)

in which F is a known vector. In GM, PGM, WRM, and LSP (4.96) mustbe subjected to all boundary conditions of the BVP i.e. the known BCsof the BVP must be described in terms of some dofs of the vector δ andimposed in (4.96). In this process δ vector can be split into

δT = δ1T , δ2T (4.97)

in which δ1 are now known due to the known BCs of the BVP and δ2contains those dofs that are yet to be determined.


4.2.6.2 GM/WF

In Galerkin method with weak form over an element Ωe integration byparts results into the definitions of PVs, SVs, EBCs, and NBCs and defini-tion of secondary variables into vector P e. This is due to the fact that overan element the concomitant can not be simplified using BCs (more specifi-cally NBCs) of the BVP. Thus, in GM/WF the assembled equations for thediscretization ΩT will be of the form

[K]δ = P+ F (4.98)

The secondary variable vector P is due to assembly of the secondary vari-ables resulting for each element of the discretization ΩT and is not known.As in other methods, in GM/WF also F is a known vector. Thus, in thismethod consideration of the BCs (EBCs and NBCs) in (4.98) will result inspecification of some elements of δ as well as some elements of P. Weconsider details in the following.

Consideration of NBCs and EBCs

(1) The NBCs only affect the sum of the secondary variable vector P inthe assembled equations.

(2) The sum of the secondary variables at a node from all connecting ele-ments must be equal to the specified external disturbance at the node.

(3) If the specified disturbance at a node is zero then the sum of the sec-ondary variables at that node must be zero.

(4) Consideration of the EBCs that are specifications of PVs on some bound-aries result in specification of some components of the degrees of freedomvector δ say δ1.

(5) If the primary variables are specified at a node then the sum of thesecondary variables is unknown at that node. Thus, the sum of thesecondary variables is known at all those grid points where the primaryvariables are unknown and their sum is not known where the primaryvariables are known. This clearly helps us in identifying the componentsof the vector P that are known, say P2, and those that are notknown, say P1.

Thus in GM/WF we have

P =

P1P2

(4.99)

δ =

δ1δ2

(4.100)


We note that in (4.99) and (4.100) P2 and δ1 are known and P1 andδ2 are unknown. We note that if a component of P is known then thecorresponding component of δ is not known and if a component of δ isknown then the corresponding component of P is not known. This rulemust always hold and there is no exception to this. As an example, in caseof solid mechanics δ can be displacements and P can be forces, hencethis rule in this case means that if at a point displacement is known then theforce is not known (reaction) and if the force is known then the displacementis not known. At any point both displacements and forces can not be knownor unknown. If one is known, then the other is defined by the response ofthe structure.

4.2.7 Computation of the solution: finite element processesbased on all methods of approximation except LSP fornon-linear operators

It is sufficient to consider the assembled equations for GM/WF as forother methods we only have to set P to null vector. The assembled equa-tions (4.94) for the whole discretization ΩT are partitioned in terms of knowndegrees of freedom δ1 and the unknown degrees of freedom δ2 to be cal-culated, [

[K11] [K12][K21] [K22]

] δ1δ2

=

P1P2

+

F1F2

(4.101)

When the differential operator A is linear, (4.101) is a system of linearsimultaneous algebraic equations in the nodal degrees of freedom. Whenthe differential operator A is non-linear, (4.101) is naturally a system ofnon-linear algebraic equations in the nodal dofs and must be solved usingiterative methods. We consider the case when (4.101) is a system of linearsimultaneous algebraic equations to illustrate the details. We note that δ1,P2, F1 and F2 are all known, whereas δ2 and P1 are unknown.From (4.101), we could solve for δ2 using

[K21]δ1+ [K22]δ2 = P2+ F2 (4.102)

or[K22]δ2 = P2+ F2 − [K21δ1 (4.103)

We can use elimination methods such as Gauss elimination to solve for δ2from (4.103). Symbolically, we can write

δ2 = [K22]−1(P2+ F2 − [K21]δ1

)(4.104)

Thus, now the entire solution [δ1T , δ2T ] is known. The unknown sec-ondary variables P1 are calculated using

P1 = [K11]δ1+ [K12]δ2 − F1 (4.105)


4.2.8 Assembly of element equations and their solution infinite element processes based on residual functional(LSP) when the differential operator A is non-linear

Let φh =⋃e φ

eh be the approximation of φ over ΩT , the discretization of

Ω. Then in LSP, we have

(1) Existence of a functional I(φh)

I(φh) = (E,E)ΩT (4.106)

E = Aφh − f (4.107)

(2) Necessary condition

δI(φh) = 2(E, δE)ΩT = 2g(φh) = 0 or g(φh) = 0 (4.108)

(3) Sufficient condition or extremum principle

δ2I(φh) = 2(δE, δE)ΩT + 2(E, δ2E)ΩT (4.109)

Since the differential operator A is non-linear, g(φh) in (4.108) is a non-linear function of φh and, hence, we must find a φh iteratively that satisfies(4.108). Let

φh =⋃e

φeh (4.110)

where

φeh =n∑i=1

Niδei (4.111)

is the local approximation. Theng(φh)

=g(δ)

= 0, δ =

⋃e

δe (4.112)

must be satisfied iteratively. Let δ0 be an assumed solution (initial guessof δ). Then

g(δ0)6= 0 (4.113)

Let ∆δ be a change in δ0 such thatg(δ0+ ∆δ)

= 0 (4.114)

Expanding g(·) in (4.114) in Taylor series about δ0 and retaining onlyup to linear terms in ∆δ (Newton’s linear method or Newton Raphsonmethod)

g(δ0+ ∆δ)

=g(δ0)

+∂g∂δ

∣∣∣∣δ0∆δ+ · · · = 0 (4.115)


Hence

∆δ ≈ −[∂g∂δ

]−1

δ0

g(δ0)

(4.116)

However,

∂g∂δ

= δg =1

2δ2I = (δE, δE)ΩT + (E, δ2E)ΩT (4.117)

From (4.116) and (4.117), we obtain

∆δ = −1

2[δ2I]−1

δ0g(δ0)

(4.118)

For a unique ∆δ from (4.118), the coefficient matrix [δ2I]δ0 must bepositive definite. This is possible if we approximate δ2I by

δ2I ≈ 2(δE, δE)ΩT > 0 (4.119)

which yields a unique extremum principle or sufficient condition, and wehave

∆δ ≈ − [(δE, δE)ΩT ]−1δ0

g(δ0)

=

1

2

[δ2I]−1

δ0g(δ0)

(4.120)

An improved solution δ is obtained using

δ = δ0+ α∆δ (4.121)

in which generally 0 < α ≤ 2 and is determined using I(δ) ≤ I(δ0).That is, for series of values of α, we determine I(δ) and choose an αfor which I(δ) is minimum. This is referred to as line search. The linesearch is helpful in accelerating the convergence of the Newton’s first-orderiterative method due to the fact that when a right direction (i.e. ∆δ) hasbeen found using (4.120), we proceed in this direction as far as possible aslong as I(δ) is less than or equal to I(δ0). The details presented abovecan be easily expressed for the discretization ΩT such that the contributionsof the individual elements are summed or assembled. We provide these inthe following. Assume a starting solution δ0. Then

(1)

I(φh) = (E,E)ΩT =

M∑e=1

(Ee, Ee)Ωe =

M∑i=1

Ie (4.122)

in which E = Aφh − f and Ee = Aφeh − f


(2)

δI(φh) = 2(E, δE)Ωe = 2

M∑e=1

(Ee, δEe)Ωe

= 2

M∑e=1

ge = 2g = 0 or g = 0 (4.123)

We must find a solution that satisfies g = 0 iteratively.

(3)

δ2I(φh) ≈ 2(δE, δE)ΩT = 2M∑e=1

(δEe, δEe)Ωe > 0 (4.124)

(4)

∆δ = −1

2[δ2I]−1

δ0 g(δ0) = −[ M∑e=1

(δEe, δEe)Ωe

]−1

δ0

g(δ0)

(4.125)

or

∆δ = −[ M∑e=1

[Ke]]−1

δ0

g(δ0)

= −[K]−1

δ0g(δ0)

(4.126)

in which [Ke] = (δEe, δEe)Ωe is the element matrix for an element e ofΩT and

[K] =M∑e=1

[Ke] (4.127)

is the assembled matrix for ΩT . We also note that g =∑M

i=1ge, i.e.g is the result of the assembly of ge for the individual elements ofΩT . The improved solution δ is given by

δ = δ0+ α∗∆δ, 0 < α∗ ≤ 2 such that I(δ) ≤ I(δ0) (4.128)

(5) Compute g using (4.123). If δ in (4.127) is the converged solution,then

g(δ)

= 0 holds. We check the absolute values of the com-

ponents of g to ensure that these are less than or equal to a presettolerance ∆, a threshold value for numerically computed zero (generally∆ ≤ 10−6 suffices) for the convergence of the iterative solution method.

(6) If the criterion in 5. is satisfied, then we have a new δ that satisfies(4.123) and, hence is the desired converged solution from the Newton’slinear method with line search. If not, then we set δ0 equal to δ,the current solution and repeat steps 2 through 6.


Remarks.

(1) The iterative solution method presented here is known as Newton’s linearmethod with line search.

(2) With the approximation (4.124), we have a least-squares finite elementformulation for a BVP in which the differential operator A is non-linearbut the integral form is variationally consistent.

(3) The motivation for neglecting (E, δ2E)ΩT in δ2I(φh) is obviously toachieve variational consistency of the integral form. This approxima-tion is not as crude or heuristic as it might appear.

(a) When we are in the close proximity to the correct solution E ≈ 0,then (E, δ2E)ΩT can be expected to be making only a small (i.e.negligible) contribution to δ2I(φh).

(b) We note that in finding the solution δ we are in fact solving fora root of g(δ) = 0 = 1

2δ(I(δ)

), hence, δ2I(δ) represents a

tangent plane to the hyper surface defined by δI(δ) = 0 at δ0.Thus, approximating δ2I amounts to changing the orientation (orslope) of the tangent plane to the hypersurface δI = 0 at δ0. Ithas no effect on the least-squares process which ends with (4.122)and (4.123).

(4) In view of (a) and (b) in (3), the approximation in (4.124) is not heuristicbut well justified especially when its major benefit is variational consis-tency of the integral form that leads to unconditional stability of thecomputations.

4.2.9 Post processing of the solution

Once the nodal values of the solution in δ are all known and since

δ =⋃e

δe (4.129)

we in fact know δe, the degrees of freedom for each element of the dis-cretization ΩT . Using local the approximation

φeh =n∑i=1

Niδei = [N ]δe (4.130)

the solution is defined everywhere over each element Ωe, i.e. Ωe as well as Γe.Using (4.130), any desired further processing of the solution can be done onan element by element basis. This may involve calculating derivatives of φehor any desired norms over Ωe. This part of the finite element computationsis referred to as the post-processing phase (the word post implying after thecalculation of the solution δ).

4.3. SUMMARY 221

4.3 Summary

Ideal and important features of a good computational method are dis-cussed. Mathematical classification of the differential operator is shown tobe essential to address finite element processes for stability of all BVPs ina consistent and rigorous manner. Basic steps in the finite element process:discretization, integral forms, element matrices resulting from the integralforms, assembly of element equations, computation of solution and postprocessing are introduced. Various methods of approximation such as GM,GM/WF, WRM, PGM, and LSP are considered for the three classes of dif-ferential operators and the VC or VIC of the resulting form is evaluated toestablish unconditional stability of the resulting computational process orlack of it.

[1–10]

References for additional reading[1] B. Jiang. The Least-Squares Finite Element Method: Theory and Applications in

Computational Fluid Dynamics and Electromagnetics. Springer, 1998.

[2] K. S. Surana, L. R. Anthoni, S. Allu, and J. N. Reddy. Strong and weak formof governing differential equations in least squares finite element processes in hpkframework. Int. J. Comp. Meth. in Eng. Sci. and Mech., 2008.






[8] O. C. Zienkiewicz. The Finite Element Method, volume 1. McGrawhill, England, 4rdedition, 1989.

[9] K. J. Bathe. Finite Element Procedures. Prentice Hall, New Jersey, 1996.

[10] J. C. Heinrich and D. W. Pepper. Intermediate Finite Element Method. Taylor andFrancis, Philadelphia, 1999.

5

Self-Adjoint DifferentialOperators

5.1 Introduction

In this chapter we consider one, two and three dimensional boundaryvalue problems in single and multi variables that are described by self-adjoint differential operators. The self-adjoint differential operators arisein many branches of engineering, science, and mathematical physics. Lin-ear elastic solid mechanics, structural mechanics, and linear heat conditionare some examples in engineering. First, we review some basic properties ofthese differential operators and the integral forms resulting from the classicalmethods of approximation.

Consider Aφ− f = 0 in Ω.

(1) The self-adjoint differential operators are linear and symmetric. Theseoperators contain even order derivatives of the dependent variables. Forthese problems the adjoint A∗ of the operator is the same as the operatorA.

(2) Based on the material presented in chapter 3, the integral forms usingGM, PGM, and WRM over Ω are always variationally inconsistent.

(3) In GM/WF, we begin with fundamental lemma and transfer half of thedifferentiation from the dependent variable to the test function to obtain

B(φn, v)Ω = l(v)Ω (5.1)

in which B(·, ·) is bilinear and symmetric and l(v) is linear. The weakform (5.1) is variationally consistent.

(4) The integral form based on residual functional (LSM or LSP) is alsovariationally consistent. Here, we use

E = Aφn − f in Ω

I = (E,E)Ω

δI = 0

(5.2)

to obtain the desired integral form.

223

224 SELF-ADJOINT DIFFERENTIAL OPERATORS

(5) Variationally consistent integral forms yield unconditionally stable com-putational processes and ensure unique φn. Based on (2) - (5) we onlyconsider finite element processes derived using GM/WF and LSP. Thedetails of classical GM/WF and LSP have been presented in chapter 3.In the classical methods, the domain of definition Ω is not discretizedand hence the approximation φn is global over Ω. The details of the finiteelement processes derived using the classical methods of approximationhave also been presented in chapter 3. In the following, we presentthe important steps involved in the finite element processes based onGM/WF and LSP (described in chapter 3).

In all finite element processes, regardless of the choice of integral form(i.e., the method of approximation), the following details are common:

Let ΩT =⋃Me=1 Ωe be the discretization of Ω in which Ωe = Ωe ∪ Γe =

closure of Ωe is the domain of an element e and Γe is the closed boundary ofΩe. Let φh be the approximation of φ over ΩT and φeh, the approximationof φ over Ωe such that φh =

⋃e φ

eh.

5.1.1 GM/WF

In this method we begin with the integral statement (based on funda-mental lemma) over ΩT ,

(Aφh − f, v)ΩT =

∫ΩT

(Aφh − f)v dΩ = 0, v = δφh (5.3)

or ∑e

(Aφeh − f, v)Ωe = 0 or∑e

(Aφeh, v)Ωe −∑e

(f, v)Ωe = 0 (5.4)

For an element e consider (Aφeh − f, v)Ωe = (Aφeh, v) − (f, v)Ωe . Since thedifferential operator is self-adjoint (implying even order derivatives of φeh),we transfer half of the differentiation from φeh to v to obtain


in which Be(φeh, v) is bilinear and symmetric and l˜e(v) is the concomitant

resulting as a consequence of integration by parts. We note that since theintegrals are over Ωe, the domain of an element e, the concomitant l˜e(v)

cannot be simplified using the boundary conditions of the BVP. Instead,the unknown secondary variables appearing in l˜e(v) need to be symbolically

identified for each node of the element. Using (5.5), for an element e we canwrite

(Aφeh − f, v)Ωe = Be(φeh, v)− l˜e(v)− (f, v)Ωe (5.6)

5.1. INTRODUCTION 225

upon substituting local approximation φeh given by

φeh =n∑i=1

Niδei = [N ]δe (5.7)

andv = δφeh = Nj , j = 1, 2, . . . , n (5.8)

into (5.6) and using the notations,

Be(φeh, v) = [Ke]δe(f, v)Ωe = F el˜e(v) = P e

(5.9)

we can write (5.6) in the form

(Aφeh − f, v)Ωe = [Ke]δe − P e − F e (5.10)

In the numerical example considered in this chapter, the details of the ele-ment matrices and the vectors on the right side of (5.10) will be considered.Using (5.10), assembly of element equations follows standard procedure de-scribed in this chapter and symbolically described in chapter 4.

5.1.2 LSP based on residual functional

In this method the residual functions E and Ee are defined as

E = Aφh − f in ΩT

Ee = Aφeh − f in Ωe (5.11)

Let the residual functional I(φh) be given by

I(φh) = (E,E)ΩT =∑e

Ie =∑e

(Ee, Ee)Ωe , Ie = Ie(φeh) (5.12)

Then the necessary condition for an extremum of I is

δI = 2(E, δE)ΩT =∑e

δIe = 2∑e

(Ee, δEe)Ωe = 0 = 2∑e

ge = 2g

(5.13)or

g =∑e

ge =∑e

(Ee, δEe)Ωe = 0 (5.14)

We consider ge = (Ee, δEe)Ωe for an element e

δEe = Av, v = δφeh (5.15)


Thus

ge = (Aφeh − f,Av)Ωe = (Aφeh, Av)Ωe − (f,Av)Ωe (5.16)

or

ge= (Aφeh − f,Av)Ωe = Be(φeh, v)− le(v) (5.17)

Using local approximations (5.7) and (5.8), (5.17) can be written as

ge = [Ke]δe − F e (5.18)

Details of [Ke], δe and F e will be considered for each of the numeri-cal examples presented in the following sections. Assembly of the elementequations (5.18) and solution follows standard procedure.

5.2 One-dimensional BVPs in a single dependentvariable

5.2.1 1D steady-state diffusion equation: finite elementprocesses based on GM/WF

Consider the following BVP:

− d

dx

(a(x)

dφ

dx

)+ c φ− q(x) = 0 ∀x ∈ Ω = (0, L) ⊂ R1 (5.19)

with the boundary conditions

φ(0) = φ0 and(adφ

dx

)∣∣∣x=L

= P (5.20)

where a(x), c, φ0 and q(x) are known (data).

This BVP describes a variety of physical processes: axial deformationof a rod, 1D heat conduction, flow through channels and pipes, transversedeflection of cables, etc. A schematic of the problem (for axial deformationof a rod) is shown in Fig. 5.1. In this case

A = − d

dx

(a(x)

d

dx

)+ c (5.21)

and

f = q(x) (5.22)


Aφ− f = 0 ∀x ∈ Ω (5.23)

In this case, we can show that the differential operator A is

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE 227

(i) linear

(ii) symmetric when φ0 = 0 and P = 0. However, we note that in

(Aφ, v) = (φ,A∗v) + 〈Aφ, v〉Γ (5.24)

A∗ = A holds and the concomitant 〈Aφ, v〉Γ only becomes zero whenφ0 and P are both zero.

The physics of the problem in Fig. 5.1 (a) can be idealized as shown inFig. 5.1 (b) due to the fact that φ = φ(x). That is, a line representation ofthe domain Ω = [0, L] is justified without any additional assumptions.

5.2.1.1 Discretization

Figure 5.1 (c) shows a typical discretization of the domain Ω = [0, L]using two-node line elements with element lengths of h1, h2, h3, h4. A typicalelement e with domain Ωe = [xe, xe+1] isolated from the discretization isshown in Fig. 5.1 (d). A map of the element e in the natural coordinatespace ξ is shown in Fig. 5.1 (e) in which the element is mapped into a twounit length with the origin of the ξ coordinate system located at the centerof the element, i.e. Ωe → Ωξ = [−1, 1]. We discuss the details of the validityof the choice of the two-node element later. Let

h = maxe

(he) (5.25)

in which h is referred to as the characteristic length of the discretization

5.2.1.2 Integral form using GM/WF (weak form) of the BVP foran element e with domain Ωe

We consider (Aφeh − f, v)Ωe in which v = δφeh.

(Aφeh − f, v)Ωe =

xe+1∫xe

(Aφeh − f)v dx (5.26)

or

(Aφeh−f, v)Ωe =

xe+1∫xe

− d

dx

(a(x)

dφehdx

)v dx+

xe+1∫xe

c φehv dx−xe+1∫xe

qv dx (5.27)


21

1−1

1 21 2 3 4 543

(d) Isolating an element e fromthe discretization in (c)

(e) Map of an element e in thenatural coordinate system

η

y

xe xe+1

he

x ξ

Node numbers 1 and 2are local node numbers

y

(b) Mathematical idealization of the problem

xP =

(a(x)dφ

dx

)∣∣∣x=L

q(x)

L

φ = φ0 = 0

y

x

element length (element characteristic length)

element number

(c) A four element discretization of the domain Ω

grid point

h4h3h2h1

y

(a) Schematic of the problem

x P =(a(x)dφ

dx

)∣∣∣x=L

q(x)

L

e

e

Figure 5.1: Schematic and discretization for problem of section 5.2.1

Transferring one order of differentiation from(a(x)

dφehdx

)to v in the first

term in (5.27)


xe+1∫xe

dv

dxa(x)

dφehdx

dx−[v

(a(x)

dφehdx

)]xe+1

xe

+

xe+1∫xe


qv dx (5.28)


In (5.28) the concomitant 〈Aφeh, v〉Γe is

〈Aφeh, v〉Γe =

[v

(a(x)

dφehdx

)]xe+1

xe

Using the concomitant in (5.28) we identify PV, SV,EBC and NBC.

PV : φeh

SV : a(x)dφehdx

EBC : φeh = φ on some Γ1

NBC : a(x)dφehdx

= q on some Γ2

(5.29)

where φ and q are known on some boundaries Γ1 and Γ2 of the element e.We note that the boundary term in (5.28) cannot be simplified using theboundary conditions of the BVP (as done in classical methods) due to thefact that the element e is any arbitrary element of the discretization and theBCs can only be used for the whole discretization i.e. the assemblage of theelement equations. Let

−(a(x)

dφehdx

)∣∣∣xe

= P e1(a(x)

dφehdx

)∣∣∣xe+1

= P e2

(5.30)

where P e1 and P e2 are the secondary variables at nodes 1 and 2 (local nodenumbers) of element e. We note that the secondary variables P e1 and P e2 areunknown. Substituting from (5.30) into (5.28)


xe+1∫xe

dv

dxa(x)

dφehdx

dx− v(xe)Pe1 − v(xe+1)P e2

+

xe+1∫xe


qv dx (5.31)

Collecting the terms containing both φeh and v and those containing only v


xe+1∫xe

(dvdxa(x)

dφehdx

+ c φehv)dx

−( xe+1∫xe

qv dx+ v(xe)Pe1 + v(xe+1)P e2

)(5.32)


We define

Be(φeh, v) =

xe+1∫xe

(dvdxa(x)

dφehdx

+ c φehv)dx (5.33)

le(v) =

xe+1∫xe

qv dx+ v(xe)Pe1 + v(xe+1)P e2 (5.34)

or


Equation (5.35) is the desired weak form of the BVP (5.19). We note that

(a) Be(φeh, v) is bilinear

(b) Be(φeh, v) is symmetric, Be(φeh, v) = Be(v, φeh)

(c) le(v) is linear in v

Due to these properties of Be(·, ·) and le(·), the quadratic functional Ie(φeh)for an element e is possible and is given by

Ie(φeh) =1

2Be(φeh, φ

eh)− le(φh) (5.36)

or

Ie(φeh) =1

2

xe+1∫xe

(a(x)

(dφehdx

)2+ c(φeh)2

)dx

−xe+1∫xe

q φeh dx− φeh(xe)Pe1 − φeh(xe+1)P e2 (5.37)

We note that δIe(φeh) yields the weak form (5.35) over an element Ωe and

δ2Ie(φeh) = δ(Be(φeh, v)− le(v)) = Be(v, v) (5.38)

or

δ2Ie(φeh) = Be(v, v)

=

xe+1∫xe

(a(x)

(dvdx

)2+ c(v)2

)dx > 0 if a > 0, c > 0 (5.39)


5.2.1.3 Approximation space Vh, test function space V and localapproximation φeh

First, we note that in GM/WF we begin with


(Aφeh − f, v) = 0, φh =⋃e

φeh (5.40)

After integration by parts for an element e with domain Ωe, we obtain


(Aφeh − f, v)Ωe =∑e

Be(φeh)−∑e

le(v) = 0 (5.41)

We consider the approximation space Vh and test function space V in thefollowing for the BVP as well as the weak form.

The local approximation φeh is generally a polynomial of degree p. Thus,φeh is of class Cp(Ωe). However, the global differentiability of φh is controlledby differentiability of φeh over Ωe as well as the differentiability at the in-terelement boundaries. Obviously, since φh =

⋃e φ

eh, the lower of these two

is the global differentiability of φh. We reflect this in defining the class ofφeh. For example, if φeh is a polynomial of degree p, but if at the interelementboundaries only φh is continuous then φh is of class C0 and we say thatφeh ∈ C0(Ωe). Thus, the class of φh is reflected in defining the class of φeh.

Vh for BVP

Since the highest order of the derivative of φ in the BVP is two (2m = 2),for φh over ΩT to be admissible in the BVP in the pointwise sense, thefollowing must hold:

φh ∈ Vh ⊂ Hk(ΩT ), k ≥ 3 hence φeh ∈ Vh ⊂ Hk(Ωe), k ≥ 3

In which k = 3 corresponds to the minimally conforming space. For thechoice of k ≥ 3, (Aφh − f, v)ΩT is in the Riemann sense and all three formsin (5.41) are precisely equivalent. That is

(Aφh−f, v)ΩT ⇔R

∑e

(Aφeh−f, v)Ωe ⇔R

∑e

(Be(φeh, v)− le(v)

)(5.42)

where R indicates that the integrals are Riemann. Since v = δφh, or v = δφeh,v ∈ Vh also holds.


Vh of one order lower than minimally conforming for the BVP

If we choose Vh to be one order lower than that of the BVP, i.e. if wechoose the order of the space Vh to be k = 2. then

(Aφh − f, v)ΩT8→L

∑e

(Aφeh − f, v)Ωe8→L

∑e


)︸︷︷︸

R

(5.43)where L indicates that the integrals are Lebesgue. For this choice, the lastform (i.e. weak form) in (5.43) holds in the Riemann sense over ΩT as itcontains only the first order derivatives of φeh and since φeh, φh are of classC1. For this choice it is possible to go from ΩT to Ωe in Lebesgue sense andthen to the weak form but not possible to return back to the integral formover ΩT .

Vh of order one, i.e. local approximations φeh of class C0(Ωe)

If we just examine the weak form Be(φeh, v)− le(v), we note that it onlycontains first order derivatives of the dependent variable and the test func-tion as the highest order derivatives. Thus, if we choose φeh of class C0, i.e.in H1(Ωe) space, then

∑e(B

e(φeh, v) − le(v)) is in the Lebesgue sense. Inthis case, obviously (Aφh − f, v)ΩT and

∑e(Aφ

eh − f, v)Ωe are not defined

and symbolically we can write

(Aφh − f, v)ΩT <∑e

(Aφeh − f, v)Ωe <∑e


)︸︷︷︸

L

(5.44)

That is, when φeh ∈ Vh ⊂ H1(Ωe), φh ∈ Vh ⊂ H1(ΩT ) all equivalences in(5.44) are lost.

A remark on the notation

The definite integrals are always over the closed domain. For example,a line integral of f(x) for a ≤ x ≤ b is independent of the path and onlydepends upon a and b. Thus, the notation used in the book is correct (i.e.integrals are over Ωe, ΩT and not over Ωe, ΩT ). However, we note that theintegrals can be in the Riemann or Lebesgue sense depending upon the choiceof the approximation space Vh. To emphasize this fact we could change theintegrals over Ωe to Ωe meaning that these could have resulted from Lebesguemeasures. This may be the case in most writings on the subject. However,in this book we adhere to the correct notation, i.e. integrals over Ωe, ΩT

etc. and not over Ωe, ΩT etc.


5.2.1.4 Local approximation φeh and mapping Ωe to Ωξ

From the weak form (5.32)–(5.35), we note that an approximation of φehof φ of class C0 (i.e. k = 1) must at least be differentiable once, hence, alinear approximation of φ over Ωe is the lowest possible requirement. Thoughsuch choice is completely inadmissible based on minimally conforming spacefor this BVP (k = 3), however we continue with this choice to illustratevarious procedural details as these become much simplified with this choice.Based on the interpolation theory in chapter 8, a two-node linear elementmay be chosen to satisfy these requirements.

Let φe1 and φe2 be the values of φ at the two nodes (local node numbers1 and 2) of a typical element e (see Fig. 5.1). Since the element is a linesegment, its map in the natural coordinate space ξ will also be a line segmentof length two units, that is, a stretch mapping suffices for mapping of Ωe

into Ωξ. That is

x(ξ) =(1− ξ

2

)xe +

(1 + ξ

2

)xe+1 (5.45)

and

φeh(ξ) =(1− ξ

2

)φe1 +

(1 + ξ

2

)φe2 = N1(ξ)φe1 +N2(ξ)φe2 (5.46)

or

φeh(ξ) =2∑i=1

Ni(ξ)φei =

2∑i=1

Ni(ξ)δei = [N ]δe (5.47)

where [N ] = [N1, N2] is a row matrix of approximation functions and δeT =[φe1, φ

e2]. Equation (5.45) describes the mapping of points from the ξ space

to the x space. φeh is the local approximation of φ over Ωe in which φei or δeior δe are called degrees of freedom for an element e. Using (5.45)

dx =dx

dξdξ = J dξ (5.48)

where J is called the Jacobian of transformation. For this particular mapping(5.45), we have

dx =dx

dξ=xe+1 − xe

2=he2

(5.49)

he being the length of the element e in the Cartesian coordinate space. We

also note that in the weak form, we requiredφehdx , i.e. dNi

dx (i = 1, 2). SinceNi = Ni(ξ), we can write

dNi

dξ=dNi

dx

dx

dξ= J

dNi

dx, i = 1, 2 (5.50)

HencedNi

dx=

1

J

dNi

dξ, i = 1, 2 (5.51)


5.2.1.5 Element equations

Recall the weak form of the BVP over Ωe


or


xe+1∫xe

(dvdxa(x)

dφehdx

+ c φehv)dx

−( xe+1∫xe

qv dx− v(xe)Pe1 − v(xe+1)P e2

)(5.53)

We have

φeh = N1(ξ)φe1 +N2(ξ)φe2, δφeh = v = Nj(ξ), j = 1, 2 (5.54)

dφehdx

=dN1

dxφe1 +

dN2

dxφe2,

dv

dx=dNj

dx, j = 1, 2, dx = J dξ (5.55)

Substituting from (5.54) and (5.55) into (5.53) and changing the limits ofintegration to (−1, 1) yield


1∫−1

(dNj

dxa(ξ)

( 2∑i=1

dNi

dxφei

)+ c( 2∑i=1

Ni φei

)Nj

)J dξ

−1∫−1

q(ξ)Nj(ξ)J dξ −Nj(−1)P e1 −Nj(1)P e2 (5.56)

for j = 1, 2. We can expand (5.56) for j = 1, 2. This gives the two relations


1∫−1

(dN1

dxa(ξ)

( 2∑i=1

dNi

dxφei

)+ c( 2∑i=1

Ni φei

)N1

)J dξ

−1∫−1

q(ξ)N1(ξ)J dξ −N1(−1)P e1 −N1(1)P e2 (5.57)

and


1∫−1

(dN2

dxa(ξ)

( 2∑i=1

dNi

dxφei

)+ c( 2∑i=1

Ni φei

)N2

)J dξ

−1∫−1

q(ξ)N2(ξ)J dξ −N2(−1)P e1 −N2(1)P e2 (5.58)


Equations (5.57) and (5.58) can be arranged in matrix and vector form togive the following element equations

(Aφeh − f, v)Ωe = [Ke]δe − F e − P e (5.59)

where Kij of [Ke], δei of δe, F ei of F e and P e are given by

Keij =

1∫−1

(dNi

dxa(ξ)

dNj

dx+ cNiNj

)J dξ, i, j = 1, 2

F ei =

1∫−1

q(ξ)Ni J dξ, i = 1, 2

δei = φei , i = 1, 2

P e = P e1 , P e2 T because

N1(−1) = 1, N1(1) = 0

N2(−1) = 0, N2(1) = 1

(5.60)

For this particular choice of local approximation (5.54), we have

dN1

dξ= −1

2,dN2

dξ=

1

2

dN1

dx=

1

J

dN1

dξ=

1

he/2

(−1

2

)= − 1

he(5.61)

dN2

dx=

1

J

dN2

dξ=

1

he/2

(1

2

)=

1

he

Substituting from (5.61) into (5.60) and assuming a(ξ) = ae, c = ce andq(ξ) = qe constant for an element e and evaluating the integrals, we obtain

[Ke] =ae

he

[1 −1−1 1

]+cehe

6

[2 11 2

], F e =

qehe2

11

=

F e1F e2

(5.62)

We note that if a = a(x), c = c(x) and q = q(x) then using the mappingx = x(ξ), we can easily obtain a = a(ξ), c = c(ξ) and q = q(ξ) which can thenbe used in (5.60) to evaluate the coefficients of [Ke] and F e. The elementmatrix [Ke] and the vectors F e and P e are valid for each element of thediscretization; that is, by using e = 1, . . . ,M we have element matrices [Ke]and the vectors F e and P e for each element of the discretization.

5.2.1.6 Assembly of element equations andcomputation of the solution

For illustrating the various steps clearly, we consider a uniform discretiza-tion of four elements (see Fig. 5.2) with ae = a, ce = 0, qe = q, e =


1, 2, . . . , 4. For this case, he = L/4 = h (e = 1, 2, . . . , 4). For each ofthe four elements of the discretization (noting that ce = 0 and using thelocal node numbers) we have the following element equations:

(Aφ1h − f, v)Ω1 =

[K1

11 K112

K121 K

122

]φ1

1

φ12

−F 1

1

F 12

−P 1

1

P 12

; K1

21 = K112

(Aφ2h − f, v)Ω2 =

[K2

11 K212

K221 K

222

]φ2

1

φ22

−F 2

1

F 22

−P 2

1

P 22

; K2

21 = K212

(Aφ3h − f, v)Ω3 =

[K3

11 K312

K321 K

322

]φ3

1

φ32

−F 3

1

F 32

−P 3

1

P 32

; K3

21 = K312

(Aφ4h − f, v)Ω4 =

[K4

11 K412

K421 K

422

]φ4

1

φ42

−F 4

1

F 42

−P 4

1

P 42

; K4

21 = K412

(5.63)

In the matrix representation of the weak forms in (5.63), the element localnode numbers are used to identify dofs at the element nodes that are onlyintrinsic to each element.

1 21 2 3 4 543

1 2 2 2 2

1 2 3 4

1 1 1

φ11 φ2

1φ12 φ2

2 φ31 φ3

2 φ41 φ4

2

P 11 P 2

1P 12 P 2

2 P 31 P 3

2 P 41 P 4

2

F 11 F 2

1F 12 F 2

2 F 31 F 3

2 F 41 F 4

2

y

x

q = constant

(a) A four element discretization of the domain Ω

P

(b) Primary variables, secondary variables and F e vectorat each element node in the element local node numberingsystem. (he = L/4 = h)

Figure 5.2: A four-element uniform discretization of problem of section 5.2.1


5.2.1.7 Inter-element continuity conditions on PVs or dependentvariables

For the discretization shown in Fig. 5.2 (a), we have four elements andfive grid points, one through five. At each grid point, the function valuesφi; i = 1, . . . , 5 are the quantities of interest. Thus, if the assembly of thefour elements of Fig. 5.2 (b) is to yield Fig. 5.2 (a) then the following inter-element behaviors of φeh must hold. These are called inter-element continuityconditions on φeh or primary variables:

at node 1 of element 1 : φ11 = φ1

at node 2 of element 1


: φ1

2 = φ21 = φ2



: φ2

2 = φ31 = φ3



: φ3

2 = φ41 = φ4

at node 2 of element 4 : φ42 = φ5

(5.64)

When the inter-element continuity conditions are substituted into (5.63), weobtain the following for elements one through four in which the connectionsof the elements to their neighbors are clearly evident due to the uniquefunction values at common nodes:

(Aφ1h − f, v)Ω1 =

[K1

11 K112

K121 K

122

]φ1

φ2

−F 1

1

F 12

−P 1

1

P 12

; K1

21 = K112

(Aφ2h − f, v)Ω2 =

[K2

11 K212

K221 K

222

]φ2

φ3

−F 2

1

F 22

−P 2

1

P 22

; K2

21 = K212

(Aφ3h − f, v)Ω3 =

[K3

11 K312

K321 K

322

]φ3

φ4

−F 3

1

F 32

−P 3

1

P 32

; K3

21 = K312

(Aφ4h − f, v)Ω4 =

[K4

11 K412

K421 K

422

]φ4

φ5

−F 4

1

F 42

−P 4

1

P 42

; K4

21 = K412

(5.65)

5.2.1.8 Rules for assembling element matrices and vectors

We note that



Be(φeh, v)−∑e

le(v) = 0 (5.66)


in which le(v) = l˜e(v) + (f, v)Ωe . l˜e(v) is the concomitant containing sec-

ondary variables. However

Be(φeh, v) = [Ke]δele(v) = F e+ P e

(5.67)

Therefore, after substituting (5.67) into (5.66), we obtain∑e

[Ke]δe =∑e

F e+∑e

P e (5.68)[∑e

Ke]δ =

∑e

F e+∑e

P e (5.69)

in which δ =⋃eδe denoting

[K] =∑e

[Ke], F =∑e

F e, P =∑e

P e (5.70)

we obtain the following from (5.69)

[K]δ = F+ P (5.71)

Equations (5.71) hold for ΩT and, hence are referred to as assembled equa-tions. (5.70) show that [Ke], F e and P e need to be added or assem-bled to obtain (5.71). For the four element discretization considered here,δ = [φ1 φ2 φ3 φ4 φ5]T ; that is, (5.71) represents a system of five linearsimultaneous algebraic equations in unknowns φi (i = 1, . . . , 5). We alsonote that P, the assembled vector of secondary variables, is also unknownat this point.

Method IWe note that element equations (5.65) must be added based on (5.66)

or (5.69) to yield (5.71). However, in (5.65) the unknowns in the elementequations, i.e. nodal values φi, change from element to element. Thus, ifwe are to entertain a brute force matrix addition then we must ensure thatequations for each element are expanded to contain all of the five unknowns,φ1, . . . , φ5. This can be easily done by introducing identities for those dofsthat are not present in the element equations keeping in mind to maintainthe same order of φ1, . . . , φ5 for each element of the discretization. The resultis that we can write

(Aφ1h − f, v)Ω1 =

K1

11 K112 0 0 0

K121 K

122 0 0 0

0 0 0 0 00 0 0 0 00 0 0 0 0

φ1

φ2

φ3

φ4

φ5

−F 1

1

F 12

000

−P 1

1

P 12

000

(5.72)


(Aφ2h − f, v)Ω2 =

0 0 0 0 00 K2

11 K212 0 0

0 K221 K

222 0 0

0 0 0 0 00 0 0 0 0

φ1

φ2

φ3

φ4

φ5

−

0F 2

1

F 22

00

−

0P 2

1

P 22

00

(5.73)

(Aφ3h − f, v)Ω3 =

0 0 0 0 00 0 0 0 00 0 K3

11 K312 0

0 0 K321 K

322 0

0 0 0 0 0

φ1

φ2

φ3

φ4

φ5

−

00F 3

1

F 32

0

−

00P 3

1

P 32

0

(5.74)

(Aφ4h − f, v)Ω4 =

0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 K4

11 K412

0 0 0 K421 K

422

φ1

φ2

φ3

φ4

φ5

−

000F 4

1

F 42

−

000P 4

1

P 42

(5.75)

Equations (5.72)–(5.75) can now be added in a straight forward manner (justplain matrix and vector addition) keeping in mind that F e add to F eand P e add to P e. The result is


(Aφeh − f, v)Ωe

=

K111 K1

12 0 0 0

K121

K122

+K211K1

12 0 0

0 K221

K222

+K311K3

12 0

0 0 K321

K322

+K411K4

12

0 0 0 K421 K4

22

φ1

φ2

φ3

φ4

φ5

−

F 11

F 12 + F 2

1

F 22 + F 3

1

F 32 + F 4

1

F 42

−

P 11

P 12 + P 2

1

P 22 + P 3

1

P 32 + P 4

1

P 42

= 0

(5.76)

K122+K2

11

K222

+K311

+K411

K322

Row φ5

Row φ1

Row φ2

Row φ3

Row φ4

K111 K1

12

K121 K2

12

K221

K412

K422K4

21

φ4

φ1

φ3

φ5φ4φ3φ2φ1

φ2

φ5

Columns

= +

P 11

P 12 + P 2

1

P 22 + P 3

1

P 32 + P 4

1

P 42

F 11

F 12 + F 2

1

F 22 + F 3

1

F 32 + F 4

1

F 42 Row φ5

Row φ1

Row φ2

Row φ3

Row φ4

0 0

0

K312 0

0

0

0

0

0 0 0

0 K321

(5.77)These are the assembled algebraic equations for the whole discretization ΩT .


Method II

In this example the differential operator is self-adjoint, hence, the quadraticfunctional I(φh) is possible

I(φh) =∑e

I(φeh) =∑e

(1

2Be(φeh, φ

eh)− le(φeh)

)(5.78)

After substituting inter-element continuity conditions in (5.78) we note thatI = I(φ1, . . . , φ5), hence, minimization of I implies

∂I

∂φi= 0, i = 1, 2, . . . , 5 (5.79)

(5.79), when arranged in the matrix and vector form, yields exactly the sameequations as in (5.77).

Method III (Preferred)

From method I (and likewise method II) we observe that the assembledequation (5.77) can be obtained in a more prudent and efficient manner.From the element equations (5.65), we note that unknown function valuesat the nodes vary from element to element. Thus, for each element we canidentify its rows and columns by using the degrees of freedom that appearin it. In this approach, for element 1 we identify the rows and column by φ1

and φ2 and similarly for the remaining three element. Thus, symbolicallywe can write

K111 K1

12

K121

φ2φ1

Row φ1

Row φ2 K122

−

F 1

2

φ1

φ2

F 11

−

P 11

P 12

Row φ1

Row φ2

Element 1:

Columns

(5.80)

K211 K2

12

K221

φ3φ2

Row φ2

Row φ3 K222

−

F 2

2

φ2

φ3

F 21

−

P 21

P 22

Row φ2

Row φ3

Element 2:

Columns

(5.81)

K311 K3

12

K321

φ4φ3

Row φ3

Row φ4 K322

−

F 3

2

φ3

φ4

F 31

−

P 31

P 32

Row φ3

Row φ4

Element 3:

Columns

(5.82)


K411 K4

12

K421

φ5φ4

Row φ4

Row φ5 K422

−

F 4

2

φ4

φ5

F 41

−

P 41

P 42

Row φ4

Row φ5

Element 4:

Columns

(5.83)

We realize that the assembled equations must correspond to the degreesof freedom φ1, . . . , φ5. Hence, the assembled [K] will be a (5×5) matrix andthe assembled F and P would be (5× 1) vectors. We set aside a (5× 5)(initialized to zero) space for assembly to obtain [K] and (5× 1) spaces forassembly to obtain F and P and identify rows and columns by φ1, . . . , φ5

(in any preferred order) as labels (just like the element matrices and vectors)giving us the following setup ready for assembly of element equations.

Row φ5

Row φ1

Row φ2

Row φ3

Row φ4 φ4

φ1

φ3

φ5φ4φ3φ2φ1

φ2

φ5

Columns

= +

Row φ5

Row φ1

Row φ2

Row φ3

Row φ4

space for [K]

space for Fspace for P

0 0 0 0 0 0 0

0 0 0 0 0

0

0

00

0

0

0

0

0

0

0

0

0

0

0 0000

0 0 0

0

(5.84)Let us consider element one, i.e. [K1], F 1 and P 1 in (5.80) based on therow and column labels for the coefficients. That is, K1

11 corresponding torow φ1 and column φ1 in [K1] is added to the corresponding position (rowφ1, column φ1) in [K] and likewise K1

12 to row φ1, column φ2; K121 to row

φ2, column φ1; and K122 to row φ2, column φ2 locations in [K]. Similarly,

F 11 and F 1

2 corresponding to rows φ1 and φ2 in F 1 were added to thecorresponding row locations in F. P 1 is assembled likewise. At the endof the assembly of (5.80) in (5.84), the locations in (5.84) now contain

Row φ5

Row φ1

Row φ2

Row φ3

Row φ4

K111 K1

12

K121

φ4

φ1

φ3

φ5φ4φ3φ2φ1

φ2

φ5

Columns

= +

P 11F 1

1

Row φ5

Row φ1

Row φ2

Row φ3

Row φ4

K122

F 12 P 1

2

0 0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0

0

0

0

0

(5.85)


Equations for element two (5.81) are now assembled in (5.85) keeping inmind that rows and columns of this element are identified (or labeled) by φ2

and φ3. After assembly of the equations for element two in (5.85), we have

K122+K2

11

Row φ5

Row φ1

Row φ2

Row φ3

Row φ4

K111 K1

12

K121 K2

12

K221

φ4

φ1

φ3

φ5φ4φ3φ2φ1

φ2

φ5

Columns

= +

P 11

P 12 + P 2

1

F 11

F 12 + F 2

1

Row φ5

Row φ1

Row φ2

Row φ3

Row φ4

P 22F 2

2K222

0 0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0

000

(5.86)Likewise, after assembling element three, i.e. (5.82), in (5.86) we have

K122+K2

11

K222

+K311

Row φ5

Row φ1

Row φ2

Row φ3

Row φ4

K111 K1

12

K121 K2

12

K221

K321 φ4

φ1

φ3

φ5φ4φ3φ2φ1

φ2

φ5

Columns

= +

P 11

P 12 + P 2

1

P 22 + P 3

1

F 11

F 12 + F 2

1

F 22 + F 3

1

Row φ5

Row φ1

Row φ2

Row φ3

Row φ4

K312

F 32 P 3

2K322

0 0 0

0

00

0

0

0

0

0

0

0

0 000

(5.87)Finally, after assembling element four, i.e. (5.83), in (5.87) we obtain thefinal assembled equations,

K122+K2

11

K222

+K311

+K411

K322

Row φ5

Row φ1

Row φ2

Row φ3

Row φ4

K111 K1

12

K121 K2

12

K221

K412

K422K4

21

φ4

φ1

φ3

φ5φ4φ3φ2φ1

φ2

φ5

Columns

= +

P 11

P 12 + P 2

1

P 22 + P 3

1

P 32 + P 4

1

P 42

F 11

F 12 + F 2

1

F 22 + F 3

1

F 32 + F 4

1

F 42 Row φ5

Row φ1

Row φ2

Row φ3

Row φ4

0 0

0

K312 0

0

0

0

0

0 0 0

0 K321

(5.88)In (5.88), we observe that the coefficients in [K] and F are known but thecoefficients of assembled P e in P are still unknown.

5.2.1.9 Inter-element continuity conditions on the sum of secondaryvariables

Following the general discussion presented in chapter 4 (repeated herefor convenience), we have the following.


(a) The sum of secondary variables is zero at a node at which there is noexternally applied disturbance.

(b) If there is an externally applied disturbance at a node, then the sumof the secondary variables at that node must be equal to the externallyapplied disturbance.

(c) The sum of all secondary variables at a node at which PVs are specified(EBCs) is unknown.

These conditions are derived based on equilibrium considerations at eachnode of the discretization. Referring to Fig. 5.2 (a), we can arrive at thefollowing conditions for the assembled secondary variables in P.

P 11 : not known because φ1 = 0 (EBC)

P 12 + P 2

1 = 0 : no externally applied disturbance at node 2

P 22 + P 3

1 = 0 : no externally applied disturbance at node 3 (5.89)

P 32 + P 4

1 = 0 : no externally applied disturbance at node 4

P 42 = P : where P is an externally applied disturbance at node 5

5.2.1.10 Imposition of EBCs

The essential boundary conditions of the BVP must be described or spec-ified in terms of the dofs at the nodes or grid points of the discretization.Referring to Fig. 5.1 (b) we have

φ1 = 0 (EBC) (5.90)

5.2.1.11 Solving for unknown degrees of freedom

Based on (5.90), the assembled equation for ΩT in nodal dofs φi (i =1, . . . , 5) have φi (i = 2, . . . , 5) as unknowns since φ1 = 0 is known dueto the EBC (5.90). We partition (5.88) in terms of known nodal dofs andunknown nodal dofs[

[K11] [K12][K21] [K22]

]δ1δ2

=

F1F2

+

P1P2

(5.91)

in which

δ1 = φ1 = 0 known primary degrees of freedom

F1 and F2 both are known

P1 = P 11 unknown due to the fact that δ1 = φ1 = 0 is known

δ2 = φ2 φ3 φ4 φ5T unknown

(5.92)


P2 =

P 1

2 + P 21

P 22 + P 3

1

P 32 + P 4

1

P 42

=

000P

known (5.93)

F2 =

F 1

2 + F 21

F 22 + F 3

1

F 32 + F 4

1

F 42

known

We note that where the φis are known, the sum of secondary variables is notknown. Likewise, where the φis are not known the sum of secondary variablesis known. We first calculate δ2 and then P1. Expanding (5.91)

[K11]δ1 + [K12]δ2 = F1 + P1[K21]δ1 + [K22]δ2 = F2 + P2

(5.94)

From the second set of equations in (5.94) in which the right side is com-pletely known, we can write

[K22]δ2 = −[K21]δ1 + F2 + P2 (5.95)

(5.95) represents a system of linear simultaneous equations in δ2, fromwhich δ2 can be calculated using elimination methods (such as Gausselimination). Symbolically, we write the following from which δ2 can becalculated:

δ2 = [K22]−1(−[K21]δ1 + F2 + P2

)(5.96)

knowing δ2, P1 can be calculated using the first set of equations in(5.94),

P1 = [K11]δ1 + [K12δ2 − F1 (5.97)

Hence, the values of φ at the nodes of the discretization ΩT are completelyknown as well as the secondary variables P1 = P 1

1 at node one whereφ = φ1 = 0 (EBC).

5.2.1.12 Special case: numerical study

Consider a, a constant, c = 0 and q(x) = qe = q, a constant, then for theuniform four element discretization we have, using he = h = L/4

4a

L

1 −1 0 0 0−1 2 −1 0 0

0 −1 2 −1 00 0 −1 2 −10 0 0 −1 1

φ1

φ2

φ3

φ4

φ5

=qL

8

12221

+

P 1

1

000P

(5.98)


The partitioned equations are

4a

L1φ1 +

4

La−1 0 0 0

φ2

φ3

φ4

φ5

=qL

81+ P 1

1 1 (5.99)

and

4

La

−1

000

φ1 +4

La

2 −1 0 0−1 2 −1 0

0 −1 2 −10 0 −1 1

φ2

φ3

φ4

φ5

=qL

8

2221

+

000P

(5.100)

Since φ1 = 0, we have for δ2 using (5.100)φ2

φ3

φ4

φ5

=L

4a

2 −1 0 0−1 2 −1 0

0 −1 2 −10 0 −1 1

−1qL8

2221

+

000P

(5.101)

or φ2

φ3

φ4

φ5

=L

4a

1 1 1 11 2 2 01 2 3 31 2 3 4

qL8

2221

+

000P

(5.102)

or φ2

φ3

φ4

φ5

=qL2

32a

7121516

+PL

4a

1234

(5.103)

And using (5.99) we have

P 11 = −qL

8+

4a

L(−φ2) (5.104)

or

P 11 = −qL

8+

4a

L

(7qL2

32a+PL

4A

)= −qL

8− 7qL

8− P (5.105)

or

P 11 = −qL− P (5.106)

Thus, (5.103) and (5.106) provide the complete solution.


5.2.1.13 Post-processing of solution

Knowing the nodal values of φ, i.e. φi; i = 1, . . . , 5, we can now post-process the solution for each element of the discretization, i.e. using elementlocal approximation functions (in natural coordinate system) and the nodalvalues of the solution, we can write the following.

Element 1: φ(ξ) =(1− ξ

2

)φ1 +

(1 + ξ

2

)φ2 =

(1 + ξ

2

)(7qL2

32a+PL

4a

),

−1≤ξ≤1

0≤x≤L/4


2

)φ2 +

(1 + ξ

2

)φ3 =

(1− ξ2

)(7qL2

32a+PL

4a

)+(1 + ξ

2

)(12qL2

32a+

2PL

4a

),−1≤ξ≤1

L/4≤x≤L/2


2

)φ3 +

(1 + ξ

2

)φ4 =

(1− ξ2

)(12qL2

32a+

2PL

4a

)+(1 + ξ

2

)(15qL2

32a+

3PL

4a

),−1≤ξ≤1

L/2≤x≤3L/4


2

)φ4 +

(1 + ξ

2

)φ5 =

(1− ξ2

)(15qL2

32a+

3PL

4a

)+(1 + ξ

2

)(16qL2

32a+

4PL

4a

),−1≤ξ≤1

3L/4≤x≤L (5.107)

Likewise, the derivatives of φ with respect to x can also be obtained for eachelement of the discretization (note that J = h/2 = L/8).

Element 1:dφ

dx=

1

J

dφ

dξ=

4

L

(7qL2

32a+PL

4a

),−1≤ξ≤1

0≤x≤L/4

Element 2:dφ

dx=

4

L

(5qL2

32a+PL

4a

),−1≤ξ≤1

L/4≤x≤L/2

Element 3:dφ

dx=

4

L

(3qL2

32a+PL

4a

),−1≤ξ≤1

L/2≤x≤3L/4

Element 4:dφ

dx=

4

L

(qL2

32a+PL

4a

),−1≤ξ≤1

3L/4≤x≤L

(5.108)

5.2.1.14 Analytical solution and comparison withfinite element solutions

When a and q are constants and φ = 0 at x = L and adφdx = P at x = L,the analytical solution φt is given by

φt =q

2a

(2Lx− x2

)+P

a, 0 ≤ x ≤ L (5.109)

dφtdx

=q

a

(L− x

)+P

a, 0 ≤ x ≤ L (5.110)


First, we compare φ and φt at the nodes of the discretization and at themid points of the elements.

(1) Table 5.1 shows a comparison of φt and the finite element solution φ.We note that the finite element solution matches exactly with the analyticalsolution at the nodes, but is in error in the interior of each of the fourelements due to the fact that φt is parabolic in x whereas the finite elementsolution is linear over each of the four elements.

Table 5.1: Comparison of φt and φ

x Analytical solution φt Finite element solution φ

node 1 0 0 0

L8

3.7532

qL2

a+ 1

8PLa

3.532

qL2

a+ 1

8PLa

node 2 L4

732qL2

a+ 1

4PLa

732qL2

a+ 1

4PLa

3L8

9.7532

qL2

a+ 3

8PLa

9.532

qL2

a+ 3

8PLa

node 3 L2

1232qL2

a+ 1

2PLa

1232qL2

a+ 1

2PLa

5L8

13.72532

qL2

a+ 5

8PLa

13.532

qL2

a+ 5

8PLa

node 4 3L4

1532qL2

a+ 3

4PLa

1532qL2

a+ 3

4PLa

7L8

15.7532

qL2

a+ 7

8PLa

15.532

qL2

a+ 7

8PLa

node 5 L 1632qL2

a+ PL

a1632qL2

a+ PL

a

(2) In the numerical studies we choose q = 1, a = 1, L = 1, c = 0, andP = 1. Thus adφdx = P = 1 at x = L = 1.0. Figure 5.3 shows φt and the finiteelement solution of class C0 graphically for the four element discretizationwith p = 1.

(3) Since φ is linear over each element, dφdx is constant over each element as

seen from (5.108), whereas dφtdx in (5.110) is clearly linear in x. Table 5.2 and

Fig. 5.4 show a comparison of dφtdx and dφdx . We observe that dφ

dx has the correctvalue only at the center of each element. A more disturbing fact is that atall the inter-element boundaries (nodes 2, 3 and 4) dφ

dx is discontinuous, adirect consequence of the fact that φeh for each element Ωe is of class C0.

(4) With mesh refinement and/or by increasing the degree of approxima-tion of φeh over each element, we hope to achieve progressively diminishing

discontinuity of dφdx at the inter-element nodes. Such behavior is referred to

as weak convergence of C0 solutions to class C1.


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.2 0.4 0.6 0.8 1

Solu

tio

n φ

Distance x

q=1, P=1, L=1, c=0

(GM/WF) C0, p=1: 4 el. mesh

theoretical

Figure 5.3: Solution φ versus x

Table 5.2: Comparison of dφtdx

and dφdx

x Analytical solution φt Finite element solution φ

node 1 0 88qLa

+ Pa

78qLa

+ Pa

L8

78qLa

+ Pa

78qLa

+ Pa

node 2 L4

68qLa

+ Pa

78qLa

+ Pa

58qLa

+ Pa

3L8

58qLa

+ Pa

58qLa

+ Pa

node 3 L2

48qLa

+ Pa

58qLa

+ Pa

38qLa

+ Pa

5L8

38qLa

+ Pa

38qLa

+ Pa

node 4 3L4

28qLa

+ Pa

38qLa

+ Pa

18qLa

+ Pa

7L8

18qLa

+ Pa

18qLa

+ Pa

node 5 L Pa

18qLa

+ Pa


1

1.2

1.4

1.6

1.8

2

2.2

2.4

0 0.2 0.4 0.6 0.8 1

dφ

/dx

Distance x

q=1, P=1, L=1, c=0

(GM/WF) C0, p=1: 4 el. mesh

theoretical

Figure 5.4: Plots of dφdx

versus x

(5) Even when C0 solutions are converged weakly to class C1, they still areinadmissible in the BVP Aφ− f = 0.

(6) It is possible to employ higher degree local approximation functions thanlinear, but of class C0 in which case, also, the converged solutions are of weakC1 class.

h-convergence: C0: p = 1 and C0: p = 2 solutions

Next we consider mesh refinement studies (h-convergence) for solutionsof class C0 with p = 1 and 2. First we consider uniform mesh refinementat p = 1 beginning with a two-element mesh. Figure 5.5 shows solution φversus x and Fig. 5.6 gives plots of dφdx versus x for various uniform meshes atp = 1. We note that the solution φ agrees reasonably well with the theoret-ical solution φt for discretization utilizing four or more elements. However,dφdx exhibits inter-element discontinuity (inherent in C0 local approximation).

Upon mesh refinement the jump in dφdx at the inter-element boundaries di-

minishes but is never eliminated.

Figures 5.5 and 5.6 also show φ versus x and dφdx versus x for one and

two element (uniform) discretizations with p = 2. Since the theoreticalsolution is quadratic in x, a single p-version element with p = 2 producesthe theoretical solution. With two-element mesh (or more refined mesh), thecomputed solution remains in exact agreement with the theoretical solution.This also holds for meshes with more than two p-version elements with p = 2.


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.2 0.4 0.6 0.8 1

Solu

tio

n φ

Distance x

C0, p=1

(GM/WF)

2 el. mesh ; p = 1 4 el. mesh ; p = 1 8 el. mesh ; p = 1

16 el. mesh ; p = 1 theoretical ; 1 el. mesh ; p = 2 theoretical ; 2 el. mesh ; p = 2


1

1.2

1.4

1.6

1.8

2

2.2

2.4

0 0.2 0.4 0.6 0.8 1

dφ

/dx

Distance x

C0, p=1

(GM/WF) 2 el. mesh ; p = 1 4 el. mesh ; p = 1 8 el. mesh ; p = 1

16 el. mesh ; p = 1 theoretical ; 1 el. mesh ; p = 2 theoretical ; 2 el. mesh ; p = 2


versus x

Remarks. (1) We note that with local approximations of class C0(Ωe),the nodal degrees of freedom at the two end nodes of each element arefunction values. Thus for C0(Ωe) local approximation, the boundarycondition adφdx = dφ

dx = 1 (in this case) can not be enforced using nodaldegrees of freedom.

(2) However, in GM/WF adφdx = dφdx (as a = 1) is a secondary variable, hence


in case of GM/WF the condition dφdx = 1 can be enforced through the

secondary variable value equal to one at x = L = 1.0 as we have donein the computations. When the computed solutions are not converged,P = 1 at x = L = 1.0, as secondary variable is satisfied due to the factthat this condition is enforced. This can be confirmed by computing thesecondary variables P e for the last element containing x = L = 1.0coordinate using the following.

P e = [Ke]δe − F e

However, when the computed solutions are not converged,dφehdx computed

at x = L = 1.0 using the local approximation φeh(ξ) for the last elementcontaining x = L = 1.0 may not be equal to one. We can confirm this

fromdφehdx versus x plots in Fig. 5.4 for the four element discretization.

We note thatdφehdx at x = 1 is not equal to one even though P = 1

is enforced, hence holds. From Fig. 5.6 we note that with C0 local

approximation with p = 1 progressive mesh refinement yieldsdφehdx at

x = 1 that progressively approaches 1.

(3) Thus in case of C0 local approximation the condition dφdx = P = 1 at

x = 1 is satisfied for all discretization as P = 1 (secondary variable)

but only weakly enforced asdφehdx = 1 at x = 1 as confirmed in Figs. 5.4

and 5.6. That is as the discretization is progressively refineddφehdx at

x = 1 progressively approaches 1. This is a serious drawback of C0

local approximations that result in inadequate description of physics incoarser discretizations.

5.2.2 1D steady-state diffusion equation

Here we consider a slightly modified form of the BVP used in the previoussection with more complex q = q(x) and different boundary conditions.Consider

− d

dx

(adφ

dx

)= q(x) ∀x ∈ (0, L) = Ω ⊂ R1 (5.111)

with the following BCs:

φ(0) = 0,(adφ

dx

)x=L

= 0 (5.112)

We consider two cases. Case (a): q(x) = xn, n is a positive integer. Case(b): q(x) = sinnπx, n is a positive integer. In all numerical studies weconsider a = 1 and L = 1. We consider finite element processes based on


Galerkin method with weak form. For an element e with domain Ωe, wehave (following Example 5.2.1):

φeh =n∑i=1

Ni(ξ) δei , δei being nodal degrees of freedom

v = δφeh = Nj(ξ), j = 1, 2, . . . , n

(5.113)

we can write the following using weak form.


(Aφeh − f, v)Ωe = [Ke]δe − P e − F e (5.115)

where

Keij =

1∫−1

adNi

dx

dNj

dxJ dξ, i, j = 1, 2, . . . , nnotag (5.116)

F ei =

1∫−1

q(ξ)Ni(ξ) J dξ, i = 1, 2, . . . , n (5.117)

P eT = P e1 , 0 , . . . , 0 , P e2 , P e1 = −(adφ

dx

)∣∣∣∣xe

, P e2 =(adφ

dx

)∣∣∣∣xe+1

anddNi

dx=

1

J

dNi

dξ, J =

he2, i = 1, 2, . . . , n (5.118)

We remark that specific forms of Ni(ξ) and δei depend upon the choice oflocal approximation, p-level, and k, the order of the approximation space.In P e the two nonzero terms are obviously the secondary variables at thetwo end nodes of the element. Details of element computations, assembly,solution and post-processing remain the same as in example 5.2.1. In thefollowing we present numerical studies for the two cases.

5.2.2.1 Case (a): a = 1, L = 1, q(x) = xn, n is a positive integer

For this choice of parameters, the theoretical solution φt(x) of (5.111)with BCs (5.112) is given by (for Ω = [0, L])

φt(x) =[− 1

a(n+ 1)(n+ 2)

]xn+2 + x

[ 1

a(n+ 1)(L)n+1

](5.119)

If we choose n = 6, then for this choice φt(x) is of class C8. Minimallyconforming space for the BVP is obviously k = 3 (solutions of class c2). Weconsider the following numerical studies.


(I) h-refinement with C0 p-version (p = 2) local approximations

(II) h-refinement with C1 p-version (p = 3, minimum for C1) local ap-proximations

(III) h-refinement with C2 p-version (p = 5, minimum for C2) local ap-proximations

Details of the studies, results and discussion are presented in the following.We note that F e for an element e can be easily calculated for this choiceof q(x) = x6 using (5.115).

I. h-refinement with C0 p-version with p = 2In this choice of local approximation, the element is a three-node element

with nodes 1, 2, 3 located at ξ = −1, 0, 1 and φeh is given by (see chapter 8)

φeh(ξ) =(1− ξ

2

)φe1 +

(1 + ξ

2

)φe3 +

(ξ2 − 1

2

)(d2φ

dξ2

)2

(5.120)

orφeh(ξ) = N1(ξ) δe1 +N3(ξ) δe3 +N2(ξ) δe2 (5.121)

and

φeh =3∑i=1

Ni(ξ) δei (5.122)

In this case, [Ke] is a (3× 3) matrix with

δeT = δe1 δe2 δe3 (5.123)

We begin with a two-element uniform mesh and perform uniform mesh re-finement, i.e. subdivide each element into two for the next discretizationleading to 2, 4, 8, 16, . . . element uniform discretizations. For each case wekeep p = 2 fixed and compute solutions. Plots of φ versus x and dφ

dx versusx for various discretizations are shown in Figs. 5.7 and 5.8. From Fig. 5.7,we note that the solution φ is reasonably good for all discretizations. InFig. 5.8, we observe that dφ

dx exhibits inter-element discontinuity which di-

minishes rapidly upon mesh refinement. The values of dφdx for sixteen-elementuniform discretization are quite close to the theoretical values for practicalpurposes.

We note that due to C0 local approximation adφdx = dφdx = 0 at x = L = 1.0

can only be enforced using secondary variable (NBC) as dφdx is not a dof

at the nodes for C0 approximation, hence can not be specified as EBC at

x = 1.0. Thus,dφehdx

∣∣∣x=1

computed using local approximation for the element

containing x = 1.0 will only result indφehdx

∣∣∣x=1

= 0 when the solution is

converged. From Fig. 5.8 we clearly note that computed solutions for 2, 4,


and 8 element discretizations do not yielddφehdx

∣∣∣x=1

= 0 but progressively

approachdφehdx

∣∣∣x=1

= 0 upon mesh refinement. For 16 element discretization

its value is close to zero.

In this case B(·, ·) is bilinear and symmetric and l(·) is linear hence thequadratic functional I(φh) is given by I(φh) =

∑e I(φeh) and minimization

of I(φh) yields ∑e

Be(φeh, v) =∑e

le(v) (5.124)

which is exactly same as what we obtain using weak form for discretiza-tion ΩT . Thus, since in this case quadratic functional I(φh) is minimized,hence behavior of I(φh) versus dofs obtained from various discretizations isinstructive to examine. Figure 5.9 shows a plot of I(φh) versus dofs. Wenote that beyond the eight-element uniform discretization, I(φh) shows nosignificant change, whereas we note that dφ

dx in Fig. 5.8 still has measurable

error for this mesh including jumps at inter-element nodes in dφdx . This indi-

cates that minimization of I(φh) is a good overall indicator of equilibriumbut may not be as sensitive to local errors in the solution. We note that inthis case L2-norm of E, the residual is not possible due to C0 nature of thelocal approximations.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 0.2 0.4 0.6 0.8 1

Solu

tio

n φ

Distance x

C0, p=2

q(x)=xn, n=6

(GM/WF)

2 el. mesh, p = 24 el. mesh, p = 28 el. mesh, p = 2

close to theoretical: 16 el. mesh, p = 2

Figure 5.7: Case (a): solution φ versus x

II. h-refinement with C1 p-version with p = 3 (minimum for C1)

With this choice of local approximation we have a three-node element ewith nodes 1, 2, 3 located at ξ = −1, 0 and 1 and the local approximation is


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 0.2 0.4 0.6 0.8 1

dφ

/dx

Distance x

C0, p=2

q(x)=xn, n=6

(GM/WF)


close to theoretical: 16 el. mesh, p = 2

Figure 5.8: Case (a): dφdx

versus x

-0.00834

-0.00833

-0.00832

-0.00831

-0.00830

-0.00829

-0.00828

-0.00827

0 5 10 15 20 25 30 35

Quadra

tic f

uncti

onal I

dofs

q(x)=xn, n=6

(GM/WF)2

4

8

C0, p = 2

Figure 5.9: Case (a): quadratic functional I versus dofs

given by (see chapter 8)

φeh(ξ) = N01 (ξ)φe1 +N1

1 (ξ)(dφdx

)ξ=−1

+N03 (ξ)φe3 +N1

3 (ξ)(dφdx

)ξ=+1

(5.125)


or

φeh =

4∑i=1

Ni(ξ) δei (5.126)

in which

N1(ξ) = N01 (ξ) =

(1− ξ2

)+(ξ3 − ξ

4

)N2(ξ) = N1

1 (ξ) =

[(ξ3 − ξ4

)−(ξ2 − 1

4

)]J

N3(ξ) = N03 (ξ) =

(1 + ξ

2

)−(ξ3 − ξ

4

)N4(ξ) = N1

3 (ξ) =

[(ξ3 − ξ4

)+(ξ2 − 1

4

)]J

(5.127)

and

δe1 = φe1 , δe2 =(dφdx

)ξ=−1

, δe3 = φe3, δe4 =(dφdx

)ξ=1

(5.128)

with J = he2 . In this case the element matrix [Ke] is (4 × 4). The center

node of each element has no degrees of freedom, only the end nodes (1 and3) have two degrees of freedom each. We begin with a two-element uniformdiscretization and continue uniform mesh refinement leading to 2, 4, 8, . . .element uniform discretizations. For each case we keep p = 3 fixed andcompute solutions.

For local approximations of class C1(Ωe) the end nodes of each ele-ment contain φ and dφ

dx as nodal degrees of freedom. In this finite ele-

ment formulation (GM/WF) we have two choices to impose BC adφdx = 0at x = L = 1.0. This BC can be imposed by setting the sum of secondaryvariables at x = L = 1.0 to zero i.e. P e3 = 0 at node three (at x = L = 1.0)of the last element of the discretization. In this approach dφ

dx imposed atx = L = 1.0 is a natural boundary condition (NBC). In the second approachdφdx = 0 at x = L = 1.0 can be imposed by setting the dof dφ

dx at grid point

located at x = L = 1 to zero. In this approach, specification of dφdx falls

into EBC. Both of these approaches have different consequences. dφdx

∣∣∣x=1

= 0

as EBC (imposed through dof) will be satisfied by the local approximation,

hence will hold regardless of the discretization whereas dφdx

∣∣∣x=1

= 0 imposed

through NBC will only be satisfied upon convergence as shown earlier incase of local approximation of class C0.

We present results for both cases here. Solutions φ versus x in Fig. 5.10computed using both choices of BC at x = 1.0 remain almost the same.Figure 5.11(a) shows plots of dφ

dx versus x using local approximations when


dφdx

∣∣∣x=1

= 0 is imposed as NBC. We note that for two element discretization

when the solution is not converged, dφdx

∣∣∣x=1

= 0 is not satisfied even though

secondary variable at x = 1 is zero. Upon mesh refinement dφdx

∣∣∣x=1

= 0

is approached. Figure 5.11(b) also shows plots of dφdx vs x for various dis-

cretizations at p = 3 for C1 solutions in which dφdx

∣∣∣x=1

= 0 is imposed as

EBC. We clearly observe that dφdx

∣∣∣x=1

= 0 holds in this case regardless of the

discretization. Figures 5.12(a) and (b) show plots of d2φdx2

versus x for pro-gressively refined discretizations at p = 3 for solutions of class C1. Influence

of weakly satisfying dφdx

∣∣∣x=1

= 0 using NBC is clearly observed.

Except the two-element discretization all other discretizations yield good

values of φ and dφdx . Even d2φ

dx2for the eight-element discretization (and be-

yond) is in good agreement with the theoretical solution for both cases using

NBC and EBC to impose dφdx

∣∣∣x=1

= 0. In this case computations of the resid-

ual functional I = (E,E)ΩT =∑

e(Aφeh − f,Aφeh − f)Ωe is possible but the

integrals are in Lebesgue sense. Computation of I is not significantly in-

fluenced by the manner in which dφdx

∣∣∣x=1

= 0 is imposed i.e. NBC or EBC.

Figure 5.17 shows a plot of√I versus dofs for both cases in which dφ

dx

∣∣∣x=1

= 0

is imposed as NBC and as EBC. Fixed slope of√I versus dofs indicates

constant convergence rate.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 0.2 0.4 0.6 0.8 1

Solu

tion φ

Distance x

C1, p=3

q(x)=xn, n=6

(GM/WF)



Figure 5.10: Case (a): solution φ versus x (for dφdx

∣∣x=1

= 0 imposed as NBC or EBC)


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 0.2 0.4 0.6 0.8 1

dφ

/dx

Distance x

C1, p=3

q(x)=xn, n=6

(GM/WF)



(a) dφdx

∣∣x=1

= 0 imposed as NBC

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 0.2 0.4 0.6 0.8 1

dφ

/dx

Distance x

C1, p=3

q(x)=xn, n=6

(GM/WF)



(b) dφdx

∣∣x=1

= 0 imposed as EBC


versus x

III. h-refinement with C2 p-version with p = 5 (minimum for C2)

With this choice of local approximation we have a three-node elementwith nodes 1,2,3 located at ξ = −1, 0 and 1. The local approximation φeh is


-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0 0.2 0.4 0.6 0.8 1

d2φ

/dx

2

Distance x

C1, p=3

q(x)=xn, n=6

(GM/WF)



(a) d2φdx2

∣∣∣x=1

= 0 imposed as NBC

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0 0.2 0.4 0.6 0.8 1

d2φ

/dx

2

Distance x

C1, p=3

q(x)=xn, n=6

(GM/WF)



(b) d2φdx2

∣∣∣x=1

= 0 imposed as EBC

Figure 5.12: Case (a): d2φdx2

versus x

given by (see chapter 8)

φeh(ξ) =N01 (ξ)φe1 +N1

1 (ξ)(dφdx

)ξ=−1

+N21 (ξ)

(d2φ

dx2

)ξ=−1

+N03 (ξ)φe3 +N1

3 (ξ)(dφdx

)ξ=1

+N23 (ξ)

(d2φ

dx2

)ξ=1

(5.129)


or

φeh(ξ) =

6∑i=1

Ni(ξ) δei (5.130)

The basis functions Ni(ξ) for this case are described in chapter 8. In this casealso the center node of each element has no degrees of freedom. We repeatstudies similar to the previous cases for uniform mesh refinement beginning

with the two-element uniform discretization. Plots of φ, dφdx , d2φ

dx2and d3φ

dx3

versus x are shown in Fig. 5.13–5.16. For solutions of class C2, the imposition

of the boundary condition dφdx

∣∣∣x=1

= 0 as NBC or EBC has virtually no effect

on the results. Even the two-element mesh shows good agreement with

the theoretical solution except d3φdx3

which improves dramatically with thefour-element uniform discretization. For this choice of local approximationfunctions the residual functional I = (E,E) =

∑e(Aφ

eh − f,Aφeh − f) is in

Riemann sense. Figure 5.17 shows a plot of√I versus dofs. In this case also,

constant slope indicates fixed convergence rate. Higher slope of√I versus

dofs graph for solutions of class C2 at p = 5 indicates faster convergence ratecompared to solutions of class C1 with p = 3. For a given dofs much lowervalues of

√I for this case are obvious from Fig. 5.17 compared to solutions

of class C1 with p = 3.

From the results presented in Fig. 5.17, it is difficult to conclude whetherthe reduction in

√I for solutions of class C2 at p = 5 compared to solutions of

class C1 at p = 3 for a fixed degrees of freedom is due to increase in the orderof the space or due to increase in p-level or both. Figure 5.18 shows plotsof√I versus dofs for solutions of class C1 and C2 at p = 5 for progressively

uniform mesh refinements in which element lengths are halved each time.From the results in Fig. 5.18 we note that for a fixed dofs the solutions ofclass C2 have lower values of

√I compared to solutions of class C1. Slopes

of√I versus dofs for both cases are almost the same, even though a slight

increase in the slope is observed for solutions of class C2 with progressivelyrefined discretizations. These results decisively demonstrate higher accuracyof the solution of class C2 compared to C1 for a fixed dofs. For all practicalpurposes we can assume approximately same convergence rate in both cases.This behavior shown in Fig. 5.18 holds true for each p-level greater than orequal to 5 (five being minimum p-level for solutions of class C2).

5.2.2.2 Case (b): a = 1, L = 1, q(x) = sinnπx, n = 4

For this choice of parameters the theoretical solution φt(x) ∀x ∈ [0, L] of(5.111) with BCs (5.112) is given by

φ =1

a(nπ)2sinnπx+ x

(− 1

anπcosnπL

)(5.131)


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 0.2 0.4 0.6 0.8 1

Solu

tio

n φ

Distance x

C2, p=5

q(x)=xn, n=6

(GM/WF)


16 el. mesh, p = 5

Figure 5.13: Case (a): solution φ versus x

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 0.2 0.4 0.6 0.8 1

dφ

/dx

Distance x

C2, p=5

q(x)=xn, n=6

(GM/WF)


16 el. mesh, p = 5


versus x

For n = 4 there are two periods of the sine wave of q(x) over [0, L] andd2φdx2

= − sinnπx is negative of q(x). For this choice of q(x) we also calculateF e using the same procedure as case (a).

We perform numerical studies similar to case (a) using progressively re-fined uniform descretizations. We keep in mind that the BC φ(0) = 0 (EBCin GM/WF) poses no problem and can be imposed using degrees of freedom


-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0 0.2 0.4 0.6 0.8 1

d2φ

/dx

2

Distance x

C2, p=5

q(x)=xn, n=6

(GM/WF)


16 el. mesh, p = 5


versus x

-6

-5

-4

-3

-2

-1

0

1

0 0.2 0.4 0.6 0.8 1

d3φ

/dx

3

Distance x

C2, p=5

q(x)=xn, n=6

(GM/WF)


16 el. mesh, p = 5


versus x

φ at the element nodes regardless of whether the local approximations are of

class C0 or higher classes. However the BC dφdx

∣∣∣x=1

= 0 can only be imposed

by setting the secondary variable (NBC) at x = 1 to be zero when using C0

approximation in which case dφdx

∣∣∣x=1

= 0 will only be satisfied upon conver-

gence. As shown earlier in case of C1(Ωe), C2(Ωe), and higher class local


-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0.5 1 1.5 2 2.5 3 3.5

log(√I)

log(dofs)

(GM/WF)NBC

EBC

C1, p=3

C2, p=5

C1, p=3

C2, p=5

Figure 5.17: Case (a): ||E||L2=√I versus dofs (for dφ

dx

∣∣x=1

= 0 imposed as NBC and

EBC)

-10

-9

-8

-7

-6

-5

-4

-3

-2

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

log(√I)

log(dofs)

(GM/WF) C1, p=5

C2, p=5

Figure 5.18: Case (a): ||E||L2=√I versus dofs (for dφ

dx

∣∣x=1

= 0 imposed as NBC)

approximations dφdx

∣∣∣x=1

= 0 can be imposed either using NBC or EBC. When

dφdx

∣∣∣x=1

= 0 is imposed using EBC this condition at x = 1 is always satis-

fied by the local approximation regardless of the discretization. We presentnumerical results for C1(Ωe) and C2(Ωe) local approximations using both

approaches of imposing dφdx

∣∣∣x=1

= 0 boundary condition. A summary of the

results is given in the following.


Figures 5.19 and 5.20 show plots of φ and dφdx versus x for C0(Ωe), p = 2

local approximation for progressively refined uniform discretizations. In this

case due to C0(Ωe) local approximations, the boundary condition dφdx

∣∣∣x=1

= 0

can only be imposed using NBC. Figure 5.21 shows a plot of quadraticfunctional I versus dofs for these solutions.

Figures 5.22–5.24 show results for solutions of class C1(Ωe) at p = 3 forprogressively refined uniform discretizations. Figure 5.22 shows graphs of φ

versus x for C1(Ωe), p = 3 for both cases in which dφdx

∣∣∣x=1

= 0 is imposed

using NBC and EBC. We do not observe any significant difference in the

results regardless of how dφdx

∣∣∣x=1

= 0 is specified. Figure 5.23(a) and (b)

and Fig. 5.24(a) and (b) show plots of dφdx versus x and d2φ

dx2versus x when

dφdx

∣∣∣x=1

= 0 is imposed either using NBC or using EBC. We clearly observe

that dφdx

∣∣∣x=1

= 0 is satisfied for all discretizations when this BC is imposed

using EBC (see Fig. 5.23(b)), whereas in Fig. 5.23(a) dφdx

∣∣∣x=1

= 0 is only

satisfied when the computed solution is converged. In Figs. 5.24(a) and (b)

we observe some differences in d2φdx2

versus x, specially at x = 1.0, for the twocases for coarser discretizations.

Figures 5.25–5.28 show results for local approximations of class C2(Ωe),p = 5 for progressively refined discretizations. Behavior of φ versus xshown in Fig. 5.25 is not affected appreciably by the two choices of definingdφdx

∣∣∣x=1

= 0. Figures 5.26(a) and (b), 5.27(a) and (b), and 5.28(a) and (b)

show graphs of dφdx ,

d2φdx2

, d3φdx3

versus x at p = 5 for progressively refined uni-form discretizations. We note that the two sets of solutions are very closewithout much appreciable, measurable difference between them.

Figure 5.29 shows plots of L2-norm of E; that is,√I versus dofs for

solutions of class C1 at p = 3 and the solutions of class C2 at p = 5 for both

NBC and EBC of imposing dφdx

∣∣∣x=1

= 0. We clearly observe that the two

choices do not influence these results significantly. From Fig. 5.29 we notethat solutions of the class C2 at p = 5 exhibit much higher convergence rate(higher slope of

√I versus dofs) compared to solutions of class C1 at p = 3

as well as lower values of√I for a given dofs confirming higher accuracy of

the solutions of class C2 at p = 5 compared to solutions of class C1 at p = 3.


√I for solutions of class C2 at p = 5 compared to solutions of

class C1 at p = 3 for a fixed degrees of freedom is due to increase in the orderof the space or due to increase in p-level or both. Figure 5.30 shows plotsof√I versus dofs for solutions of class C1 and C2 at p = 5 for progressively

uniform mesh refinements in which element lengths are halved each time.


From the results in Fig. 5.30 we note that for a fixed dofs the solutions ofclass C2 have lower values of

√I compared to solutions of class C1. Slopes

of√I versus dofs for both cases are almost the same, even though a slight

increase in the slope is observed for solutions of class C2 with progressivelyrefined discretizations. These results decisively demonstrate higher accuracyof the solution of class C2 compared to C1 for a fixed dofs. For all practicalpurposes we can assume approximately same convergence rate in both cases.This behavior shown in Fig. 5.30 holds true for each p-level greater than orequal to 5 (five being minimum p-level for solutions of class C2).

-0.09

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0 0.2 0.4 0.6 0.8 1

Solu

tion φ

Distance x

C0, p=2

q(x)=sin nπx, n=4

(GM/WF) 2 el. mesh, p = 24 el. mesh, p = 28 el. mesh, p = 2

16 el. mesh, p = 264 el. mesh, p = 2

Figure 5.19: Case (b): solution φ versus x

5.2.3 Least-squares finite element formulation

Consider the 1D steady state diffusion equation (same as in example 5.2.2).

− d

dx

(adφ

dx

)+ cφ = q(x) ∀x ∈ (0, L) = Ω ⊂ R1 (5.132)

with the following boundary conditions.

φ(0) = 0(adφ

dx

)x=L

= 0(5.133)

a, c and q(x) are given data. We consider the least-squares finite elementformulation of this BVP based on residual functional. In this case

A = − d

dx

(ad

dx

)+ c

f = q(x)(5.134)


-0.2

-0.15

-0.1

-0.05

0

0.05

0 0.2 0.4 0.6 0.8 1

dφ

/dx

Distance x

C0, p=2

q(x)=sin nπx, n=4


16 el. mesh, p = 264 el. mesh, p = 2

Figure 5.20: Case (b): dφdx

versus x

-0.005

-0.0045

-0.004

-0.0035

-0.003

-0.0025

-0.002

0 20 40 60 80 100 120 140

Quad

rati

c f

uncti

onal I

dofs

q(x)=sin nπx, n=4

(GM/WF) C0, p = 2

Figure 5.21: Case (b): quadratic functional I versus dofs

and we can writeAφ− f = 0 ∀x ∈ Ω (5.135)

Let φeh(x) or φeh(ξ) be local approximation of φ over Ωe or Ωξ, an element eof discretization ΩT =

⋃e Ωe. Then the residual function Ee over Ωe can be

defined byEe = Aφeh − f ∀x ∈ Ωe (5.136)


-0.09

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0 0.2 0.4 0.6 0.8 1

Solu

tio

n φ

Distance x

C1, p=3

q(x)=sin nπx, n=4


16 el. mesh, p = 3

Figure 5.22: Case (b): solution φ versus x ( dφdx

∣∣x=1

= 0 as NBC or EBC)

The residual or least-squares functional I for ΩT is constructed using

I =∑e

Ie =∑e

(Ee, Ee)Ωe , existence of I (5.137)

in which Ie is the least-squares functional for Ωe. First variation of I set tozero gives necessary conditions.

δI =∑e

δIe =∑e

2(Ee, δEe)Ωe = 2∑e

ge = g = 0 (5.138)

in which

ge = (Ee, δEe)Ωe (5.139)

The element relations can be derived using ge. We note that

Ee = Aφeh − f = − d

dx

(adφehdx

)+ cφeh − q(x) (5.140)

δEe = − d

dx

(adv

dx

)+ cv, v = δφeh (5.141)

Let

φeh =

n∑i=1

Ni(ξ) δei (5.142)

in which Ni(ξ) are approximation functions and δei are nodal degrees offreedom. We could choose a three-node element with nodes 1, 2, 3 located


-0.2

-0.15

-0.1

-0.05

0

0.05

0 0.2 0.4 0.6 0.8 1

dφ

/dx

Distance x

C1, p=3

q(x)=sin nπx, n=4


16 el. mesh, p = 3

(a) dφdx

∣∣x=1

= 0 as NBC

-0.2

-0.15

-0.1

-0.05

0

0.05

0 0.2 0.4 0.6 0.8 1

dφ

/dx

Distance x

C1, p=3

q(x)=sin nπx, n=4


16 el. mesh, p = 3

(b) dφdx

∣∣x=1

= 0 as EBC


versus x

at ξ− = 1, 0, and 1. Explicit forms of Ni(ξ) and δei depend upon the classof φeh, i.e. the order of the approximation space Vh ⊂ Hk(Ωe). We discussdetails a little later. First, based on local approximation φeh, we have

v = δφeh = Nj(ξ), j = 1, 2, . . . , n (5.143)


-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

d2φ

/dx

2

Distance x

C1, p=3

q(x)=sin nπx, n=4


16 el. mesh, p = 3

(a) d2φdx2

∣∣∣x=1

= 0 as NBC

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

d2φ

/dx

2

Distance x

C1, p=3

q(x)=sin nπx, n=4


16 el. mesh, p = 3

(b) d2φdx2

∣∣∣x=1

= 0 as EBC

Figure 5.24: Case (b): d2φdx2

versus x

Substituting φeh and v in ge we have gej of ge

gej =

(− d

dx

(a

n∑i=1

dNi

dxδei

)+c

n∑i=1

Ni δei−f,−

d

dx

(adNj

dx

)+cNj

)Ωe

(5.144)


-0.09

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0 0.2 0.4 0.6 0.8 1

Solu

tio

n φ

Distance x

C2, p=5

q(x)=sin nπx, n=4


16 el. mesh, p = 5

Figure 5.25: Case (b): solution φ versus x ( dφdx

∣∣x=1

= 0 as NBC or EBC)

for j = 1, 2, . . . , n. Since the differential operator A is linear, gej (j =1, 2, . . . , n) can be written as

gej =

(n∑i=1

(− d

dx

(adNi

dx

)+ cNi

)δei − f,−

d

dx

(adNj

dx

)+ cNj

)Ωe

(5.145)

which is same as

gej =

( n∑i=1

(ANi) δei − f,ANj

)Ωe

; j = 1, 2, . . . , n (5.146)

orge = [Ke]δe − F e (5.147)

in which

Keij = (ANj , ANi)Ωe (5.148)

F ei = (f,ANj)Ωe (5.149)

Substituting for A

Keij =

(− d

dx

(adNj

dx

)+ cNj ,−

d

dx

(adNi

dx

)+ cNi

)Ωe, i, j = 1, 2, . . . , n

(5.150)

F ei =

(f,− d

dx

(adNi

dx

)+ cNi

)Ωe, i = 1, 2, . . . , n (5.151)


-0.2

-0.15

-0.1

-0.05

0

0.05

0 0.2 0.4 0.6 0.8 1

dφ

/dx

Distance x

C2, p=5

q(x)=sin nπx, n=4


16 el. mesh, p = 5

(a) dφdx

∣∣x=1

= 0 as NBC

-0.2

-0.15

-0.1

-0.05

0

0.05

0 0.2 0.4 0.6 0.8 1

dφ

/dx

Distance x

C2, p=5

q(x)=sin nπx, n=4


16 el. mesh, p = 5

(b) dφdx

∣∣x=1

= 0 as EBC


versus x

Using element map in the natural coordinate space and noting that dx =J dξ, we obtain (i, j = 1, 2, . . . , n)

Keij =

1∫−1

(− d

dx

(adNj

dx

)+ cNj

)(− d

dx

(adNi

dx

)+ cNi

)Jdξ (5.152)

F ei =

1∫−1

f

(− d

dx

(adNi

dx

)+ cNi

)J dξ (5.153)


-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

d2φ

/dx

2

Distance x

C2, p=5

q(x)=sin nπx, n=4


16 el. mesh, p = 5

(a) d2φdx2

∣∣∣x=1

= 0 as NBC

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

d2φ

/dx

2

Distance x

C2, p=5

q(x)=sin nπx, n=4


16 el. mesh, p = 5

(b) d2φdx2

∣∣∣x=1

= 0 as EBC


versus x

Since Ni = Ni(ξ),dmNi

dxm=

1

JmdmNi

dξm, m = 1, . . . (5.154)

and J = he2 . Hence, Ke

ij and F ei are explicitly defined. We use gauss quadra-ture to calculate numerical values of the coefficients of [Ke] and F e. Know-


-25

-20

-15

-10

-5

0

5

10

15

20

0 0.2 0.4 0.6 0.8 1

d3φ

/dx

3

Distance x

C2, p=5

q(x)=sin nπx, n=4

(GM/WF)


16 el. mesh, p = 5

(a) d3φdx3

∣∣∣x=1

= 0 as NBC

-25

-20

-15

-10

-5

0

5

10

15

20

0 0.2 0.4 0.6 0.8 1

d3φ

/dx

3

Distance x

C2, p=5

q(x)=sin nπx, n=4

(GM/WF)


16 el. mesh, p = 5

(b) d3φdx3

∣∣∣x=1

= 0 as EBC


versus x

ing

ge = [Ke]δe − F e (5.155)

for an element Ωe, we can write the following for the whole discretizationΩT

e∑i=1

ge =∑e

[Ke]δe −∑e

F e = 0 (5.156)


-8

-7

-6

-5

-4

-3

-2

-1

0

0.5 1 1.5 2 2.5 3 3.5

log(√I)

log(dofs)

(GM/WF)NBC

EBC

C1, p=3

C2, p=5

C1, p=3

C2, p=5

Figure 5.29: Case (b): ||E||L2=√I versus dofs

-8

-7

-6

-5

-4

-3

-2

-1

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

log(√I)

log(dofs)

(GM/WF) C1, p=5

C2, p=5

Figure 5.30: Case (b): ||E||L2=√I versus dofs (for dφ

dx

∣∣x=1

= 0 imposed as NBC)

or ∑e

[Ke]δe =∑e

F e (5.157)

or [∑e

[Ke]]δ = F; δ =

⋃e

δe (5.158)


or[K]δ = F for ΩT (5.159)

in which

[K] =∑e

[Ke]

F =∑e

F e(5.160)

[K] and F are obtained from [Ke] and F e using usual process of assem-bly described in example 5.2.1.

5.2.3.1 Approximation space Vh

Since the highest order of the derivative is two in the operator A andhence φeh ∈ Vh ⊂ H3,p(Ωe) is minimally conforming choice for which allintegrals remain Riemann in the entire computational process. In this choiceφeh is of class C2(Ωe). If we choose φeh ∈ Vh ⊂ H2,p(Ωe) then all integralsin the entire least-squares process are in Lebesgue sense. φeh of class C0 arenot admissible in this finite element formulation.

5.2.3.2 Numerical studies

In this section we consider numerical studies. We choose the followingparameters a = 1, L = 1, c = 0 and q(x) = sinnπx, n = 4. This choice issame as used in example 5.2.2 case (b). This will permit us to compare thesolutions from GM/WF with those computed here using least-squares finiteelement formulations.

When a is constant, (5.152) and (5.153) reduce to

Keij =

1∫−1

(−ad

2Nj

dx2+ cNj

)(−ad

2Ni

dx2+ cNi

)J dξ (5.161)

F ei =

1∫−1

f(−ad

2Ni

dx2+ cNi

)J dξ (5.162)

for i, j = 1, 2, . . . , n. Using (5.154) and J = he

2

Keij =

1∫−1

[−a( 2

he

)2d2Nj

dξ2+ cNj

][−a( 2

he

)2d2Ni

dξ2+ cNi

]he2dξ (5.163)

F ei =

1∫−1

f

[−a( 2

he

)2d2Ni

dξ2+ cNi

]he2dξ (5.164)


Numerical values of Keij and F ei are obtained using gauss quadrature. We

consider the following numerical studies:

I. h-convergence study with φeh of class C1 using p = 3

II. h-convergence study with φeh of class C2 using p = 5

III. p-convergence study with φeh of class C1 for a fixed discretization withp = 3, 4, 5, . . .

IV. p-convergence study with φeh of class C2 for a fixed discretization withp = 5, 6, 7, . . .

Results are summarized in the following:We note that in this formulation for solutions of classes C1(Ωe) and higher

the boundary condition dφdx

∣∣∣x=1

= 0 is satisfied regardless of the discretization

as it is imposed using nodal degree of freedom dφdx . Figures 5.31 to 5.33

show plots of φ, dφdx and d2φ

dx2versus x for solutions of class C1 at p = 3

for progressively refined discretizations. Figures 5.34 to 5.37 show plots

of φ, dφdx , d2φ

dx2and d3φ

dx3versus x for solutions of class C2 at p = 5. With

progressive mesh refinement, solutions of both classes show convergence to

the theoretical solutions. Interelement jumps of d2φdx2

for solutions of class

C1 and those of d3φdx3

for solutions of class C2 shown in Figs. 5.33 and 5.37

progressively diminish with mesh refinement. Figure 5.38 shows plots of√I

versus dofs for solutions of both classes. Higher convergence rate and betteraccuracy of the solutions of class C2 at p = 5 is quite clear from Fig. 5.38.


√I for solutions of class C2 at p = 5 compared to solutions

of class C1 at p = 3 for a fixed degrees of freedom is due to increase in theorder of the space or due to increase in p-level both. Figure 5.39 shows plotsof√I versus dofs for solutions of class C1 and C2 at p = 5 for progressively

uniform mesh refinements in which element lengths are halved each time.From the results in Fig. 5.39 we note that for a fixed dofs the solutions ofclass C2 have lower values of

√I compared to solutions of class C1. Slopes of√

I versus dofs for both cases are almost the same. These results decisivelydemonstrate higher accuracy of the solution of class C2 compared to C1 fora fixed dofs. For all practical purposes we can assume approximately sameconvergence rate in both cases. This behavior shown in Fig. 5.39 holds truefor each p-level greater than or equal to 5 (five being minimum p-level forsolutions of class C2).

In the last study we consider a four-element uniform discretization withsolutions of classes C1 and C2 and present a p-convergence study by progres-sively increasing the p-levels by one in both cases beginning with minimallyconforming p-level. Better accuracy and higher convergence rates of the


solution of class C2 with progressively increasing p-level is quite clear (seeFig. 5.40).

-0.09

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0 0.2 0.4 0.6 0.8 1

φ

x

C1, p=3

(LSP: single equation) 2 el. mesh4 el. mesh8 el. mesh

16 el. mesh32 el. mesh64 el. mesh


-0.2

-0.15

-0.1

-0.05

0

0.05

0 0.2 0.4 0.6 0.8 1

dφ

/dx

x

C1, p=3




versus x


-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

d2φ

/dx

2

x

C1, p=3

(LSP: single equation)



Figure 5.33: Plots of d2φdx2

versus x

-0.09

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0 0.2 0.4 0.6 0.8 1

φ

x

C2, p=5




5.2.4 LSFEP using auxiliary variables and auxiliaryequations

In the least-squares finite element formulation of the second order dif-fusion equation we note C0 local approximations are not admissible. Thepurpose of the material presented in this section is to provide an alternative


-0.2

-0.15

-0.1

-0.05

0

0.05

0 0.2 0.4 0.6 0.8 1

dφ

/dx

x

C2, p=5




versus x

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

d2φ

/dx

2

x

C2, p=5





versus x

least-squares finite element formulation that permits C0 local approxima-tion. First, we note that admissibility of C0 local approximation in theleast-squares process is possible only if the GDEs are a system of first or-der ordinary or partial differential equations. Consider the same BVP as in


-25

-20

-15

-10

-5

0

5

10

15

20

0 0.2 0.4 0.6 0.8 1

d3φ

/dx

3

x

C2, p=5





versus x

-8

-7

-6

-5

-4

-3

-2

-1

0

0 0.5 1 1.5 2 2.5 3 3.5

log(√I)

log(dofs)


(GM/WF)

C1, p=3

C2, p=5

C1, p=3

C2, p=5

Figure 5.38: Plots of ||E||L2=√I versus dofs

example 5.2.3.

− d

dx

(adφ

dx

)+ cφ = q(x) ∀x ∈ (0, L) = Ω ⊂ R1 (5.165)

φ(0) = 0(adφ

dx

)x=L

= 0(5.166)


-8

-7

-6

-5

-4

-3

-2

-1

0

0.5 1 1.5 2 2.5 3

log(√I)

log(dofs)


(GM/WF)

C1, p=5

C2, p=5

C1, p=5

C2, p=5


-14

-12

-10

-8

-6

-4

-2

0

0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

log(√I)

log(dofs)

p-convergence study

4-element uniform mesh

C1

C2


We convert the second order ODE into a system of two first order ODEsusing auxiliary variable and auxiliary equation. Let

τ = adφ

dx(5.167)


Then we have the following for the BVP:

−dτdx

+ cφ = q(x)

τ = adφ

dx

(5.168)

and

φ(0) = 0(adφ

dx

)x=L

= τ(L) = 0(5.169)

τ is called auxiliary variable and τ = adφdx is called auxiliary equation. Thus,(5.165), a second order ODE in φ, has been converted into two first orderODEs in dependent variables φ and τ in (5.168). We also note the appear-ance of τ in the boundary conditions in (5.168). We approximate φ and τby φeh and τ eh over Ωe independent of each other (even though the two are

related by τ = adφdx ).We can now consider least-squares finite element formulation of (5.168)

and (5.169) instead of (5.165) and (5.166). Let φeh and τ eh be local approxi-mations of φ and τ over Ωe. Then we have

Ee1 = −dτ ehdx

+ cφeh − q(x)

Ee2 = τ eh − a(x)dφehdx

∀x ∈ Ωe (5.170)

in which Ee1 and Ee2 are residuals corresponding to the two ODEs in (5.168).We define

Ie1 = (Ee1, Ee1)Ωe

Ie2 = (Ee2, Ee2)Ωe

(5.171)

as least-squares functionals corresponding to the residual equations Ee1 andEe2. Then for the discretization ΩT , we can write

I =∑e

( 2∑i=1

Iei

)=∑e

Ie ; Ie =2∑i=1

Iei =2∑i=1

(Eei , Eei )Ωe (5.172)

This establishes existence of the least-squares functional I. The first varia-tion of I set to zero provides the necessary conditions:

δI =∑e

δIe =∑e

2∑i=1

δIei = 2∑e

ge = 2g = 0 (5.173)


Since the differential operator A is linear

δIe =

2∑i=1

δIei = 2

2∑i=1

(Eei , δEei )Ω = 2ge (5.174)

Hence

δI = 2

(∑e

2∑i=1

(Eei , δEei )Ωe

)= 2

∑e

ge = 0 (5.175)

where

ge =

2∑i=1

(Eei , δEei )Ωe (5.176)

Let

φeh =

n∑i=1

Nφi (ξ) φδei

τ eh =

n∑i=1

N τi (ξ) τδei

(5.177)

in which Nφi (ξ) and N τ

i (ξ) are local approximation functions and φδei andτδei are corresponding nodal degrees of freedom for φeh and τ eh, we note n andn˜ are total degrees of freedom for φeh and τ eh. Thus, local approximations

for φ and τ can be chosen differently. For convenience, we write the localapproximations φeh and τ eh as follows:

φeh = [Nφ]φeτ eh = [N τ ]τ e

(5.178)

Substitution of φeh and τ eh in the residual equations Ee1 and Ee2 yields

Ee1 = −[dN τ

dx

]τ e+ c[Nφ]φe − q(x)

Ee2 = [N τ ]τ e − a(x)[dNφ

dx

]φe

(5.179)

Let

δe =

φeτ e

(5.180)

Therefore

δEe1 =

∂Ee1∂φe∂Ee1∂τe

=

c[Nφ]T

−[dNτ

dx

]T (5.181)


and

δEe2 =

∂Ee2∂φe∂Ee2∂τe

=

−a(x)

[dNφ

dx

]T[N τ ]T

(5.182)

Recall that

ge =2∑i=1

(Eei , δEei )Ωe (5.183)

or

ge =

[ 2∑i=1

[δEei , δEei ]

]δe −

2∑i=1

(q(x), δEei ) (5.184)

Since the differential operator A is linear, we can write

ge = [Ke]δe − F e (5.185)

where

[Ke] =2∑i=1

(δEei , δEei )Ωe (5.186)

or

[Ke] =2∑i=1

∑ 1∫−1

(δEei )(δEei )T J dξ (5.187)

and

F e =

2∑i=1

1∫−1

q(x)(δEei ) J dξ (5.188)

For the discretization ΩT

g =∑e

ge =∑e

[Ke]δe −∑e

F e = 0 (5.189)

or [∑e

[Ke]]δ = F (5.190)

or[K]δ = F (5.191)

where[K] =

∑e

[Ke], F =∑e

F e (5.192)

andδ =

⋃e

δe (5.193)

Coefficients of [Ke] and F e are computed using gauss quadrature. Theprocess of assembly for [K] and [F ] in (5.192) follows the usual procedure(as in example 5.2.1).


5.2.4.1 Approximation spaces for φeh and τ eh

Generally one chooses equal order, equal degree local approximations forboth φ and τ . That is

Nφ = N τ = N and n = n˜ = n

It can be shown [1] that with this choice the least-squares process remainsconvergent. Thus, we can choose C0 p-version local approximations of samedegree (and same order) for both φeh and τ eh. Hence, we have φeh, τ

eh ∈

Vh ⊂ H1,p(Ωe). For this approximation space the integrals in the LSP arein Lebesgue sense. Based on the usual definition of minimally conformingspaces we must choose φeh, τ

eh ∈ Vh ⊂ H2,p(Ωe). For this choice all integrals

in the LSP are Riemann. The problems associated with this approach usingauxiliary variables and auxiliary equations in LSP has been pointed out bySurana et al. [2].

This approach of constructing least-squares finite element processes usingfirst order systems of GDEs has been a common practice in computationalmechanics prior to the introduction of k-version of the finite element methodby Surana et al. [3–5]

5.2.4.2 Numerical studies

We consider the same data as used in example 5.2.3 and present the follow-ing numerical studies: First, we discuss the manner in which the boundaryconditions are imposed. φ(0) = 0 is rather straightforward to specify as φis a degree of freedom at the end nodes of the element regardless of the or-der of local approximation. However, imposition of the boundary conditiondφdx

∣∣∣x=1

= 0 requires some considerations. We note that in case of C0 local

approximation for φ and τ this boundary condition can not be imposed us-ing nodal degrees of freedom for φ as dφ

dx is not a degree of freedom at the

end nodes of the element. However, τ = dφdx is an auxiliary equation, thus

τ |x=1= 0 imposes dφdx

∣∣∣x=1

= 0 indirectly or in the weak sense due to the fact

that τ = dφdx only holds when the residuals approach zero i.e. when the so-

lutions are sufficiently converged. Thus, in case of C0 solutions dφdx

∣∣∣x=1

= 0

calculated using element local approximations for φ will not be zero eventhough τ |x=1= 0 is a specified boundary condition. This is a serious draw-back of local approximations based on approximation spaces that are notminimally conforming as dictated by the BVP.

When the local approximations are of classes C1(Ωe), C2(Ωe) or higher

we have two choices. In the first choice we can elect to impose dφdx

∣∣∣x=1

= 0


using τ |x=1= 0 as in case of C0(Ωe) local approximation. This of course willhave similar consequences as in case of C0(Ωe). In the second choice we can

impose dφdx

∣∣∣x=1

= 0 using degree of freedom dφdx in the local approximation

for φ. In this case dφdx calculated at x = 1 using local approximations for φ

will naturally yield dφdx

∣∣∣x=1

= 0 but τ |x=1= 0 will only hold when solutions

are converged. That is when the residual functional corresponding to theauxiliary equation is sufficiently close to zero. In the computations of thenumerical solutions we consider both approaches of imposing the boundary

condition dφdx

∣∣∣x=1

= 0. We consider the following numerical studies.

I. h-convergence study with φeh and τ eh of class C0 using p = 2

II. h-convergence study with φeh and τ eh of class C1 using p = 3

III. h-convergence study with φeh and τ eh of class C2 using p = 5

IV. p-convergence study (fixed discretization) with φeh and τ eh of class C1

and C2

Figures 5.41 and 5.42 show plots of φ and dφdx versus x for solutions of class C0

at p-level of two using progressively refined discretizations in which elementlength is halved each time. In Fig. 5.41 we note that except two elementdiscretization all others yield good values of solution φ. In Fig. 5.42 accuracy

of dφdx improves with progressive mesh refinement. We note that dφ

dx

∣∣∣x=1

= 0

is not satisfied for coarser discretizations as expected.

Figures 5.43, 5.44(a) and (b), and 5.45(a) and (b) show plots of φ, dφdx andd2φdx2

versus x for solutions of class C1(Ωe) at p-level of 3 using progressivemesh refinement. The behavior of φ versus x (see Fig. 5.43) is not influencedsignificantly whether the derivative boundary condition at x = 1 is specifiedusing τ or dφ

dx as degrees of freedom. Figures 5.44(a) and (b) show plots

of dφdx versus x for τ |x=1= 0 and dφ

dx

∣∣∣x=1

= 0, respectively, used for defining

derivative boundary condition at x = 1. We note that dφdx

∣∣∣x=1

= 0 is not

satisfied in figure 5.44(a) for coarser discretizations whereas in Fig. 5.44(b)this boundary condition holds for all discretizations. Likewise τ |x=1= 0holds for all plots in Fig. 5.44(a) but not in Fig. 5.44(b). However, when thediscretizations are sufficiently refined the results are same for both cases. In

Fig. 5.45(a) and (b) graphs of d2φdx2

show differences in the solution only forcoarser discretizations.

Graphs of φ, dφdx , d2φ

dx2and d3φ

dx3versus x for solutions of class C2 at p = 5

are shown in Figs. 5.46–5.49. In these studies choice of τ |x=1= 0 or dφdx

∣∣∣x=1

= 0

has virtually insignificant influence on the computed solutions.


-0.09

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0 0.2 0.4 0.6 0.8 1

φ

x

C0, p=2

(LSP: first order system) 2 el. mesh4 el. mesh8 el. mesh



-0.2

-0.15

-0.1

-0.05

0

0.05

0 0.2 0.4 0.6 0.8 1

dφ

/dx

x

C0, p=2




versus x

Figure 5.50 shows plots of√I versus dofs for solutions of classes C1 and

C2. Higher convergence rate and better accuracy of the solutions of class C2


-0.09

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0 0.2 0.4 0.6 0.8 1

φ

x

C1, p=3



Figure 5.43: Solution φ versus x (τ |x=1= 0 or dφdx

∣∣x=1

= 0)

at p = 5 is quite clear from the figure. From the results presented in Fig. 5.50,it is difficult to conclude whether the reduction in

√I for solutions of class

C2 at p = 5 compared to solutions of class C1 at p = 3 for a fixed degrees offreedom is due to increase in the order of the space or due to increase in p-level both. Figure 5.51 shows plots of

√I versus dofs for solutions of class C1

and C2 at p = 5 for progressively uniform mesh refinements in which elementlengths are halved each time. From the results in Fig. 5.51 we note that fora fixed dofs the solutions of class C2 have lower values of

√I compared to

solutions of class C1. Slopes of√I versus dofs for both cases are almost the

same, even though a slight increase in the slope is observed for solutions ofclass C2 with progressively refined discretizations. These results decisivelydemonstrate higher accuracy of the solution of class C2 compared to C1 fora fixed dofs. For all practical purposes we can assume approximately sameconvergence rate in both cases. This behavior shown in Fig. 5.51 holds truefor each p-level greater than or equal to 5 (five being minimum p-level forsolutions of class C2).

In the last study we consider a four-element uniform discretization withsolutions of classes C1 and C2 and present a p-convergence study by progres-sively increasing the p-levels by one in both cases beginning with minimallyconforming p-level. Better accuracy and higher convergence rates of the so-lution of class C2 with progressively increasing p-level is quite clear fromFig. 5.52.


-0.2

-0.15

-0.1

-0.05

0

0.05

0 0.2 0.4 0.6 0.8 1

dφ

/dx

x

C1, p=3



(a) τ |x=1= 0

-0.2

-0.15

-0.1

-0.05

0

0.05

0 0.2 0.4 0.6 0.8 1

dφ

/dx

x

C1, p=3



(b) dφdx

∣∣x=1

= 0


versus x

5.2.5 One-dimensional heat conduction withconvective boundary

Here we consider 1D heat conduction problem (also a diffusion equa-tion) with a convective boundary condition (Neumann boundary condition)containing unknown function (temperature). The purpose of this example


-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

d2φ

/dx

2

x

C1, p=3



(a) τ |x=1= 0

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

d2φ

/dx

2

x

C1, p=3



(b) dφdx

∣∣x=1

= 0


versus x

is to illustrate how the finite element computations are effected due to theNeumann boundary conditions containing unknown temperature. Consider

− d

dx

(kadT

dx

)= Q ∀x ∈ (0, L) = Ω ⊂ R1 (5.194)


-0.09

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0 0.2 0.4 0.6 0.8 1

φ

x

C2, p=5



Figure 5.46: Solution φ versus x (τ |x=1= 0 or dφdx

∣∣x=1

= 0)

-0.2

-0.15

-0.1

-0.05

0

0.05

0 0.2 0.4 0.6 0.8 1

dφ

/dx

x

C2, p=5




versus x (τ |x=1= 0 or dφdx

∣∣x=1

= 0)

with

T (0) = T0

kadT

dx+ β(T − T∞) + q = 0 at x = L

(5.195)


-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

d2φ

/dx

2

x

C2, p=5





∣∣x=1

= 0)

-25

-20

-15

-10

-5

0

5

10

15

20

0 0.2 0.4 0.6 0.8 1

d3φ

/dx

3

x

C2, p=5

(LSP: first order system)





∣∣x=1

= 0)

where k is thermal conductivity, a is area of cross section, β is heat transfercoefficient, T∞ is ambient temperature, q is heat flux and Q is internal heatgeneration. These are data and are given. T = T (x) is temperature.


-14

-12

-10

-8

-6

-4

-2

0

0 1 2 3 4 5 6

log(√I)

log(dofs)



C1, p=3

C2, p=5

C1, p=3

C2, p=5

Figure 5.50: Plots of ||E||L2=√I versus dofs (τ |x=1= 0 or dφ

dx

∣∣x=1

= 0)

-14

-12

-10

-8

-6

-4

-2

0

0 1 2 3 4 5 6

log(√I)

log(dofs)



C1, p=5

C2, p=5

C1, p=5

C2, p=5


We consider the Galerkin method with weak form. In this case

A = − d

dx

(kadT

dx

)f = Q

(5.196)


-14

-12

-10

-8

-6

-4

-2

0

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

log(√I)

log(dofs)

p-convergence study

4-element uniform mesh

C1

C2



AT −Q = 0 ∀x ∈ Ω (5.197)

In this case A∗ = A. Let T eh be approximation of T over Ωe, an elemente of the discretization ΩT =

⋃e Ωe of Ω. Then, if v is variation of T eh , we

consider the following:

(AT eh −Q, v)Ωe =

∫Ωe

[− d

dx

(kadT ehdx

)−Q

]v dx (5.198)

in which Ωe = [xe, xe+1] mapped into Ωξ = [−1, 1]. Using integration byparts for the first term in the integrand of (5.198)


∫Ωe

(dvdxkadT ehdx

)dx−

[v(kadT ehdx

)]xe+1

xe

−∫Ωe

Qv dx (5.199)

In (5.199), the concomitant 〈AT eh −Q, v〉Γe is (with φeh = T eh and f = Q)

〈AT eh −Q, v〉Γe = −[v(kadT ehdx

)]xe+1

xe

From (5.199), we conclude that

T is PV⇒ T = T on some Γ1 is EBC

kadT ehdx

is SV⇒ kadT ehdx

= q on some Γ2 is NBC(5.200)


Let

−[kadT ehdx

]xe

= P e1[kadT ehdx

]xe+1

= P e2

(5.201)

where P e1 and P e2 are the secondary variables at the two end nodes of theelement located at ξ = ∓1 in Ωξ. Substituting from (5.201) into (5.199)


∫Ωe

(dvdxkadT ehdx

)dx− v(1)P e2 − v(−1)P e1

−∫Ωe

Qv dx (5.202)

or

(AT eh −Q, v)Ωe = Be(T eh , v)− le(v) (5.203)

where

Be(T eh , v) =

∫Ωe

dv

dxkadT ehdx

dx (5.204)

le(v) = v(−1)P e1 + v(1)P e2 +

∫Ωe

Qv dx (5.205)

Let

T eh =n∑i=1

Ni δei = [N ]δe (5.206)

be local approximation of T over Ωe. Then

v = δT eh = Nj , j = 1, 2, . . . , n (5.207)

Substituting from (5.206) and (5.207) into (5.204) and (5.205)

Be(T eh , v) =

∫Ωe

dNj

dxka

( n∑i=1

dNi

dxδei

)dx, j = 1, 2, . . . , n (5.208)

le(v) = Nj(−1)P e1 +Nj(1)P e2 +

∫Ωe

QNj dx, j = 1, 2, . . . , n (5.209)


(5.203) is the desired weak form resulting from the Galerkin method withweak form. (5.208) and (5.209) can be arranged in the matrix and vectorform

Be(T eh , v) = [Ke]δe (5.210)

le(v) = P e+ F e (5.211)

in which

Keij =

1∫−1

dNj

dxkadNi

dxJ dξ = Ke

ji

P eT = [P e1 , 0 , . . . , 0 , P e2 ]

, i, j = 1, 2, . . . , n (5.212)

and

F ei =

1∫−1

QNi J dξ, i = 1, 2, . . . , n (5.213)

We also note that

dmNi

dxm=

1

JmdmNi

dξn; J =

he2, m = 1, 2, . . . , i = 1, 2, . . . , n (5.214)

Thus, knowing explicit expressions for Ni(ξ), the local approximation func-tions, the coefficients Ke

ij and F ei can be calculated using Gauss quadrature.


Since the BVP contains second order derivative of temperature T and theweak form that contains only the first order derivatives of the temperature,we have the following for minimally conforming spaces:

(i) Admissibility of Th =⋃e T

eh in AT − f = 0 in the pointwise sense in

ΩT requires T eh ∈ Vh ⊂ Hk,p(Ωe); k = 3 is minimally conforming. Forthis choice, the integrals in the following are all Riemann:

(ATh −Q, v)ΩT =∑e

(AT eh −Q, v)Ωe =∑e

[Be(T eh , v)− le(v)

]= 0

(5.215)Thus, if we choose T eh of class C2(Ωe), then all integrals are Riemannand all forms in (5.215) are precisely equivalent.


(ii) Based on the weak form, if we choose T eh of class C1(Ωe), that is, if wechoose T eh ∈ Vh ⊂ H2,p(Ωe), then the following holds:

(AT eh −Q, v)ΩT ⇔L

∑e

(AT eh −Q, v)Ωe

⇔L

∑e

(Be(T eh , v)− le(v)

)︸︷︷︸

R

= 0 (5.216)

That is, for this choice of T eh ,∑

e[Be(·, ·)−le(·)] holds in Riemann sense

but all other integral forms in (5.216) are in Lebesgue sense.

(iii) If we choose T eh ∈ Vh ⊂ H1,p(Ωe), then∑

e(Be(·, ·)− le(·)) holds only in

Lebesgue sense and the other two integral forms in (5.216) are mean-ingless.

5.2.5.2 Numerical study

For illustrating the details of the computations it suffices to choose T eh ofclass C0 with p = 1, i.e. a two-node element with linear approximation ofT eh over Ωe or Ωξ. Let the element local nodes 1 and 2 be located at ξ = −1and ξ = 1 in Ωξ map of Ωe defined by

x = x(ξ) =(1− ξ

2

)xe +

(1 + ξ

2

)xe+1 (5.217)

and

T eh(ξ) =(1− ξ

2

)T e1 +

(1 + ξ

2

)T e2 (5.218)

Thus, in this case

N1(ξ) =1− ξ

2, N2(ξ) =

1 + ξ

2(5.219)

using (5.212) we can obtain the following for an element e with domainΩe → Ωξ (with the assumption that ka = keae = constant over Ωe andQ = f0, also a constant).

(AT eh , v)Ωe =keaehe

[1 −1−1 1

]T e1T e2

−P e1P e2

− f0he

2

11

(5.220)


Equations (5.220) are valid for each element of the discretization. Let uschoose a two-element non-uniform discretization,

L = 10

h1 = 4, h2 = 6

k1a1 = 76, k2a2 = 96

β = 10, T∞ = 30, q = 0

T (0) = T0 = 100

Then, for the two element discretization shown below

321 21

x = 0 x = 4 x = 10

64

x

we have

Ele. 1: (AT 1h , v)Ω1 = 19

[1 −1−1 1

]T 1

1

T 12

−P 1

1

P 12

− 2f0

11

(5.221)

Ele. 2: (AT 2h , v)Ω2 = 16

[1 −1−1 1

]T 2

1

T 22

−P 2

1

P 22

− 2f0

11

(5.222)

Inter-element continuity conditions on PVs

T 11 = T1 = 100

T 12 = T 2

1 = T2

T 22 = T3

not known

Element equations after imposing inter-element continuity conditions on PVsbecome

Ele. 1: (AT 1h , v)Ω1 = 19

[1 −1−1 1

]T1

T2

−P 1

1

P 12

− 2f0

11

(5.223)

Ele. 2: (AT 2h , v)Ω2 = 16

[1 −1−1 1

]T2

T3

−P 2

1

P 22

− 2f0

11

(5.224)


Assembled element equations

∑e

(AT eh−f0, v)ΩT = 0⇒

19 −19 0−19 35 −16

0 −16 16

T1

T2

T3

= 2f0

121

+

P 1

1

P 12 + P 2

1

P 22

(5.225)

Inter-element continuity conditions on the sum of secondary vari-ables

T1 = 100 ⇒ P 11 is unknown

P 12 + P12 = 0

P 22 =

(kdT ehdx

)∣∣∣∣x=L

= −β(L)(T (L)− T∞) + q(L) = −10(T3) + 300

(5.226)

Substituting from (5.226) into (5.225) 19 −19 0−19 35 −16

0 −16 16

T1

T2

T3

= 2f0

121

+

P 1

1

0−10T3 + 300

(5.227)

We can transfer −10T3 on the left side of (5.227), substitute T1 = 100 andif we choose f0 = 0, then we have 19 −19 0

−19 35 −160 −16 26

100T2

T3

=

P 1

1

0300

(5.228)

or [35 −16−16 26

]T2

T3

=

0

300

(5.229)

giving T2 = 82.88, T3 = 62.54 and

P 11 =

[19 −19 0

]100

82.8862.54

= 325.28 (5.230)

Post processing: computations of T (x) and dTdx for 0 ≤ x ≤ 10

Temperature

T (ξ) =(1− ξ

2

)T1 +

(1 + ξ

2

)T2, −1 ≤ ξ ≤ 1, 0 ≤ x ≤ 4

T (ξ) =(1− ξ

2

)T2 +

(1 + ξ

2

)T3, −1 ≤ ξ ≤ 1, 4 ≤ x ≤ 10


Derivatives of temperature

dT

dξ=dT

dx

dx

dξ⇒ dT

dx=

1

J

dT

dξ

Element 1: J = h12 = 4

2 = 2

dT

dx=

1

4(T2 − T1), 0 ≤ x ≤ 4

Element 2: J = h22 = 6

2 = 3

dT

dx=

1

6(T3 − T2), 4 ≤ x ≤ 10

5.2.6 1D axisymmetric heat conduction

Consider axisymmetric heat conduction independent of circumferentialcoordinate Θ and axial coordinate z in r,Θ, z cylindrical coordinate systemdescribed in Example 3.3. Following Example 3.3, this BVP is described byradial heat conduction equation

− 1

r

d

dr

(krdθ

dr

)− f = 0 ∀ri ≤ r ≤ ro (5.231)

with BCs:θ(ri) = θ0 (5.232)

krdθ

dr+ β(θ − θ∞) = 0 at r = ro (5.233)

where θ is temperature, k is conductivity, r is radius, β is film or convectiveheat transfer coefficient, and θ∞ is ambient temperature. The differentialoperator A is defined as

A = −1

r

d

dr

(kr

d

dr

)(5.234)

Thus we can write (5.231) as

Aθ − f = 0 (5.235)

We can show that A is linear and that the adjoint of A, that is, A∗ is sameas A.

Let θeh be an approximation of θ over Ωe, an element of the discretizationΩT = ∪

eΩe of Ω = [ri, ro]. We consider Galerkin method with weak form and

the LSP based on residual functional for an element e of ΩT with domainΩe.



Let v = δθeh, then we consider the following using dΩ = 2πr dr.

(Aθeh − f, v) = 2π

∫Ωe

[−1

r

d

dr

(krdθehdr

)− f

]vr dr (5.236)

or

(Aθeh − f, v) = 2π

∫Ωe

− d

dr

(krdθehdr

)v dr − 2π

∫Ωe

fvr dr (5.237)

Transfer one order of differentiaion from θeh to v in the first term on the rightside of (5.237) (using Ωe = [re, re+1]):

(Aθeh − f, v) = 2π

∫Ωe

kdv

dr

dθehdr

r dr −[v

(2πrk

dθehdr

)]re+1

re

− 2π

∫Ωe

fvr dr

(5.238)In (5.238), concomitant is given by

〈Aθeh, v〉Γe =

[v

(2πrk

dθehdr

)]re+1

re

(5.239)

Let

−[2πrk

dθehdr

]re

= P e1[2πrk

dθehdr

]re+1

= P e2

(5.240)

Using (5.240) in (5.239) we can write

〈Aθeh, v〉Γe = v(re+1)P e2 + v(re)Pe1 (5.241)

Using (5.241) we can write (5.238) as follows

(Aθeh − f, v) = 2π

re+1∫re

kdv

dr

dθehdr

r dr − v(re+1)P e2 − v(re)Pe1 − 2π

re+1∫re

fvr dr

(5.242)or

(Aθ)he − f, v) = Be(θeh, v)− le(v) (5.243)

in which

Be(θeh, v) = 2π

re+1∫re

kdv

dr

dθehdr

r dr (5.244)


and

le(v) = v(re+1)P e2 + v(re)Pe1 + 2π

re+1∫re

fvr dr (5.245)

Let θeh, the local approximation of θ over Ωe, be given by (using the elementΩe map in Ωξ = [−1, 1]):

θeh =n∑i=1Ni(ξ)δ

ei = [N ]δe

v = δθeh = Nj(ξ), j = 1, 2, . . . , n

(5.246)

in which Ni(ξ) are approximation functions and δe are nodal degrees offreedom and

r(ξ) =

(1− ξ

2

)re +

(1 + ξ

2

)re+1

dr

dξ= J =

re+1 − re2

=he2

(5.247)

dr = J dξ

dNi

dr=

1

J

dNi

dξ(5.248)

Substituting from (5.246)–(5.248) into (5.244) and (5.245) we obtain (v =Nj)

Be(θeh, Nj) = 2π

∫Ωξ

(kdNj

dr

n∑i=1

dNi

drδei

)r(ξ)J dξ

= 2π

1∫−1

k1

J

dNj

dξ

(1

J

n∑i=1

dNi

dξδei

)r(ξ)J dξ

= [Ke]δe (5.249)

where

Keij =

2π

J

1∫−1

kdNj

dξ

dNi

dξr(ξ) dξ, i, j = 1, 2, . . . , n (5.250)

Also, we have

le(Nj) = Nj(re)Pe1 +Nj(re+1)P e2 + 2π

1∫−1

fNj(ξ)r(ξ)J dξ (5.251)


for j = 1, 2, . . . , n. We note that

N1(re) = N1

(ξ|−1

)= 1, Nn(re) = Nn

(ξ|−1

)= 0

N1(re+1) = N1

(ξ|+1

)= 0, Nn(re+1) = Nn

(ξ|+1

)= 1

(5.252)

Hence

le(Nj) =

P e10...0P e2

+ F e (5.253)

in which

F ei =

1∫−1

fNi(ξ)2πr(ξ)he2dξ = πhe

1∫−1

fNi(ξ)r(ξ) dξ (5.254)

for i = 1, 2, . . . , n. Hence, we have

(Aθeh, v)Ωe = Be(θeh, v)− le(v) = [Ke]δe −

P e10...0P e2

− F e (5.255)

Equations (5.255) are the element equations resulting from GM/WF over anelement e with domain Ωe = Ωξ = [−1, 1].

5.2.6.2 LSM based on residual functional

If θeh is approximation of θ over Ωe = [re, re+1] or Ωξ = [−1, 1] then(assuming k to be constant ke over Ωe) the residual function Ee over Ωe isgiven by

Ee = Aθeh − f ∀r ∈ Ωe (5.256)

The residual functional I for ΩT is constructed using

I =∑eIe =

∑e

(Ee, Ee)Ωe , existence of I (5.257)

in which Ie is the least-squares functional for Ωe. First variation of I set tozero gives necessary condition.

δI =∑eδIe =

∑e

2(Ee, δEe)Ωe = 2∑ege = g = 0 (5.258)


in which

ge = (Ee, δEe)Ωe (5.259)

The element relation can be derived using ge. We note that

Ee =1

r

d

dr

(krdθehdr

)+ f(r) (5.260)

δEe =1

r

d

dr

(krdv

dr

), v = δθeh (5.261)

Let

θeh =n∑i=1Ni(ξ)δ

ei (5.262)

in which Ni(ξ) are approximation functions and δei are nodal degrees offreedom. We choose a three node element with nodes 1, 2, 3 located atξ = −1, 0, 1. Explicit forms of Ni(ξ) and δei depend upon the class of θeh i.e.the order of the approximation space Vh ⊂ Hk(Ωe). We discuss details in alater section. First, based on local approximation θeh, we have

v = δθeh = Nj(ξ), j = 1, 2, . . . , n (5.263)

Substituting θeh and v in ge we obtain gej (j = 1, 2, . . . , n) as

gej =

(1

r

d

dr

(kr

n∑i=1

dNi

drδei

)+ f(r),

1

r

d

dr

(krdNj

dr

))Ωe

(5.264)

Since the differential operator A is linear, gej can be written as

gej =

(n∑i=1

1

r

d

dr

(krdNi

dr

)δei + f(r),

1

r

d

dr

(krdNj

dr

))Ωe

(5.265)

which is same as

gej =

(n∑i=1

(ANi)δei + f, ANj

)Ωe

(5.266)

or

ge = [Ke]δe+ F e (5.267)

in which

Keij = (ANj , ANi)Ωe (5.268)

F ei = (f, ANi)Ωe (5.269)


Substituting for A we can obtain explicit forms of Keij and F ei (i, j =

1, 2, . . . , n):

Keij =

(1

r

d

dr

(krdNj

dr

),

1

r

d

dr

(krdNi

dr

))Ωe

(5.270)

F ei =

(f,

1

r

d

dr

(krdNi

dr

))Ωe

(5.271)

Using element map in natural coordinate space ξ i.e. Ωe = [re, re+1]→ Ωξ =[−1, 1] we can write

r(ξ) =

(1− ξ

2

)re +

(1 + ξ

2

)re+1 (5.272)

dr

dξ= J =

1

2(re+1 − re) =

he2

(5.273)

dr = J dξ (5.274)

dmNi

drm=

1

JmdmNi(ξ)

dξm=

(2

he

)m dmNi(ξ)

dξm(5.275)

and dΩ = 2πr dr = 2πr(ξ) dr. Additionally (assuming k to be constant)

1

r

d

dr

(krdNi

dr

)=

1

rkdNi

dr+ k

d2Ni

dr2(5.276)

Hence, we can write the following for Keij and F ei in (5.270) and (5.271).

Keij = 2π

1∫−1

(1

r(ξ)kdNj

dr+ k

d2Nj

dr2

)(1

r(ξ)kdNi

dr+ k

d2Ni

dr2

)r(ξ)J dξ

(5.277)

F ei = 2π

1∫−1

f(ξ)

(1

r(ξ)kdNi

dr+ k

d2Ni

dr2

)r(ξ)J dξ (5.278)

dNidr and d2Ni

dr2can be substituted in (5.277) and (5.278) using (5.275). The

usual process of assembly of (5.267) follows the standard procedure.

5.2.7 A 1D BVP governed by a fourth-order differentialoperator

Consider the following BVP describing bending of beams:

d2

dx2

(EI

d2w

dx2

)−Q = 0 ∀x ∈ (0, L) = Ω ⊂ R1 (5.279)


with

w(0) =dw

dx

∣∣∣∣x=0

= 0

d

dx

(bd2w

dx2

)∣∣∣∣x=L

= FL

EId2w

dx2

∣∣∣∣x=L

= ML

(5.280)

where EI is the product of the modulus of elasticity and bending momentof inertia, Q is distributed load along the length of the beam, FL and ML

are shear force and bending moments at x = L. w = w(x) is the transversedeflection of the beam.

L

w(0) = 0

dwdx

∣∣x=0

FL

ML

Q(x)

A schematic of the BVP is shown in the figure above. In this case thedifferential operator A is given by

A =d2

dx2

(EI

d2

dx2

)and f = Q (5.281)

The operator is linear and symmetric (if the boundary conditions are homo-geneous) and A∗ = A. Thus, we can write (5.279) as

Aw −Q = 0 ∀x ∈ Ω (5.282)

Let weh be approximation of w over Ωe, an element of the discretizationΩT =

⋃e Ωe of Ω. We consider Galerkin with weak form. If v = δweh then

we consider the following for Ωe:

(Aweh −Q, v)Ωe =

(d2

dx2

(EI

d2

dx2

)−Q, v

)Ωe

(5.283)

in which Ωe = [xe, xe+1] is mapped into Ωξ = [−1, 1]. We can also write

(Aweh −Q, v)Ωe =

∫Ωe

(d2

dx2

(EI

d2

dx2

)v −Qv

)dx (5.284)


Transfer two orders of differentiation to v in the first term of the integrandin (5.284).

(Aweh −Q, v)Ωe =

∫Ωe

(d2v

dx2EI

d2wehdx2

)dx+

[v

(d

dx

(EI

d2wehdx2

))]xe+1

xe

−[dv

dx

(EI

d2wehdx2

)]xe+1

xe

−∫Ωe

Qv dx (5.285)

In (5.285), the concomitant 〈Aweh, v〉Γe is given by

(Aweh, v)Γe =

[v

(d

dx

(EI

d2wehdx2

))]xe+1

xe

−[dv

dx

(EI

d2wehdx2

)]xe+1

xe

Using concomitant

weh is a PV; hence, weh = w on some Γ1 is an EBC

dwehdx

is also a PV; hence,dwehdx

= θ on some Γ2 is an EBC

d

dx

(EI

d2wehdx2

)is a SV; hence,

d

dx

(EI

d2wehdx2

)= F on some Γ3 is a NBC

EId2wehdx2

is a SV; hence, EId2wehdx2

= M on some Γ4 is a NBC

Introducing the following notations (assuming the end nodes of the elementsto be 1 and 2 and located at ξ = −1 and +1 in its map Ωξ)

Qe1 =

(d

dx

(EI

d2wehdx2

))x=xe

Qe2 =

(d

dx

(EI

d2wehdx2

))x=xe+1

M e1 =

(bd2wehdx2

)x=xe

M e2 =

(bd2wehdx2

)x=xe+1

(5.286)


(Aweh −Q, v)Ωe =

∫Ωe

(d2v

dx2EI

d2wehdx2

)dx− v(xe)Q

e1 − v(xe+1)Qe2

−

(dv

dx

∣∣∣∣xe

)M e

1 −

(dv

dx

∣∣∣∣xe+1

)M e

2 −∫Ωe

Qv dx (5.287)


or(Aweh −Q, v)Ωe = Be(weh, v)− le(v) (5.288)

where

Be(weh, v) =

∫Ωe

(d2v

dx2EI

d2wehdx2

)dx (5.289)

le(v) = v(xe)Qe1 + v(xe+1)Qe2 +

dv

dx

∣∣∣∣xe

M e1 +

dv

dx

∣∣∣∣xe+1

M e2

+

∫Ωe

Qv dx (5.290)

Let weh be local approximation of w over Ωe

weh =n∑i=1

Ni(ξ) δei = [N ]δe (5.291)

Thereforeδweh = v = Nj(ξ), j = 1, 2, . . . , n (5.292)


Be(weh, v) =

∫Ωe

(d2Nj

dx2EI( n∑i=1

d2Ni

dx2δei

))dx (5.293)

le(v) = Nj(xe)Qe1 +Nj(xe+1)Qe2 +

dNj

dx

∣∣∣∣xe

M e1 +

dNj

dx

∣∣∣∣xe+1

M e2

+

∫Ωe

Qv dx (5.294)

where j = 1, 2, . . . , n. Equation (5.288) is the weak form resulting from theGalerkin method with weak form. Equations (5.293) and (5.294) can bewritten in the matrix and vector form as

Be(weh, v) = [Ke]δe, le(v) = P e+ F e (5.295)

in which (i, j = 1, 2, . . . , n)

Keij =

1∫−1

(d2Nj

dx2EI

d2Ni

dx2

)J dξ = Ke

ji

P eT =[Qe1 M

e1 0 0 . . . Qe2 M

e2

]T(5.296)

F ej =

∫Ωe

QNj J dξ


We also note thatdmNj

dxm=

1

JmdmNj

dξm, J =

he2

(5.297)

Thus, knowing explicit expressions for Ni(ξ), the local approximation func-tions, the coefficients of [Ke] and F e can be calculated using Gauss quadra-ture.


Since the BVP that contains fourth order derivative of the deflection wand the weak form that only contains second order derivative of the deflec-tion, we have the following for the minimally conforming spaces:

(i) Admissibility of wh =⋃ew

eh in Aw −Q = 0 in the pointwise sense in

ΩT requires weh ∈ Vh ⊂ Hk,p(Ωe); k = 5 is minimally conforming. Forthis choice, the integrals in the following are all Riemann:

(Awh −Q, v)ΩT =∑e

(Aweh −Q, v)Ωe =∑e

(Be(weh, v)− le(v)

)= 0

(5.298)Thus, if we choose weh of class C4(Ωe), then all integrals are Riemannand all forms in (5.298) are precisely equivalent.

(1) Based on the weak form, if we choose weh of class C2(Ωe); that is, ifweh ∈ Vh ⊂ H3,p(Ωe), then the following holds:

(Awh −Q, v)ΩT and∑e

(Aweh −Q, v)Ωe are not defined, but

∑e

[Be(·, ·)− le(·)

]= 0

holds in Riemann sense. For this choice, obviously there is no equiva-lence between the three integral forms in (5.298), as the first two arenot defined.

(2) If we choose weh of class C1(Ωe) then∑e

[Be(·, ·)− le(·)

]= 0

holds in Lebesgue sense but the first two integral forms in (5.298) arenot defined.


5.3 Two-dimensional boundary value problems

In this section we consider two dimensional boundary value problems insingle and multiple dependent variables described by self-adjoint differentialoperators. The finite element processes for these boundary value problemsremain the same as those for one dimensional boundary value problems inconcepts and basic steps except that for two dimensional boundary valueproblems:

(i) The domain of definition of the BVP is in R2, i.e. two dimensional (sayin x, y) and, hence, consists of an area. Therefore, the discretizationof the domain consisting of subdomains that is finite elements containsarea elements that may be triangular, quadrilateral or any other desiredshape.

(ii) The boundary of the domain of definition of the BVP is a closed contouror curve Γ, hence, in the process of integration by parts the concomitantconsists of boundary integrals as opposed to the boundary terms as inthe case of one dimensional boundary value problems.

First, we consider a two-dimensional BVP in a single dependent variable.This is followed by the BVPs in multivariables.

5.3.1 A general 2D BVP in a single dependent variable

We consider a typical 2D boundary value problem to illustrate variousconcepts, principles and procedures involved in deriving finite element pro-cesses for such BVPs. Consider the boundary value problem

− ∂

∂x

(a11

∂φ

∂x+ a12

∂φ

∂y

)− ∂

∂y

(a21

∂φ

∂x+ a22

∂φ

∂y

)+ a00φ− f = 0 (5.299)

for all (x, y) ∈ Ω ⊂ R2, with boundary conditions

φ = φ0 on Γ1 (5.300)(a11

∂φ

∂x+ a12

∂φ

∂y

)nx +

(a21

∂φ

∂x+ a22

∂φ

∂y

)ny = q on Γ2 (5.301)

in which Ω = Ω ∪ Γ and Γ = Γ1 ∪ Γ2 is the closed boundary of the domainΩ. nx and ny are direction cosines of a unit exterior normal to the boundaryΓ2 (see Fig. 5.53).

From (5.299) we note that the differential operator A is defined by

A = − ∂

∂x

(a11

∂

∂x+ a12

∂

∂y

)− ∂

∂y

(a21

∂

∂x+ a22

∂

∂y

)+ a00 (5.302)

5.3. TWO-DIMENSIONAL BOUNDARY VALUE PROBLEMS 311

x

y

Ω

nx

ny

n2x + n2

y = 1

n

Γ2

Γ1

Figure 5.53: A two-dimensional domain


Aφ− f = 0 ∀(x, y) ∈ Ω (5.303)

where a00, aij , aij = aji are known functions of position (x, y). We canshow that the differential operator A is linear and A∗ = A. Therefore, theGalerkin method with weak form will yield a variationally consistent integralform if the functional B(·, ·) is bilinear and symmetric and the functionall(·) is linear. We consider the finite element process for this boundary valueproblem based on Galerkin method with weak form. For an element e withdomain Ωe = Ωe ∪ Γe of the discretization ΩT = ∪eΩe of Ω, we consider(Aφeh − f, v)Ωe in which φeh is the local approximation of φ over Ωe, v = δφehand Γe is a closed boundary of Ωe. Let

Qx = a11∂φ

∂x+ a12

∂φ

∂y, Qy = a21

∂φ

∂x+ a22

∂φ

∂y(5.304)

Then we can write (5.299) and (5.301) as

Aφ− f = −∂Qx∂x− ∂Qy

∂y+ a00φ−Q = 0 in Ω (5.305)

and

Qxnx +Qyny = q on Γ2 (5.306)

Thus, with new notation, we have


∫Ωe

(−∂Q

ex

∂x−∂Qey∂y

+ a00φ− f)v dx dy (5.307)


or


∫Ωe

(−∂Q

ex

∂x−∂Qey∂y

)v dx dy

+

∫Ωe

a00 φehv dx dy −

∫Ωe

fv dx dy (5.308)

Using integration by parts once for the first term and using

Qex = a11∂φeh∂x

+ a12∂φeh∂x

, Qey = a21∂φeh∂x

+ a22∂φeh∂x

we obtain


∫Ωe

(∂u∂xQex +

∂v

∂yQey

)dx dy −

∮Γev(Qexnx +Qeyny) dΓ

+

∫Ωe

a00 φehv dx dy −

∫Ωe

fv dx dy (5.309)

In which the concomitant 〈Aφeh, v〉Γe is given by

〈Aφeh, v〉Γe = −∮

Γev(Qexnx +Qeyny) dΓ

nx and ny are direction cosines of a unit exterior normal to the boundaryΓe of an element Ωe. Using the concomitant in (5.309) we can identify PVs,SVs, EBCs and NBCs.

φeh is PV⇒ φeh = φ on Γ1 is EBC (5.310)

(Qexnx +Qeyny) is SV⇒ (Qexnx +Qeyny) = q on Γ2 is NBC (5.311)

where φ and q are specified or known. Let

qen = Qexnx +Qeyny (5.312)

be the flux normal to Γe. Substituting from (5.312) into (5.309)

(Aφeh−f, v)Ωe =

∫ (∂v∂xQex+

∂v

∂yQey+a00φ

ehv)dx dy−

∮Γe

vqen dΓ−∫Ωe

fv dx dy

(5.313)or



where

Be(φeh, v) =

∫Ωe

(∂v∂xQex +

∂v

∂yQey + a00 φ

ehv)

(5.315)

le(v) =

∮Γe

vqen dΓ +

∫Ωe

fv dx dy (5.316)

The right side of (5.314) is the weak form of the BVP over Ωe resulting fromthe Galerkin method with weak form. We note that Be(·, ·) is bilinear andsymmetric and le(·) is linear. These properties are obviously due to the factthat A∗ = A and the integration by parts. Let

φeh =

n∑i=1

Niδei = [N ]δe (5.317)

be the local approximation of φ over Ωe, element e of the discretization ΩT .Then

v = δφeh = Nj , j = 1, 2, . . . , n (5.318)


Be(φeh, v) =

∫Ωe

[∂Nj

∂xQex +

∂Ni

∂yQey + a00Nj

( n∑i=1

Niδei

)]dx dy (5.319)

le(v) =

∮Γe

Njqen dΓ +

∫Ωe

fNj dx dy (5.320)

for (i, j = 1, 2, . . . , n) and

Qex = a11

n∑i=1

∂Ni

∂xδei + a12

n∑i=1

∂Ni

∂yδei (5.321)

Qey = a21

n∑i=1

∂Ni

∂xδei + a22

n∑i=1

∂Ni

∂yδei (5.322)

or

Be(φeh, v) = [Ke]δe, le(v) = P e+ F e (5.323)


in which

Kij =

∫Ωe

(∂Ni

∂x

(a11

∂Nj

∂x+ a12

∂Nj

∂y

)

+∂Ni

∂y

(a21

∂Nj

∂x+ a22

∂Nj

∂y

)+ a00NjNi

)dΩ (5.324)

P ei =

∮Γe

Niqen dΓ (5.325)

F ei =

∫Ωe

fNi dΩ (5.326)

We note that Keij = Ke

ji when aαβ = aβα (α, β = 1, 2).

Remarks.

1. P e is yet to be determined which requires definition of Ωe, i.e. thedomain of definition (geometry) of an element e.

2. Once Ωe is defined, we need to establish the local approximations φeh.

3. The approximation space Vh containing local approximation functions Ni

needs to be defined as well.

5.3.1.1 Definition of Ωe: element geometry

The element geometry in the physical coordinate space (say x, y) couldbe a triangular domain or a quadrilateral domain with distorted sides. Theseshapes enable discretizations of irregular domains Ω with minimum geomet-ric approximations. Figure 5.54 show such elements in the physical spacex, y. We consider a distorted quadrilateral element for presenting details forthis model problem (see Fig. 5.55).

Following the details of mapping for 2D quadrilateral domain given inchapter 8, for the mapping of points we have

x = x(ξ, η) =

9∑i=1

Ni(ξ, η)xi (5.327)

y = y(ξ, η) =9∑i=1

Ni(ξ, η)yi (5.328)

Mapping of length (dx, dy) and (dξ, dη) is given bydxdy

= [J ]

dξdη

(5.329)


23

4

567

8

1

9

y

x

Quadrilateral element in x, yspace

y

x

Triangular element in x, yspace

Figure 5.54: Triangular and quadrilateral elements in x, y space

1 2 3

567

8 94

2

2

23

4

567

8

9

1

η

ξ

Map of Ωe in the natural co-ordinate space ξ, η (i.e. Ωξη)

Ωe Ωξη

Quadrilateral element in x, yspace

y

x

Figure 5.55: A quadrilateral element Ωe and its map Ωξη in natural coordinate spaceξ, η

where

[J ] =

[∂x∂ξ

∂x∂η

∂y∂ξ

∂y∂η

](5.330)

is the Jacobian of transformation

dx dy = |J | dξ dη (5.331)

and ∂Ni∂x∂Ni∂y

= [JT ]−1

∂Ni∂ξ∂Ni∂η

(5.332)


in which Ni(ξ, η) are the local approximation functions (used in (5.317)).We also note that (xi, yi) are the coordinates of the element nodes in the(x, y) space, and the shape functions Ni(ξ, η) have the following properties(see chapter 8)

Ni(ξj , ηj) =

1 ; j = i

0 ; j 6= i

9∑i=1

Ni(ξ, η) = 1

(5.333)

We have intentionally chosen an element geometry with nine nodes. Mid-side nodes permit quadratic geometry description. Nine-node configurationpermits easy determination of Ni(ξ, η) using tensor product (see chapter 8).


The governing differential equation describing the boundary value prob-lem contains second order derivatives of the dependent variable in x and ywhereas the weak form only contains first order derivatives of the dependentvariable and the test function v.

(i) Admissibility of φh =⋃e φ

eh in Aφ−f = 0 in ΩT in the pointwise sense

requires

φeh ∈ Vh ⊂ Hk,p(Ωe); k = 3 is minimally conforming (5.334)

For this choice, the integrals in the following are Riemann.


(Aφeh − f, v)Ωe

=∑e


)Ωe

= 0 (5.335)

Thus, if we choose φeh of class C2(Ωe) then all integrals are Riemannand all three forms in (5.335) are precisely equivalent.

(ii) Based on the weak form, if we choose φeh of class C1(Ωe) then

(Aφh − f, v)ΩT8→L

∑e

(Aφeh − f, v)Ωe

8→L

(∑e

Be(φeh, v)− le(v))

︸︷︷︸R

= 0 (5.336)


That is, for this choice of φeh,∑

e(Be(·, ·)− le(·)) holds in the Riemann

sense, but all other integral forms in (5.336) are in the Lebesgue sense.

(iii) If we choose φeh ∈ Vn ⊂ H1,p(Ωe), then∑

e(Be(·, ·) − le(·)) only holds

in the Lebesgue sense and the other two integral forms in (5.336) aremeaningless.

5.3.1.3 Computation of the element matrix [Ke] and vector F e

Using the element map Ωξη, we can transform integrals for [Ke] and F eover Ωξη. Using (5.336), we can write

Keij =

1∫−1

1∫−1

[∂Ni

∂x

(a11

∂Nj

∂x+ a12

∂Nj

∂y

)+∂Ni

∂x

(a21

∂Nj

∂x+ a22

∂Nj

∂y

)(5.337)

+ a00NjNi

]|J | dξ dη (5.338)

F ei =

1∫−1

1∫−1

f Ni |J | dξ dη (5.339)

The derivatives of Nis with respect to x and y can be obtained using (5.332),∂Ni∂ξ , ∂Ni

∂η can be easily obtained as Ni = Ni(ξ, η); det[J ] = |J | is calculatedusing (5.330). The derivatives of x(ξ, η) and y(ξ, η) with respect to ξ andη are obtained using (5.327). Numerical values of Ke

ij and F ei are obtainedusing Gauss quadrature.

5.3.1.4 Details of secondary variable vector P e

We note that P e is defined by∮

Γe v qen dΓ where v = Nj ; j = 1, 2, . . . , n.

Details of P e obviously requires definition of Ωe = Ωe⋃

Γe. Consider afour node quadrilateral element in x, y space mapped into Ωξη (or Ωm, themaster element) shown in Fig. 5.56. In this case

φeh =

4∑i=1

Ni(ξ, η)φei (5.340)

is a C0 bilinear local approximation (see chapter 8), in which Ni(ξ, η) arethe approximation functions and φei are nodal values of φ for element e. Theboundary Γe of Ωe consists of

Γe = Γe1⋃

Γe2⋃

Γe3⋃

Γe4 (5.341)


3

1 2

34

2

2

2

1

4

η

ξ

y

x

Γe1

Γe2

Γe3

Γe4

A four node quadrilateral el-ement

(b)(a)

Ωξη

Ωe

Element map in the naturalcoordinate space ξη

Figure 5.56: A four-node quadrilateral element in x, y and ξ, η space

For this particular choice of Ωe and φeh, we have

P e =

∮Γe

N1

N2

N3

N4

qen dΓ (5.342)

we can write the integral in (5.342) over the closed boundary Γe as the sumof the four integrals over Γei ; i = 1, 2, . . . , 4

P e =

∫Γe1

N1

N2

N3

N4

qen dΓ+

∫Γe2

N1

N2

N3

N4

qen dΓ+

∫Γe3

N1

N2

N3

N4

qen dΓ+

∫Γe4

N1

N2

N3

N4

qen dΓ

(5.343)We note that

N1(ξ, η) =(1− ξ

2

)(1− η2

)N2(ξ, η) =

(1 + ξ

2

)(1− η2

)N3(ξ, η) =

(1 + ξ

2

)(1 + η

2

)N4(ξ, η) =

(1− ξ2

)(1 + η

2

)(5.344)


Hence, we have

on Γe1 : N3 = N4 = 0

on Γe2 : N1 = N4 = 0

on Γe3 : N1 = N2 = 0

on Γe4 : N2 = N3 = 0

(5.345)

Substituting these into (5.343)

P e =

∫Γe1

N1

N2

00

qen dΓ+

∫Γe2

0N2

N3

0

qen dΓ+

∫Γe3

00N3

N4

qen dΓ+

∫Γe4

N1

00N4

qen dΓ

(5.346)Let us define

P e =

P e11

P e21

00

+

0P e22

P e32

0

+

00P e33

P e43

+

P e14

00P e44

(5.347)

In (5.347), the superscript e is for element e, the first subscript is the nodenumber and the second subscript represents the side (or boundary) numberi = 1, 2, . . . , 4 for Γi; i = 1, 2, . . . , 4. since qen is normal to Γe (positive whenpointing in the same direction as the unit exterior normal), P e11 and P e21 areat nodes 1 and 2 of element e, normal to the boundary Γe1. Similarly, P e33

and P e43 are at nodes 3 and 4 of element e normal to the boundary Γe3 andso on. Symbolically, we can write

P e =

P e1P e2P e3P e4

=

P e11 + P e14

P e21 + P e22

P e32 + P e33

P e43 + P e44

(5.348)

We must keep in mind that the sum in (5.348) is only symbolic as the direc-tions of the quantities in the sum are normal to the corresponding elementsides on which they are defined.

Remarks.

(1) When the φeh of class C0(Ωe), then φeh ∈ Vh ⊂ H1,p(Ωe).

(2) C0(Ωe) local approximation (of any degree p) permit inter-element con-tinuity of the function φ and its derivative tangent to the inter-elementboundaries, but the derivative of φ normal to the inter-element bound-aries is discontinuous. Referring to Fig. 5.57, we note that φeh and φe+1

h


are local approximations for elements e and e+1 with a common bound-ary (Γe2 for element e and Γe+1

4 for element e + 1). Let n and t be thenormal and tangent directions at a point on the common boundary be-tween elements e and e+1, then, for each point on this common boundarywe have

φeh|Γe1 = φe+1h

∣∣Γe+14

∂φeh∂t

∣∣∣∣Γe1

=∂φe+1

h

∂t

∣∣∣∣∣Γe+14

∂φeh∂n

∣∣∣∣Γe1

6=∂φe+1

h

∂n

∣∣∣∣∣Γe+14

(5.349)

(5.349) are intrinsic properties of C0(Ωe) local approximations in two-dimensional space x, y for each inter-element boundary like the oneshown in Fig. 5.57.

(3) When the local approximations φeh are of class C1(Ωe), then in (5.349)the first two conditions hold but additionally equality also holds in thecase of the third conditions, i.e. derivatives of the function or dependentvariable normal to the inter-element boundary is continuous as well.

5.3.2 2D Poisson’s equation: numerical studies

We consider a special case of the model problem in section 5.3.1.1 fornumerical studies. If we choose a11 = 1, a12 = 0, a21 = 0, a22 = 1 anda00 = 0 and Ω a two unit square, then we have

∂2φ

∂x2+∂2φ

∂y2+ f(x, y) = 0 ∀(x, y) ∈ Ω = (−1, 1)× (−1, 1) ⊂ R2 (5.350)

5.3.2.1 Case (a): f = 1 with BCs φ(±1, y) = φ(x,±1) = 0; GM/WF

For this case we present details of element computations, assembly, andsolution procedure using GM/WF.

Galerkin method with weak form: Following section 5.3.1.1, if wechoose a four-node quadrilateral element with φeh ∈ Vh ⊂ H1,1(Ωe), i.e.k = 1, p = 1, a bilinear C0(Ωe) approximation of φ over Ωe. Then

φeh(ξ, η) =4∑i=1

Ni(ξ, η)φei (5.351)


2 3

65

1 2

54

1

4

2 3

65

Γe+13

Γe4 Γe+12

Γe+11

Γe+14

Γe2∂φeh∂η

∣∣∣Γe+14

Γe2

Γe1

Γe3

e

e

e+ 1

Γe+14 e+ 1∂φeh

∂η

∣∣∣Γe2

∂φeh∂t

∣∣∣Γe+14

∂φeh∂t

∣∣∣Γe2

Figure 5.57: A four-node quadrilateral element with inter-element boundary in x, y

Ni(ξ, η) are given by (5.344). The element matrix [Ke] and vector F e are

Keij =

1∫−1

1∫−1

(∂Ni

∂x

∂Nj

∂x+∂Ni

∂y

∂Nj

∂y

)|J | dξ, dη, i, j = 1, 2, . . . , 4 (5.352)

F ei =

1∫−1

(1)Ni |J | dξ dη (5.353)

A schematic of the domain of definition Ωe and the boundary conditionsare shown in Fig. 5.58 (a), and a four element uniform discretization isshown in Fig. 5.58 (b). Figure 5.58 shows each of the four elements of thediscretization with local node numbers and the nodal dofs (using local nodenumbers). For each of the four elements of the discretization we calculate[Ke] and F e using (5.352) and (5.353). We can write

Be(φeh, v) = [Ke]δe =1

6

4 −1 −2 −1−1 4 −1 −2−2 −1 4 −1−1 −2 −1 4

φe1φe2φe3φe4

, e = 1, 2, . . . , 4 (5.354)


7

1 3

654

8 9

2

1 2

3 4

2

2

x

yy

x

φ = 0

φ = 0

φ = 0

φ = 0

A four-element uniform dis-cretization

(b)Schematic of Ω(a)

Figure 5.58: Domain Ω and a four-element uniform discretization

21

e

P e21P e

11

P e43

Γe1

Γe2Γe4

34

P e33

P e14

P e44

Γe3

P e22

P e32

Figure 5.59: Secondary variables for an element

le(v) = P e+ F e =

P e11 + P e14

P e21 + P e22

P e32 + P e33

P e43 + P e44

+1

4

1111

; e = 1, 2, . . . , 4 (5.355)

Details of P e for an element e are shown in Fig. 5.59. We note thatthe secondary variables are normal to the faces of the element as shown inFig. 5.59.

Inter-element continuity conditions on the nodal variables (PVs)of the elements

Comparing the four element mesh with node numbers 1-9 with the ele-ments with local node numbers (Fig. 5.60), if φ1 to φ9 are the nodal valuesof φ for the nine nodes of the discretization, then we have the followingcorrespondence:


φ11 = φ1; φ1

2 = φ21 = φ2; φ2

2 = φ3

φ14 = φ3

1 = φ4; φ13 = φ3

2 = φ5; φ23 = φ4

2 = φ6

φ34 = φ7; φ3

3 = φ44 = φ8; φ4

3 = φ9

(5.356)

φ32φ3

1

φ33φ3

4

φ12φ1

1

φ13φ1

4

φ42φ4

1

φ43φ4

4

φ22φ2

1

φ23φ2

4

21

34

21

34

21

34

4

21

34

3

1 2

Figure 5.60: Each element of the discretization with local node numbers and degrees offreedom using local node numbers

Substituting (5.356) in (5.354) we obtain the following for each of thefour elements (noting that [K1] = [K2] = [K3] = [K4]).

Element 1 : B1(φ1h, v) = [K1]

φ1

φ2

φ5

φ4

= [K1]δ1 (5.357)


φ2

φ3

φ6

φ5

= [K2]δ2 (5.358)


φ4

φ5

φ8

φ7

= [K3]δ3 (5.359)



φ5

φ6

φ9

φ8

= [K4]δ4 (5.360)

(5.355) remains valid for e = 1, 2, . . . , 4.

Boundary conditions

Using the boundary conditions of the BVP, we have

φ1 = φ2 = φ3 = φ4 = φ6 = φ7 = φ8 = φ9 = 0 (5.361)

Thus, only φ5 is unknown.

Assembly of element equations

The assembly of element matrices and vectors follow the standard pro-cedure, i.e.




)=∑e

[Ke]δe −∑e

P e −∑e

F e = [K]δ − P − F = 0 (5.362)

where

[K] =∑e

[Ke], P =∑e

P e, F =∑e

F e, δ =⋃δe (5.363)

In this case, [K] is a (9 × 9) matrix, F is a (9 × 1) vector with knowncoefficients (from assembly of [Ke] and F e). However, P is a vector ofsecondary variables containing unknown secondary variables from each ofthe four elements. We consider details in the following.

Conditions on the sum of secondary variables

Figure 5.61 shows secondary variables at the nodes of each of the fourelements. The node numbers inside each element are local node numbers,whereas those outside are global node numbers (i.e. node numbers of thegrid points of ΩT ).

We recall that

(i) The sum of secondary variables at a node must be equal to the exter-nally applied disturbance at that node.

(ii) If the externally applied disturbance at a node is zero, then the sum ofsecondary variables at that node must be zero.


8

21

1

34

21

3

34

21

4

34

21

2

34

P 111 P 1

21

P 122Γ1

1P 1

14

P 144

Γ13 P 1

32

P 133P 1

43

P 214

P 211 P 2

21

Γ21

P 222

Γ22Γ2

4

P 244

Γ23 P 2

32

P 233P 2

43

P 421P 4

11

Γ41

P 422P 4

14

P 444

Γ43 P 4

32

P 433P 4

43

Γ44 Γ4

2

P 322

P 332

P 333

Γ32

Γ31

Γ33

P 343

P 344

P 314

P 311

Γe2Γ14

Γ34

5

7

4

3

65

22

54

1

6

98

5P 321

Figure 5.61: Secondary variables at the element nodes for the four-element discretization

(iii) At nodes where the dependent variable (or PV in this case) is specified,the sum of secondary variables is not known.

An important fact to keep in mind is that secondary variables shown inFig. 5.61 are normal to the faces of the elements. Thus, care must be takenin their sum at a node. In view of the BCs (5.361) we have the following

At node 1:

P 111 : not known because φ = 0 on Γ1

1, i.e. φ = 0 at node 1


4

At node 2:

P 122 + P 2

14 = 0 : equilibrate

(P 121 + P 2

11) not known because φ = 0 at node 2

At node 3:


1, i.e. φ = 0 at node 3


2, i.e. φ = 0 at node 3

At node 4:

P 143 + P 3


(P 144 + P 3



At node 5:P 1

32 + P 244 = 0

P 133 + P 3

21 = 0

P 243 + P 4

11 = 0

P 422 + P 4

14 = 0

equilibrate

At node 6:

P 233 + P 4


(P 232 + P 4


At node 7:


4


3

At node 8:

P 332 + P 4


(P 333 + P 4


At node 9:


2


3

Then we have

δT =[φ1 φ2 φ3 φ4 φ5 φ6 φ7 φ8 φ9

]or

δT =[0 0 0 0 φ5 0 0 0 0

](5.364)

and

PT =[P1 P2 P3 P4 P5 P6 P7 P8 P9

]or

PT =[0 0 0 0 P5 0 0 0 0

](5.365)

Solving for φ5 using the assembled equations gives φ5 = 0.375. Secondaryvariables can be easily calculated using the standard procedure discussedearlier.


5.3.2.2 Case (b): BCs φ(±1, y) = φ(x,±1) = 1.0; GM/WF

Consider the same BVP as in (5.350) with boundary conditions

φ(±1, y) = φ(x,±1) = 1.0 (5.366)

We choose f(x, y) such that the theoretical solution φt(x, y) of (5.350) and(5.366) is

φ(x, y) = e(1−x2)(1−y2) (5.367)

Galerkin method with weak form: Details of the element equationsremain the same as in case (a). For numerical studies we consider a 4 × 4uniform discretization of Ω using nine-node p-version two-dimensional finiteelements. We consider local approximation φeh of φ over an element e withdomain Ωe to be in Hk,p(Ωe); k = 1, 2, 3; p ≥ 2k− 1 i.e. solutions of classesCj(ΩT ); j = 0, 1, 2. For each class of local approximation the discretizationis kept fixed and the p-levels are uniformly increased (pξ = pη = p) for eachelement. As seen in earlier examples the quadratic functional cannot be usedto quantitatively assess the approximation errors but the residual functionalcan be. Unfortunately for local approximations of class C0 it cannot becomputed. Figure 5.62 show graphs of square root of quadratic functionalversus dofs for solutions of class C1(ΩT ) at p = 3, 5, 7, 9 and of class C2(ΩT )at p = 5, 7, 9. We clearly observe that for a given dofs,

√I for solutions

of class C2(ΩT ) is considerably lower than for the solution of class C1(ΩT )confirming better accuracy of the solutions of class C2. The same slopes ofthe two curves show that the rate of convergence of

√I for the two classes of

local approximations are same. Figure 5.63 shows plots of solution φ versusx at y = 0 (same for y at x = 0) for solutions of classes C0, C1 and C2

for various p-levels. At p = 3 and beyond local approximation of all three

classes produce quite accurate results. Graphs of dφdx , d2φdx2

and d3φdx3

versus x at

y = 0 are shown in Figs. 5.64–5.66. Interelement discontinuity of dφdx dimin-

ishes with increasing p-level. At p = 3 and beyond all three classes of localapproximations produce converged solutions that are in excellent agreementwith the theoretical solution. In Fig. 5.65 solutions of class C1(ΩT ) at p = 3

show interelement jumps in d2φdx2

which diminish upon increasing p-level. d3φdx3

versus x for solutions of class C2(ΩT ) show interelement jumps that alsodiminish and eventually converge with increasing p-levels. Importance ofthe higher order global differentiability approximations in achieving conver-gence and accurate values of the higher order solution derivatives is clearlydemonstrated in the numerical studies presented in this example.


-14

-12

-10

-8

-6

-4

-2

0

1.8 2 2.2 2.4 2.6 2.8 3 3.2

log(√I)

log(dofs)

(GM/WF)

p=5 p=5

p=7 p=7

p=9 p=9

C1

C2

Figure 5.62: Case (b): square root of residual functional versus dofs

1

1.2

1.4

1.6

1.8

2

2.2

2.4

-1 -0.5 0 0.5 1

Solu

tion

φ

x

(GM/WF) C0, p=1

C0, p=2

C0, C

1, C

2, p≥3

theoretical

Figure 5.63: Case (b): solution φ versus x at y = 0 (same for y at x = 0)

5.3.3 Two-dimensional boundary value problemsin multi-variables: 2D plane elasticity

In this section we consider a boundary value problem containing morethan one dependent variable. Examples of such BVPs are in linear elasticitysuch as plane stress, plane strain, axisymmetric deformation, bending of


-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-1 -0.5 0 0.5 1

dφ

/dx

x

(GM/WF) C0, p=1

C0, p=2

C0, C

1, C

2, p≥3

theoretical


versus x at y = 0 (same for y at x = 0)

-3

-2

-1

0

1

2

-1 -0.5 0 0.5 1

d2φ

/dx

2

x

(GM/WF) C1, p=3

C1, p=4

C1, C

2, p≥5

theoretical



plates and shells, etc. Here, we specifically consider plane stress and planestrain problems in linear elasticity. We have the following.


-6

-4

-2

0

2

4

6

8

10

12

-1 -0.5 0 0.5 1

d3φ

/dx

3

x

(GM/WF) C2, p=5

C2, p=6

C2, p≥7

theoretical



1. Governing differential equations in terms of stresses

∂σx∂x

+∂τxy∂y

+ fx = 0

τxy∂x

+∂σy∂y

+ fy = 0

∀x, y ∈ Ω ⊂ R2 (5.368)

2. Strain displacement relations

εx =∂u

∂x, εy =

∂v

∂y, γxy =

∂u

∂y+∂v

∂x(5.369)

3. Constitutive equations

σx = D11 εx +D12 εy

σy = D12 εx +D22 εy

τxy = D33 γxy

(5.370)

in which u, v are displacements in 0 − x and O − y directions of a fixedCartesian coordinate frame O − xy, εx, εy and γxy are strains, σx, σy andτxy are stresses and Dij = Dji are coefficients containing material constantssuch as modulus of elasticity E, Poisson’s ratio v and shear modulus G (forthe isotropic case), we can also use matrix and vector representation for (1)and (5.370). Let

σ = [D]ε (5.371)


It is perhaps easier to express strains in terms of stresses, i.e.

εx = C11 σx + C12 σy

εy = C12 σx + C22 σy

γxy = C33 τxy

(5.372)

or

ε = [C]σ; [D] = [C]−1 (5.373)

in which for plane stress case (assuming isotropic material) we have

C11 =1

E, C12 =

−vE

= C21, C22 =−1

E, C33 =

1

G, G =

E

2(1 + v)(5.374)

Boundary conditions

u = u

v = v

on Γ2 (5.375)

andσxnx + τxyny = tx

τxyny + σny = ty

on Γ1 (5.376)

in which nx and ny are the direction cosines of a unit exterior normal tothe boundary Γ1. tx, ty are specified tractions on boundary Γ1. u and v arespecified values of displacements on boundary Γ2. We note that Ω = Ω

⋃Γ

in which Γ = Γ1⋃

Γ2 is a closed contour constituting the boundary of thedomain Ω.

GDEs in u and v

We can substitute stresses in terms of strains and then strains in termsof derivatives of displacements u and v in the equation of equilibrium as wellas boundary conditions (5.376).

− ∂

∂x

(D11

∂u

∂x+D12

∂v

∂y

)− ∂

∂y

(D33

(∂u∂y

+∂v

∂y

))= fx

− ∂

∂x

(D33

(∂u∂y

+∂v

∂y

))− ∂

∂x

(D21

∂u

∂x+D22

∂v

∂y

)= fy

∀x, y ∈ Ω ⊂ R2

(5.377)

u = u

v = v

on Γ2 (5.378)

332 SELF-ADJOINT DIFFERENTIAL OPERATORS(D11

∂u

∂x+D12

∂v

∂y

)nx +D33

(∂u∂y

+∂v

∂y

)ny = tx

D33

(∂u∂y

+∂v

∂y

)nx +

(D21

∂u

∂x+D22

∂v

∂y

)ny = ty

(5.379)

(5.377) are the desired GDEs in terms of displacements for 2D plane elas-ticity. We can also write (5.377) as

Aφ− f = 0 (5.380)

in which the different operator A and φ are given by

A = [A] =

− ∂∂x(D11

∂∂x)− ∂

∂y (D33∂∂y ) − ∂

∂x(D12∂∂y )− ∂

∂y (D33∂∂x)

− ∂∂x(D33

∂∂y )− ∂

∂y (D12∂∂x) − ∂

∂x(D33∂∂x)− ∂

∂y (D22∂∂y )

(5.381)

φ = φ =

uv

(5.382)

f = f =

fxfy

(5.383)

Equation (5.380) can also be written as

A11 u+A12 v = fx

A21 u+A22 v = fy(5.384)

We can show that the operator A is linear. We can also show that A∗ = A.Therefore, Galerkin method with weak form yielding variationally consistentintegral forms is a desirable approach to develop finite element process forthis BVP.


Let w1 = δu and w2 = δv be the test functions. Then, for an element Ωe

of the discretization ΩT =⋃e Ωe, we can write

(A11 ueh +A12 v

eh − fx, w1)Ωe =∫

Ωe

(∂

∂x

(D11

∂u

∂x+D12

∂v

∂y

)+

∂

∂y

(D33

(∂u∂y

+∂v

∂x

))− fx

)w1 dx dy (5.385)

and

(A21 ueh +A22 v

eh − fy, w2)Ωe =∫

Ωe

(∂

∂x

(D33

(∂u∂y

+∂v

∂x

))+

∂

∂y

(D21

∂u

∂x+D22

∂v

∂y

)− fy

)w2 dx dy (5.386)


Transferring one order of differentiation with respect to x and y in each termin (5.385) and (5.386) to w1 and w2

(A11 ueh +A12 v

eh − fx, w1)Ωe

=

∫Ωe

[∂w1

∂x

(D11

∂u

∂x+D12

∂v

∂y

)+∂w1

∂y

(D33

(∂u∂y

+∂v

∂x

))]dx dy

−∮Γe

w1

[(D11

∂u

∂x+D12

∂v

∂x

)nx +D33

(∂u∂y

+∂v

∂x

)ny

]dΓ−

∫Ωe

fxw1 dx dy

(5.387)

In (5.387), the concomitant is

〈A11ueh +A12v

eh − fx, w1〉Γe

= −∮Γe

w1

[(D11

∂u

∂x+D12

∂v

∂x

)nx +D33

(∂u∂y

+∂v

∂x

)ny

]dΓ

and

(A21 ueh +A22 v

eh − fy, w2)Ωe

=

∫Ωe

[∂w2

∂x

(D33

(∂u∂y

+∂v

∂x

))+∂w2

∂y

(D21

∂u

∂x+D22

∂v

∂y

)]dx dy

−∮Γe

w2

[D33

(∂u∂y

+∂v

∂x

)nx +

(D21

∂u

∂x+D22

∂v

∂x

)ny

]dΓ−

∫Ωe

fy w2 dx dy

(5.388)

Using concomitants in (5.387) and (5.388) we identify PVs, SVs, EBCs, andNBCs.

u, v are PVs, hence u = u, v = v on some boundary Γ2 are EBCs (5.389)

In which the concomitant is

〈A21ueh +A22v

eh − fx, w2〉Γe

= −∮Γe

w2

[D33

(∂u∂y

+∂v

∂x

)nx +

(D21

∂u

∂x+D22

∂v

∂x

)ny

]dΓ

(D11

∂u

∂x+D12

∂v

∂x

)nx +D33

(∂u∂y

+∂v

∂x

)ny

D33

(∂u∂y

+∂v

∂x

)nx +

(D21

∂u

∂x+D22

∂v

∂x

)ny

are SVs (5.390)


Let us introduce the following notation(D11

∂u

∂x+D12

∂v

∂x

)nx +D33

(∂u∂y

+∂v

∂x

)ny = qenx

D33

(∂u∂y

+∂v

∂x

)nx +

(D21

∂u

∂x+D22

∂v

∂x

)ny = qeny

(5.391)

Then qenx and qeny with some given values on some boundary Γ1 are NBCs.Using the notations (5.391), we can rewrite (5.387) and (5.388) in the fol-lowing form.

(A11 ueh +A12 v

eh − fx, w1)Ωe =∫

Ωe

[∂w1

∂x

(D11

∂u

∂x+D12

∂v

∂y

)+∂w1

∂y

(D33

(∂u∂y

+∂v

∂x

))]dx dy −

∮Γe

w1 qenx dΓ

−∫Ωe

fxw1 dx dy = Be1(ueh, v

eh;w1)− l˜e1(w1)−

∫Ωe

fxw1 dx dy (5.392)

and

(A21 ueh +A22 v

eh − fy, w2)Ωe =∫

Ωe

[∂w2

∂x

(D33

(∂u∂y

+∂v

∂x

))+∂w2

∂y

(D21

∂u

∂x+D22

∂v

∂y

)]dx dy −

∮Γe

w2qeny dΓ

−∫Ωe

fy w2 dx dy = Be2(ueh, v

eh;w2)− l˜e2(w2)−

∫Ωe

fyw2 dx dy (5.393)

Integrals (5.392) and (5.393) together constitute the desired weak form re-sulting from the Galerkin method with weak form. We combine (5.392) and(5.393) to give

(Aφeh − f, w) = Be(ueh, veh;w1, w2)− le(w1, w2) (5.394)

in which

Be(ueh, veh;w1, w2) =

Be

1(ueh, veh;w1)

Be2(ueh, v

eh;w2)

(5.395)

le(lw1, w2) =

l˜e

1(w1)

l˜e2(w2)

+

∫

Ωefxw1 dx dy∫

Ωefyw2 dx dy

(5.396)

and

l˜e(lw1, w2) =

∮Γew1 q

enx dΓ∮

Γew2 q

eny dΓ

(5.397)


Approximation space

The boundary value problem Aφ − f = 0 in Ω contains second orderderivatives of displacements u and v, but the weak form only contains firstorder derivatives u and v. Furthermore, equal order, equal degree localapproximations of u and v is justified based on the same order derivatives ofboth in the governing differential equations as well as the weak form. Thus,we have the following.

(i) Admissibility of ueh and veh in Aφ− f = 0 in ΩT in the pointwise senserequires

ueh, veh (i.e. φeh) ∈ Vh ⊂ Hk,p(Ωe), k = 3 is minimally conforming

For this choice, the integrals in the following are Riemann.

(Aφ− f, w)ΩT =∑e

(Aφeh − f, w)Ωe =∑e

(Be(φeh, w)− le(w)) = 0

(5.398)That is, if we choose φeh of class C2(Ωe), then all integrals are Riemannand all forms in (5.398) are equivalent.

(ii) Based on the weak form, if we choose φeh of class C1(Ωe), that is, ifφeh ∈ Vh ⊂ H2,p(Ωe), then

(Aφ− f, w)ΩT ⇔L

∑e

(Aφeh − f, w)Ωe ⇔L

∑e

(Be(φeh, w)− le(w))︸︷︷︸R

= 0

(5.399)Thus, for this choice of φeh,

∑e(B

e(·, ·) − le(·)) holds in the Riemannsense, but all other integral forms in (5.399) are in the Lebesgue sense.

(iii) If we choose φeh ∈ Vh ⊂ H1,k(Ωe), then∑

e(Be(·, ·) − le(·)) only holds


Local approximation ueh and veh

Consider equal order, equal degree local approximations of class C0 withp = 1 for a four node quadrilateral element in xy space with its map Ωξη inthe natural coordinate space. Then, we can write

ueh =

4∑i=1

Ni(ξ, η)uei = [N ]ue

veh =

4∑i=1

Ni(ξ, η) vei = [N ]ve

(5.400)


in which Ni(ξ, η) are standard local approximation functions for the fourcorner nodes of Ωe mapped in Ωξη in a two unit square with the origin ofξη coordinate system at the center of the element. ue and ve are nodaldegrees of freedom for ueh and veh at the four corner nodes, i.e. the values ofueh and veh. From (5.400)

w1 = δueh = Nj , j = 1, 2, . . . , 4

w2 = δveh = Nj , j = 1, 2, . . . , 4(5.401)

Element matrix [Ke] and vector F e

Substituting from (5.400) and (5.401) in the weak form, we obtain

Be1(ueh, v

eh, w1) =

∫Ωe

[∂Nj

∂x

(D11

4∑i=1

∂Ni

∂xuei +D12

4∑i=1

∂Ni

∂xvei

)

+∂Nj

∂y

(D33

( 4∑i=1

∂Ni

∂yuei +

4∑i=1

∂Ni

∂xvei

))]dx dy, j = 1, 2, . . . , 4 (5.402)

Be2(ueh, v

eh, w2) =

∫Ωe

[∂Nj

∂x

(D33

( 4∑i=1

∂Ni

∂yuei +

4∑i=1

∂Ni

∂xvei

))

+∂Nj

∂y

(D21

4∑i=1

∂Ni

∂xuei +D22

4∑i=1

∂Ni

∂yvei

)]dx dy, j = 1, 2, . . . , 4 (5.403)

F e =

∫

ΩeNj fx dΩ∫

ΩeNj fy dΩ

, j = 1, 2, . . . , 4 (5.404)

We can write these in the matrix and vector form

(Aφ− f, w)Ωe =

Be

1(ueh, veh, w1)

Be2(ueh, v

eh, w2)

− P e − F e (5.405)

or

(Aφeh − f, w)Ωe = [Ke]δ − P e − F e (5.406)

The specific form of [Ke] depends upon how the dofs for ueh and veh arearranged in δe for the four nodes of the element. If we assume that

δ3 = [ue1, ve1;ue2, v

e2;ue3, v

e3;ue4, v

e4]t (5.407)


then

φeh = [N˜ ]δe =

[N1 0 . . . Ni 0 . . .0 N1 . . . 0 Ni . . .

]

ue1ve1...ueivei...

(5.408)

and [Ke]ij , i.e. for node i and node j is a 2× 2 matrix

[Ke]ij =

[Ke

11 Ke12

Ke21 K

222

]ij

=

[(Ke

11)ij (Ke12)ij

(Ke21)ij (Ke

22)ij

](5.409)

where

(Ke11)ij =

∫Ωe

(∂Ni

∂xD11

∂Nj

∂x+∂Ni

∂yD33

∂Nj

∂y

)dΩ, i, j = 1, 2, . . . , 4 (5.410)

(Ke12)ij =

∫Ωe

(∂Ni

∂xD12

∂Nj

∂y+∂Ni

∂xD33

∂Nj

∂y

)dΩ, i, j = 1, 2, . . . , 4 (5.411)

(Ke21)ij =

∫Ωe

(∂Ni

∂xD33

∂Nj

∂y+∂Ni

∂yD21

∂Nj

∂x

)dΩ, i, j = 1, 2, . . . , 4 (5.412)

(Ke22)ij =

∫Ωe

(∂Ni

∂xD33

∂Nj

∂x+∂Ni

∂yD22

∂Nj

∂y

)dΩ, i, j = 1, 2, . . . , 4 (5.413)

(F e)i =

∫Ωe Ni fx dΩ∫Ωe Ni fy dΩ

, i = 1, 2, . . . , 4 (5.414)

We note that [Ke] is symmetric. For this 4-node element [Ke] is 8 × 8 andF e is 8 × 1. Details of P e for each boundary integral follows the samedetails as presented for the model problem in section 5.3.1. Numerical valuesof the coefficients of [Ke] and F e are obtained using gauss quadrature. Theintegrals in (5.410) to (5.414) over Ωe are converted to Ωξη domain (as insection 5.3.1) and then integrated using Gauss quadrature.

5.3.3.2 Least-squares method using residualfunctional

In this section we present a least-squares finite element formulation of the2D plane elasticity model problem defined by the equations (5.368)–(5.376).We rewrite these equations in a slightly different form. The momentumequations in the absence of body forces and the constitutive relations (linear


elasticity) are all we need. Using the mathematical model in (5.368)–(5.376),we can write the following:

∂σx∂x

+∂τxy∂y

= 0

∂τxy∂x

+∂σy∂y

= 0

(5.415)

σx = D11∂u

∂x+D12

∂v

∂y

σy = D21∂u

∂x+D22

∂v

∂y

τxy = D33

(∂u

∂y+∂v

∂x

) (5.416)

Details of [D], Dij = Dji are given in (5.373) and (5.374). Equations (5.415)and (5.416) are a system of first order differential equations in the dependentvariables u, v, σx, σy, and τxy. We can also substitute stresses from (5.416)into (5.415) and thereby obtain only two partial differential equations in uand v but at the expense of the appearance of up to second order derivativesof u and v them. These would obviously require higher order global differen-tiability local approximations. In LSP is is beneficial to non-dimensionalizethe mathematical model so that all dependent variables have numerical val-ues in the same range during computations. We rewrite (5.415) and (5.416)with ∧ (hat) on all variables (dependent as well as independent) implyingthat they have their usual dimension. We use the dimensionless length L0,force F0, and time t0 (t0 unnecessary for stationary problems i.e. BVPs)in the mathematical model to nondimenstionalize it. For the mathematicalmodel (5.415) and (5.416) we choose:

L0 = L, x =x

L0, y =

y

L0, E =

E

E0, σx =

σxτ0

σy =σyτ0, τxy =

τxyτ0, E0 = τ0, u =

u

L0, v =

v

L0

(5.417)

Using (5.417) in (5.415) and (5.416) we find that these are in fact also thedimensionless form.

We consider nine node p-version 2D element mapped in ξη space (twounit square). Let ueh, veh, (σx)eh, (σy)

eh, and (τxy)

eh be the local approximations

of u, v, σx, σy, and τxy for an element Ωe of the discretization ΩT =⋃e

Ωe of

the domain Ω ⊂ R2. For generality, we consider unequal order and unequal


degree local approximation.

ueh =nu∑i=1

Nui u

ei = [Nu]ue

veh =nv∑i=1

Nvi v

ei = [Nv]ve

(σx)eh =nσx∑i=1

Nσxi (σx)ei = [Nσx ](σx)e

(σy)eh =

nσy∑i=1

Nσyi (σy)

ei = [Nσy ](σy)e

(τxy)eh =

nτxy∑i=1

Nτxyi (τxy)

ei = [N τxy ](τxy)e

(5.418)

By replacing u, v, σx, σy, and τxy in (5.415) and (5.416) with their localapproximations ueh, veh, (σx)eh, (σy)

eh, and (τxy)

eh we obtain the residual equa-

tions ∀x, y ∈ Ωe ⊂ R2:

Ee1 =∂(σx)eh∂x

+∂(τxy)

eh

∂y

Ee2 =∂(τxy)

eh

∂x+∂(σy)

eh

∂y

Ee3 = (σx)eh −D11∂ueh∂x−D12

∂veh∂y

Ee4 = (σy)eh −D21

∂ueh∂x−D22

∂veh∂y

Ee5 = (τxy)eh −D33

(∂ueh∂y

+∂veh∂x

)∀x, y ∈ Ωe ⊂ R2 (5.419)

Upon substituting the local approximations 5.418 into 5.419 we obtain theexplicit form of the residual equations. Let us define nodal degrees of freedomδe for an element e.

δeT =[ueT, veT, (σx)eT, (σy)eT, (τ exyT

](5.420)

Then

δEiT =

[∂Ei∂ue

T,

∂Ei∂ve

T,

∂Ei

∂(σx)e

T,

∂Ei

∂(σy)e

T,

∂Ei

∂(τxy)e

T], i = 1, 2, . . . , 5 (5.421)


and the element matrix [Ke] is given by

[Ke] =

5∑i=1

∫Ω

∂Eei∂δe

∂Eei∂δe

TdΩ (5.422)

Remaining details of numerical integration for [Ke], assembly of elementmatrices, etc. follow the standard procedure.

Minimally conforming order of the space k is two (local approximationsof class C1(Ωe)) for all dependent variables for which the integrals over ΩT

are Riemann. However, if the theoretical solution is smooth then k = 1(local approximations of class C0(Ωe)) may suffice, keeping in mind that forthis choice of k the integrals over ΩT are in the Lebesgue sense.

Simply supported and clamped-clamped 2D beams:numerical studies

We consider a thin and narrow plate of length l of 20 inches, width b of2.5 inches, and thickness t of 0.1 inches (geometrically, it is a beam). WithL0 = 10 inches the dimensionless plate is 2× 0.05× 0.01. We consider loadsapplied in the plane of the plate. We choose E = 30×106 psi, E0 = 27.3×106

hence E = 1.0989; shear modulus G = 10.5×106 psi, henceG = GE0

= 0.3846;and Poisson’s ratio ν = 0.

Model problem 1: simply supported (SS) beam

In this case we consider the plate to be “simply supported” as shown inFig. 5.67. Points A and B are constrained in the y-direction but are free tomove in the x-direction. On face AB of the plate σy = 10−6 and on faceCD σy = −10−6 is applied causing deflection of the plate in the negativey-direction. At the center plane (EF ) the x-displacement (u) is constraineddue to symmetry about x = 0.5l. Since b and t are much smaller than l, thedeformation behavior is like a simply supported slender beam (shear defor-mation is not significant). The domain (l × b) 2 × 0.05 is modeled using atwenty element uniform discretization (ten elements along the length l andtwo elements along the width b) using nine-node p-version hierarchical planestress elements with higher order global differentiability local approximationin Hk,p(Ωe) scalar product space. Boundary conditions on the four bound-aries of the domain ABCD of the plate are also shown in Fig. 5.67. Thenine-node elements are mapped into a square of two units with the originof the natural coordinate system ξ, η at the center of the element. The el-ement local approximation as well as all computations are performed usingnatural coordinate system ξ, η. The degrees of local approximation in ξ andη (pξ, pη) are chosen to be equal p = pξ = pη and are chosen to be the same


for all dependent variables. Due to smoothness of the solution for this modelproblem both C0 and C1 solutions are expected to work well. We expectC0 solutions to approach C1 solutions upon convergence. In the numericalsolutions presented here we choose k = 1; that is, local approximation ofclass C0(Ωe).

v = 02

v = 0

0.05

τxy = 0

σy = −10−6

u = 0τxy = 0

σy = 10−6

A F

DE

0.01

τxy = 0

τxy = 0σx = 0

y

xB

C

Figure 5.67: Schematic and boundary conditions used for a simply supported plate

A p-convergence study with p = pξ = pη = 3, 5, . . . shows that at p =9 the residual functional I is of the order O(10−16), confirming that theequations are satisfied accurately in the pointwise sense. This is confirmed bythe similar studies with the solutions of class C1(Ωe) and their comparisonswith C0(Ωe) studies. Thus, in the following we present results for p = pξ =pη = 9 and local approximations of class C0(Ωe) for all dependent variablesusing 20 element uniform discretization described earlier.

Model problem 2: clamped-clamped (CC) beam

This model problem consists of the same plate as used in model prob-lem 1 but is considered clamped at the two ends (x = 0 and x = 2) asshown in Fig. 5.68. The boundary conditions on the boundaries AC andBD (clamped boundaries) are u = v = 0 (shown in Fig. 5.68). The detailsof dicretization, choice of p-levels, choice of order of space, etc. are the samefor this model problem as described on the model problem 1. In this casealso a p-convergence study for the solutions of class C0(Ωe) yields the resid-ual functional I of the order of O(10−16 as is the case of model problem 1.Thus, for this model problem also p = pξ = pη = 9 and C0(Ωe) local ap-proximation for all dependent variables yield very accurate solutions, henceare used to compute the results presented here.

Solutions for model problems 1 and 2

In the solutions presented here for model problems 1 and 2 the dimen-sionless modulus of elasticity E corresponds to E = 30 × 106 psi. Figure5.69 shows a plot of v versus x at the centerline (y = 0.025) for the simplysupported beam. The displacements v of the bottom and the top faces of


2

0.05

σy = −10−6

u = 0

σy = 10−6

A F

DE

0.01

τxy = 0

τxy = 0y

x

u = 0v = 0

u = 0v = 0

C

B

Figure 5.68: Schematic and boundary conditions used for a clamped-clamped plate

the beam (y = 0.0 and y = 0.05) are virtually the same as the displacementv at the centerline (y = 0.025) as expected for a slender beam like what isused here (validating the assumption of inextensibility of transverse normalsin a beam theory). Graphs of v versus x at the centerline of the clamped(CC) beam of Fig. 5.68 for the same values of σy on boundaries AB and CDand the same value of E are shown in Fig. 5.70. These are plotted usingthe same x, y scales as Fig. 5.69 so that displacements of the two can becompared. We observe substantially reduced v displacement in the case of

CC beam as expected. Rotation θz = 12

(∂u∂y −

∂v∂x

)versus x for SS and CC

plates at the centerline are shown in Figs. 5.71 and 5.72. In both modelproblems θz is antisymmetric about x = 1.0. Much larger values of θz incase of SS plate (due to unconstrained θz at x = 0.0 and x = 2.0) comparedto CC plate are clearly observed. Graph of τxy versus x at the centerline ofthe SS and CC plates are shown in Figs. 5.73 and 5.74. We observe that τxyis antisymmetric about x = 1.0 for both SS and CC plates.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−0.03

−0.02

−0.01

0

Distance x

Dis

pla

cem

entv

Figure 5.69: Displacement v at y = 0.025 versus distance x (simply supported plate)


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−0.03

−0.02

−0.01

0

Distance x

Dis

pla

cem

entv

Figure 5.70: Displacement v at y = 0.025 versus distance x (clamped-clamped plate)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−0.04

−0.02

0

0.02

0.04

Distance x

Rot

atio

nθ z

Figure 5.71: Rotation θz at y = 0.025 versus distance x (simply supported plate)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−0.04

−0.02

0

0.02

0.04

Distance x

Rot

atio

nθ z

Figure 5.72: Rotation θz at y = 0.025 versus distance x (clamped-clamped beam)


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−0.00004

−0.00002

0

0.00002

0.00004

Distance x

Sh

ear

Str

essτ xy

Figure 5.73: Shear stress τxy at y = 0.025 versus distance x (simply supported plate)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−0.00004

−0.00002

0

0.00002

0.00004

Distance x

Sh

ear

Str

essτ xy

Figure 5.74: Shear stress τxy at y = 0.025 versus distance x (clamped-clamped plate)

5.4 Three-dimensional boundary value problems

5.4.1 Three-dimensional boundary value problems in asingle dependent variable

Consider 3D steady-state Fourier heat conduction equation for an anisotropicmedium with symmetric conductivity matrix:

−∂qx∂x− ∂qy∂y− ∂qz∂z

+Q = 0, ∀x, y, z ∈ Ω ⊂ R3 (5.423)

where (qx, qy, qz) are the components of the heat flux vector q, which isrelated to the gradient of temperature ∇T by the Fourier heat conductionlaw, q = −k ·∇T :

5.4. THREE-DIMENSIONAL BOUNDARY VALUE PROBLEMS 345

−qx = kxx∂T

∂x+ kxy

∂T

∂y+ kxz

∂T

∂z

−qy = kyx∂T

∂x+ kyy

∂T

∂y+ kyz

∂T

∂z

−qz = kzx∂T

∂x+ kzy

∂T

∂y+ kzz

∂T

∂z

(5.424)

in which T is the temperature and Ω = Ω ∪ Γ, closure of Ω with kij = kji(i, j = x, y, z) and the boundary conditions

T = T0 on Γ2 (5.425)

qxnx + qyny + qznz + β(T − T∞) + q = 0 on Γ1 (5.426)

where (nx, ny, nz) are the direction cosines of a unit exterior normal toboundary Γ1, (qx, qy, qz) are heat fluxes in x, y and z directions, respectively,β is a film coefficient, T∞ is the ambient temperature, and q is the appliedheat flux normal to the boundary. If we substitute qx, qy and qz from (5.424)in (5.423), then we obtain a single second order partial differential equationin temperature T . One could show that for this differential equation

(a) the differential operator is linear,

(b) the adjoint A∗ of the differential operator is same as the operator, andthus

(c) the Galerkin method with weak form will yield variationally consistentintegral forms.

In the following we consider finite element processes based on Galerkinmethod with weak form.


For an element e with domain Ωe of the discretization ΩT =⋃e

Ωe with

local approximation T eh of T over Ωe, we consider

(AT eh +Q, v)Ωe =

∫Ωe

(AT eh +Q)v dΩ = −∫Ωe

( ∂∂x

(qx)

+∂

∂y(qy) +

∂

∂z(qz)

)v dx dy dz +

∫Ωe

Qv dx dy dz (5.427)


in which v = δφeh. Transferring one order of differentiation to v from eachterm in the integrand in (5.427)

(AT eh +Q, v)Ωe =

∫Ωe

(∂v∂x

(qex) +∂v

∂y(qey) +

∂v

∂z(qez)

)dx dy dz

−∮

Γev(qexnx + qeyny + qeznz) dΓ +

∫Ωe

Qv dΩ (5.428)

The concomitant 〈AT eh +Q, v〉Γe in (5.428) is given by

〈AT eh +Q, v〉Γe = −∮

Γev(qexnx + qeyny + qeznz) dΓ

Using concomitant (5.428) we can determine PVs, SVs, EBCs and NBCs; Tis the PV and T = T on some boundary Γ1 is the EBC and (qexnx + qeyny +

qeznz) is the SV. Hence, (qexnx + qeyny + qeznz) = q on some boundary Γ2 isthe NBC. As seen before, we note that the secondary variable in (5.428) isnot known. If we let

qexnx + qeyny + qeznz = qen (= n · qe) (5.429)

in which qen is the flux normal to the element boundary Γe, then we can write(5.428) as

(AT eh +Q, v)Ωe =

∫Ωe

(∂v∂xqex +

∂v

∂yqey +

∂v

∂zqez

)dx dy dz

−∮

Γevqen dΓ +

∫Ωe

Qv dx dy dz (5.430)

or

(AT eh +Q, v)Ωe = Be(T eh , v)− le(v) (5.431)

in which

Be(T eh , v) =

∫Ωe

(∂v∂xqex +

∂v

∂yqey +

∂v

∂zqez

)dx dy dz (5.432)

le(v) =

∮Γe

vqen dΓ−∫Ωe

Qv dx dy dz (5.433)

(5.431) is the weak form of the boundary value problem (5.423) and (5.424).


5.4.1.2 Approximation space

The boundary value problem contains up to second order derivatives ofthe temperature T and the test function v.

(i) Admissibility of Th is AT − f = 0 (f = −Q) in ΩT in the pointwisesense requires, T eh ∈ Vh ⊂ Hk,p(Ωe); k = 3 is minimally conforming.For this choice, the integrals in the following are Riemann

(ATh − f, v)ΩT =∑e

(AT eh − f, v)Ωe =∑e

[Be(T eh , v)− le(v)] = 0

(5.434)Thus, if we choose T eh of class C2(Ωe), then all integrals are Riemannin all forms in (5.434) and, hence we say that all three forms in (5.434)are precisely equivalent.

(ii) Based on the weak form, if we choose T eh of class C1(Ωe), if T eh ∈ Vh ⊂H2,p(Ωe), then

(ATh − f, v)ΩT ⇔L

∑e

(AT eh − f, v)Ωe ⇔L

∑e

[Be(T eh , v)− le(v)]︸︷︷︸R

= 0

(5.435)Thus, for this choice

∑e[B

e(·, ·) − le(·)] holds in the Riemann sense,but all other integral forms in (5.435) are in the Lebesgue sense.

(iii) If we choose, T eh ∈ Vh ⊂ H1,p(Ωe), then∑

e[Be(·, ·) − le(·)] only holds

in the Lebesgue sense and the other two integral forms in (5.435) aremeaningless

5.4.1.3 Local approximation T eh

Let

T eh =n∑i=1

Ni(ξ, η, ζ) δei = [N ]δe (5.436)

be the local approximation of T over an element e with domain Ωξηζ . Then

v = δT eh = Nj(ξ, η, ζ), j = 1, 2, . . . , n (5.437)

Ni(ξ, η, ζ) are the local approximation functions and δei are the nodal degreesof freedom. Substituting (5.436) and (5.437) into the weak form (5.431)

Be(T eh , Nj) =

∫Ωe

(∂Nj

∂xqex +

∂Nj

∂yqey +

∂Nj

∂zqez

)dx dy dz (5.438)

le(Nj) =

∮ΓeNj q

en dΓ−

∫Ωe

QNj dx dy dz (5.439)


for i, j = 1, 2, . . . , n, where

−qex = kxx

n∑i=1

∂Ni

∂xδei + kxy

n∑i=1

∂Ni

∂yδei + kxz

n∑i=1

∂Ni

∂zδei

−qey = kyx

n∑i=1

∂Ni

∂xδei + kyy

n∑i=1

∂Ni

∂yδei + kyz

n∑i=1

∂Ni

∂zδei

−qez = kzx

n∑i=1

∂Ni

∂xδei + kzy

n∑i=1

∂Ni

∂yδei + kzz

n∑i=1

∂Ni

∂zδei

(5.440)

Thus, we have

Be(T eh , Nj) = −[Ke]δe, le(vNj) = P e − F e (5.441)

where (i, j = 1, 2, . . . , n)

Keij =

∫Ωe

[∂Ni

∂x

(kxx

∂Nj

∂x+ kxy

∂Nj

∂y+ kxz

∂Nj

∂z

)+∂Ni

∂y

(kyx

∂Nj

∂x+ kyy

∂Nj

∂y+ kyz

∂Nj

∂z

)+∂Ni

∂z

(kzx

∂Nj

∂x+ kzy

∂Nj

∂y+ kzz

∂Nj

∂z

)]dx dy dz (5.442)

P ei =

∮Γe

qenNi dΓ (5.443)

F ei =

∫Ωe

QNi dx dy dz (5.444)

Remarks.

(1) P e is yet to be determined, which requires the definition of Ωe, i.e.the domain of definition (geometry) of the element e.

(2) Once the geometry is established, explicit form of φeh, the local approx-imation can be determined.

(3) The approximation space Vh containing the local approximation func-tions Ni needs to be defined as well.

5.4.1.4 Definition of Ωe: Element geometry

The element geometry in the physical coordinate space could be a tetra-hedron with flat faces and straight edges or it could have curved faces andedges. The element geometry could also consist of a hexahedron with curved


faces and edges or flat faces and straight line edges. We consider hexahedronelements for illustrating the details. Fig. 5.75 (a) shows a twenty-seven-nodethree-dimensional hexahedron element in the physical coordinate space. Theelement faces and the edges are distorted. The element geometry in Fig. 5.75(a) is mapped into a two-unit cube in Fig. 5.75 (b) with the origin of thecoordinate system at the center of the element. Following the details inchapter 8, we have the following for mapping of points.

x(ξ, η, ζ) =27∑i=1

Ni(ξ, η, ζ)xi

y(ξ, η, ζ) =27∑i=1

Ni(ξ, η, ζ) yi (5.445)

z(ξ, η, ζ) =

27∑i=1

Ni(ξ, η, ζ) zi

A 27-node element in x, y, z space Element map in natural coordinate space ξ, η, ζ space

ζ

ξ

η

x

y

z

Figure 5.75: A 27-node three-dimensional hexahedron element

Mapping of length (dx, dy, dz) and (dξ, dη, dζ) is described by (can bederived following details similar to 2D case, see chapter 8)

dxdydz

= [J ]

dξdηdζ

; [J ] =

∂x∂ξ

∂x∂η

∂x∂ζ

∂y∂ξ

∂y∂η

∂y∂ζ

∂z∂ξ

∂z∂η

∂z∂ζ

(5.446)

[J ] is the Jacobian of transformation and dx dy dz = |J | dξ dη dζ (can alsobe derived using details similar to 2D case, see chapter 8). We note that theshape functions Ni(ξ, η, ζ) in (5.445) have the properties

Ni(ξj , ηj , ζj) =

1, j = i

0, j 6= i,

27∑i=1

Ni(ξ, η, ζ) = 1 (5.447)


For the nodal configurations used in Figs. 5.75 (a) and (b), Ni(ξj , ηj , ζj) canbe easily derived using tensor product (see chapter 8)

5.4.1.5 Computations of element matrix [Ke] and vector F e

Using the element map Ωξηζ in the natural coordinate space ξ, η, ζ wecan transform integrals for [Ke] and F e over Ωξη and (5.442) and (5.444)can be written as

Keij =

1∫−1

1∫−1

1∫−1

[∂Ni

∂x

(kxx

∂Nj

∂x+ kxy

∂Nj

∂y+ kxz

∂Nj

∂z

)+∂Ni

∂y

(kyx

∂Nj

∂x+ kyy

∂Nj

∂y+ kyz

∂Nj

∂z

)+∂Ni

∂z

(kzx

∂Nj

∂x+ kzy

∂Nj

∂y+ kzz

∂Nj

∂z

)]|J | dξ dη dζ (5.448)

F ei =

1∫−1

1∫−1

1∫−1

QNi dΩ (5.449)

The derivatives of the approximation functions Ni(ξ, η, ζ) with respect tox, y, z can be easily obtained using (see chapter 8)

∂Ni∂x∂Ni∂y∂Ni∂z

= [JT ]−1

∂Ni∂ξ∂Ni∂η∂Ni∂ζ

(5.450)

Numerical values of [Ke] and F e can be obtained using gauss quadrature.

5.4.1.6 Details of secondary variable vector P e

Note that P e is defined by∮

Γe vqen dΓ, v = Nj(ξ, η, ζ); j = 1, 2, . . . , n.

Thus, details of P e obviously require a clear definition of Γe in Ωe =Ωe ∪ Γe. For simplicity, we consider an eight-node hexahedron element inthe x, y, z space (5.76. The element has flat faces and straight edges but theelement shape may not have to be a prism. The element is mapped into atwo-unit cube in ξ, η, ζ natural coordinate space.

The boundary Γe of the element consists of six faces

Γe =6⋃i=1

Γei


x

y

z

Eight-node element in R3 Map of an eight-node elementin ξ, η, ζ space

ζ

φe3

ξ

η

φe1

φe5

φe8 φe7

φe2

1 2

5 6

8

φe4

7

3

φe6

Figure 5.76: An eight-node three-dimensional hexahedron element

let us define Γei (i = 1, . . . , 6) as follows.

Γe1(ξ=−1)

has nodes: 1, 4, 8, 5 (5.451)

Γe2(ξ=+1)

has nodes: 2, 3, 7, 6 (5.452)

Γe3(η=−1)

has nodes: 1, 2, 6, 5 (5.453)

Γe4(η=+1)

has nodes: 3, 4, 8, 7 (5.454)

Γe5(ζ=−1)

has nodes: 1, 2, 3, 4 (5.455)

Γe6(ζ=+1)

has nodes: 5, 6, 7, 8 (5.456)

The integral over Γe can be written as the sum of the integrals over Γei ; i =1, . . . , 6.

P e =

∮Γe

NT qen dΓ =6∑i=1

∫Γei

NT qen dΓ =6∑i=1

P e|Γei (5.457)

We consider the integrals over Γei in (5.457). First, we note that

N = N1, . . . , N8 (5.458)


[N ]Γe1 =N1, 0, 0, N4, N5, 0, 0, N8

ξ=−1

[N ]Γe2 =

0, N2, N3, 0, 0, N6, N7, 0ξ=1

[N ]Γe3 =N1, N2, 0, 0, N5, N6, 0, 0

η=−1

[N ]Γe4 =

0, 0, N3, N4, 0, 0, N7, N8

η=1

[N ]Γe5 =N1, N2, N3, N4, 0, 0, 0, 0

ζ=−1

[N ]Γe6 =

0, 0, 0, 0, N5, N6, N7, N8

ζ=1

(5.459)

Using (5.459) for each of the integrals in (5.457). Consider

P eΓe1T =

∫Γe1

N1, 0, 0, N4, N5, 0, 0, N8

Tξ=−1

qen dΓ (5.460)

Let us define P e1 as

P eΓ1T =

P e11, 0, 0, P

e41, P

e51, 0, 0, P

e81

= P e1 T (5.461)

and similarly

P eΓ2T =

[0, P e22, P

e32, 0, 0, P

e62, P

e72, 0

P eΓ3

T =P e13, P

e23, 0, 0, P

e53, P

e63, 0, 0

P eΓ4

T =

0, 0, P e34, Pe44, 0, 0, P

e74, P

e84

P eΓ5

T =P e15, P

e25, P

e35, P

e45, 0, 0, 0, 0

P eΓ6

T =

0, 0, 0, 0, P e56, Pe66, P

e76, P

e86

(5.462)

We note that the secondary variables for each of the six faces are perpendic-ular to the faces. Symbolically, we can write

P e =

6∑i=1

P eΓei =

P e1...P e8

(5.463)

where P ei (i = 1, . . . , 8) are the sum of the secondary variables at the nodesof the elements in (5.462). We keep in mind that the sum of the secondaryvariables is symbolic as they are normal to the element faces.


Remarks.

(1) When T eh is of class C0(Ωe), then T eh ∈ Vh ⊂ H1,p(Ωe)

(2) C0(Ωe) local approximations (of any degree, i.e. p) permit inter-elementcontinuity of Th and its derivatives in the plane of the mating elementfaces, but the derivative of Th normal to the inter-element boundaries(faces) is discontinuous. This is exactly parallel to the 2D case exceptthat in this case the inter-element boundaries are the surfaces.

(3) When the local approximations are of class C1(Ωe), the inter-elementcontinuity of the first derivative of Th holds, but the second derivativeof Th normal to the inter-element plane is discontinuous.

(4) Element computation, assembly and solution follow standard procedure.

5.4.2 Three-dimensional boundary value problemsin multivariables

In this section we consider three dimensional boundary value problemsin multivariables. The simplest examples of such boundary value problemsare in 3D linear elasticity.

Let u, v, w be the displacements of a point in the body Ω(x, y, z) in x, yand z directions. In the following we first derive the governing differentialequations.

Equations of equilibrium

Consider force equilibrium in x, y and z directions: If τij ; i = x, y, z; j =x, y, z are the stresses at a point x, y, z such that τij = τji, then the forceequilibrium in x, y and z directions yields

∂σx∂x

+∂τxy∂y

+∂τxz∂z

+ fx = 0

∂τxy∂x

+∂σy∂y

+∂τyz∂z

+ fy = 0

∂τxz∂x

+∂τyz∂y

+∂σz∂z

+ fz = 0

∀(x, y, z) ∈ Ω ⊂ R3 (5.464)

in which fx, fy and fz are body forces per unit mass.

Strain displacement relations (linear elasticity)

εx =∂u

∂x, εy =

∂v

∂y, εx =

∂w

∂z

γyz =∂v

∂z+∂w

∂y, γzx =

∂u

∂z+∂w

∂x, γxy =

∂u

∂y+∂v

∂x

(5.465)


Constitutive equations: stress-strain relations

Using generalized Hooke’s law, stresses can be expresses as a linear com-bination of strains or strains can be expressed as a linear combination ofstresses. Expressing stresses in terms of strains we have (for orthotropic orisotropic material behavior).

σx = D11 εx +D12 εy +D13 εz



τyz = D44

(∂v∂z

+∂w

∂y

)τxz = D55

(∂w∂x

+∂u

∂z

)τxy = D66

(∂u∂y

+∂v

∂x

)(5.466)

Governing differential equations

When the stresses from (5.466) are substituted into the equations of equi-librium (5.464) and (5.465) for strains, we obtain three partial differentialequations in u, v and w that are second order in u, v and w. These equa-tions constitute the mathematical model which is starting point in the finiteelement formulation.

F (u, v, w) + fx = 0, G(u, v, w) + fy = 0, H(u, v, w) + fz = 0 (5.467)

Boundary conditions

u = u0, v = v0, w = w0 on Γ2 (5.468)

σxnx + τxyny + τxznz = tx

τxynx + σyny + τyznz = ty

τxznx + τyzny + σznz = tz

on Γ1 (5.469)

in which stresses are defined by (5.466) and the strains in the stresses by(5.465).

Remarks.

(1) One could show that the operator A is

(a) linear(b) the adjoint A∗ of the operator is same as this operator A.


(c) and, hence, self-adjoint for this case

(2) Thus, a variationally consistent finite element formulation of this bound-ary value problem is possible using Galerkin method with weak form.


Let w1 = δu, w2 = δv, w3 = δw be the test functions. Then for anelement e of the discretization with domain Ωe, we can write

Φ1 =

∫Ωe

(F (u, v, w) + fx

)w1 dx dy dz

Φ2 =

∫Ωe

(G(u, v, w) + fy

)w2 dx dy dz

Φ3 =

∫Ωe

(H(u, v, w) + fz

)w3 dx dy dz

(5.470)

Substituting for F , G and H from (5.464) (in terms of stresses) and per-forming integration by parts yields

Φ1 = −∫Ωe

(∂w1

∂xσx +

∂w1

∂yτxy +

∂w1

∂zτxz

)dx dy dz

+

∮Γe

w1(σxnx + τxyny + τxznz) dΓ +

∫Ωe

w1 fx dx dy dz = 0 (5.471)

In which the concomitant 〈φ1〉Γe is given by

〈φ1〉Γe =

∮Γe

w1(σxnx + τxyny + τxznz) dΓ

Φ2 = −∫Ωe

(∂w2

∂xτxy +

∂w2

∂yσy +

∂w2

∂zτyz

)dx dy dz

+

∮Γe

w2(τxynx + σyny + τyznz) dΓ +

∫Ωe

w2 fy dx dy dz = 0 (5.472)


〈φ2〉Γe =

∮Γe

w2(τxynx + σyny + τyznz) dΓ


Φ3 = −∫Ωe

(∂w3

∂xτxz +

∂w3

∂yτyz +

∂w3

∂zσz

)dx dy dz

+

∮Γe

w3(τxznx + τyzny + σznz) dΓ +

∫Ωe

w3 fz dx dy dz = 0 (5.473)


〈φ3〉Γe =

∮Γe

w3(τxznx + τyzny + σznz) dΓ

We identify PVs, SVs, EBCs and NBCs from the concomitants in (5.471)–(5.473).

PV: u, v, w

SV:

σxnx + τxyny + τxznz

τxynx + σyny + τzynz

τxznx + τyzny + σznz

EBCs:

u = u0

v = v0

w = w0

on some Γ1

NBCs:

σxnx + τxyny + τxznz = qenxτxynx + σyny + τzynz = qenyτxznx + τyzny + σznz = qenz

on some Γ2

(5.474)

Substituting NBCs from (5.474) into (5.471)–(5.473) and changing sign through-out yields

−Φ1 =

∫Ωe

(∂w1

∂xσx +

∂w1

∂yτxy +

∂w1

∂zτxz

)dx dy dz

−∮Γe

w1 qenx dΓ−

∫Ωe

w1 fx dx dy dz = 0 (5.475)

−Φ2 =

∫Ωe

(∂w2

∂xτxy +

∂w2

∂yσy +

∂w2

∂zτyz

)dx dy dz

−∮Γe

w2 qeny dΓ−

∫Ωe

w2 fy dx dy dz = 0 (5.476)


−Φ3 =

∫Ωe

(∂w3

∂xτxz +

∂w3

∂yτyz +

∂w3

∂zσz

)dx dy dz

−∮Γe

w3 qenz dΓ−

∫Ωe

w3 fz dx dy dz = 0 (5.477)

Equations (5.475)–(5.477) represent the desired weak form. When the stressesare substituted into (5.475)–(5.477) we can write

−Φ1

−Φ2

−Φ3

=

Be

1(u, v, w;w1, w2, w3)− le1(w1, w2, w3)Be

2(u, v, w;w1, w2, w3)− le2(w1, w2, w3)Be

3(u, v, w;w1, w2, w3)− le3(w1, w2, w3)

(5.478)

= Be(u, v, w;w1, w2, w3)− le(w1, w2, w3) (5.479)

We note that

(a) Be(·, ·) is bilinear and symmetric

(b) le(·) is linear

5.4.2.2 Approximation spaces

Since in this case the differential operator is second order (when thegoverning differential equations are expressed in terms of displacements)the discussion of the approximation spaces remains the same as in previousmodel problem (section 5.4.1) and is not repeated here. The governingdifferential equations contain second order derivatives of u, v and w and,hence, equal order, equal degree local approximations for u, v and w are avalid choice.

5.4.2.3 Local approximation

If we let

φehT =[ueh veh weh

](5.480)

Then for a choice of Ωe (tetrahedron or hexahedron) we can write

ueh =n∑i=1

Ni(ξ, η, ζ)uei = [N ]ue

veh =n∑i=1

Ni(ξ, η, ζ) vei = [N ]ve

weh =n∑i=1

Ni(ξ, η, ζ)wei = [N ]we

(5.481)


and

w1 = δueh = Nj(ξ, η, ζ), j = 1, 2, . . . , n

w2 = δveh = Nj(ξ, η, ζ), j = 1, 2, . . . , n

w3 = δweh = Nj(ξ, η, ζ), j = 1, 2, . . . , n

(5.482)

Element matrix [Ke] and vector F e

By substituting (5.481) and (5.482) into (5.478), we can write−Φ1

−Φ2

−Φ3

= [Ke]δe − P e − F e (5.483)

Explicit expressions for Keij of [Ke] and F ei of F e can be easily obtained

using (5.475)–(5.477) and (5.466). Numerical values of Keij and F ei are ob-

tained using Gauss quadrature. Details of the secondary variables follow thesame as for previous model problem in section 5.4.1.

5.5 Summary

In this chapter, details of the finite element method for various 1D and2D boundary value problems have been presented. Details of discretization,local approximation, integral forms, their variational consistency, higher de-gree as well as higher order local approximations, element equations, theirassembly, the solutions of assembled equations, and post-processing are pre-sented for each model problem considered in this chapter. Special empha-sis is placed on GM/WF and LSP based on residual functional due to thefact that these methods of approximations yield VC integral forms, henceunconditionally stable finite element computational processes. Importanceand benefit of higher degree as well as higher order global differentiabilityapproximations is illustrated in the numerical studies.

It is shown that if our objective is to satisfy the governing differentialequations in the pointwise sense then the LSP based on the differential mod-els with the highest orders of derivatives of the dependent variables and thelocal approximations in higher order spaces is most meritorious. Such com-putational process yield better accuracy than the GM/WF.

Problems

5.1 Consider a fully developed steady flow of a Newtonian fluid between two parallelplates. Let the distance between the plates be 2b. Let u and v be the velocity componentsin x and y directions.

PROBLEMS 359

u = 0

ux

u = 0

v

u(y)

2b

y

Figure 5.77: Schematic of flow between parallel plates

For this simple flow u = u(y) and v = 0 and the governing differential equations reduce tothe following:

dp

dx= µ

d2u

dy2, −b ≤ x ≤ b (1)

Since ∂p∂y

= 0, ∂p∂x

is constant (say f0), (1) reduces to

µd2u

dy2= f0, µ is viscosity (2)

with the following boundary conditions:

u = 0 at y = ±b (3)

(a) Construct the weak form of (2) using GM/WF. Give details of PV, SV, EBC and NBCas well as the nature of the resulting functionals. Establish VC or VIC of the integral form.

(b) Consider a two-node element with linear approximation of u. Derive the elementequations for a typical element e using the element map in the natural coordinate space.

Part I:

Part Ia. Consider a four element uniform mesh for the entire domain (−b ≤ y ≤ b). Usethe element equations derived in (b) to construct assembled equations in partitioned formand solve for the unknown nodal values of u. Compare your calculated solution with theanalytical solution given by

u = − 1

2µf0(b2 − y2), f0 =

dp

dx(4)

Comment on your observations and findings. Plot graphs of theoretical solution u andcomputed solution uh versus y and compare.

Part Ib. Because of symmetry it is only necessary to model half of the domain (0 ≤ y ≤b). Construct a uniform discretization of the half domain (0 ≤ y ≤ b) using two two-nodelinear elements. Compare your calculated results with the analytical solution as well asthe solution computed in Part Ia using the whole domain. Comment on your observationsand findings. Plot graphs of theoretical solution u and computed solution uh versus y andcompare.


Part II:

Consider the problem of steady state Couette flow between parallel plates. The bottomplate in stationary and the top plate in moving in its own plane with a constant velocity ofu0. The GDE for this case is also given by (2) but the boundary conditions are as follows(instead of (3)):

2b

u0

x

y

u(y)

Figure 5.78: Schematic of Couette flow

u(0) = 0, u(2b) = u0 = 1 (5)

The discretized equations derived in (b) for an element e are valid here also.

Part IIa. Consider a non-uniform discretization of four linear elements beginning withy = 0 (h1 = 0.8b, h2 = 0.5b, h3 = 0.3b, and h4 = 0.4b). Compute numerical solutions forthis discretization and compare them with the analytical solution given by

u(y) =u0 y

2b− 2b2

µf0y

2b

(1− y

2b

)(6)

Plot graphs of theoretical solution u and computed solution uh versus y and compare thetwo. Discuss your findings.

Part IIb. Consider a four-element uniform discretization. Compute a numerical solutionfor this discretization. Compare this solution with the theoretical solution (6) and thenumerical solution in Part IIa. Discuss your results. Plot graphs of theoretical solution uand computed solution uh versus y and compare the two.

Note: while performing numerical calculations you may choose the following values forvarious constants (if it is more convenient).

b = 1, µ = 1, f0 = 1, u0 = 1

5.2 Consider an axial rod of length 12 units and cross-sectional areas A1 and A2 as shownin Fig. 5.79. The rod consists of two different materials of Young’s moduli E1 = 15× 106

and E2 = 30× 106. It is subjected to an axial load P = 10, 000 at the free end. It is alsosubjected to axially distributed loads as shown in the figure.

h-version:

(a) Discretize the stepped rod of Fig. 5.79 using two-node linear axial elements. Constructuniform meshes consisting of 2, 4, 6, 8, . . . elements and compute displacements, strains,stresses and the quadratic functional.

p-version:

(b) Discretize the rod using only two three-node p-version axial elements. Compute resultsfor p-levels of 2, 3, 4, 5, . . ..


6 6

Axial loadf(x) for

6 ≤ x ≤ 12

600

y

Fixed endP

1200

1200

A1 = 4E1 = 15× 106

µ1 = 0.0

A2 = 2E1 = 30× 106

µ2 = 0.0

Axial loadf(x) for

0 ≤ x ≤ 6

Figure 5.79: Schematic of axial deformation of a rod

Results:

(I) Plot a graph of the quadratic functional (I) versus degrees of freedom (dofs) forboth h- and p-version approximations using regular or semi-log scales.

(a) Comment on the dofs needed for convergence of I for h- and p-versions.(b) Comment on the rate of convergence of the quadratic functional I for h- and

p-versions.

(II) Plot graphs of:

(a) displacement u versus axial distance x(b) strain εxx versus axial distance x(c) stress σxx versus axial distance x

for both h- and p-models to observe their convergence as h, the element characteris-tic length, is reduced in the h-version with progressive discretization refinement fora fixed p-level (p = 1) and as p-level is increased in the p-version for fixed elementcharacteristic length h.

(i) Is the convergence of the quadratic functional I (for both h- and p-versions) mono-tonic, nonmonotonic or anything else? What is the expected behavior?

(ii) Are the convergence of displacements, strains and stresses monotonic, nonmonotonicor something else? What is the expected behavior?

(iii) Prepare a short write-up containing details of discretizations and description as wellas discussion of results including the answers to the questions raised here.

You may use FINESSE, ANSYS, or any other software for performing computations.GNUPlot or any other xy graphs display program may be helpful in making xy plots.

References for additional reading[1] B. Jiang. The Least-Squares Finite Element Method: Theory and Applications in

Computational Fluid Dynamics and Electromagnetics. Springer, 1998.


[2] K. S. Surana, L. R. Anthoni, S. Allu, and J. N. Reddy. Strong and weak form of gov-erning differential equations in least squares finite element processes in hpk framework.Int. J. Comp. Meth. in Eng. Sci. and Mech., 2008.




6

Non-Self-Adjoint DifferentialOperators

6.1 Introduction

In this chapter we consider boundary value problems described by non-self-adjoint differential operators. These could be single or multi-variableboundary value problems in single or multi-dimensional space. First, we re-view some basic properties of these differential operators and the propertiesof the integral forms resulting from various classical methods of approxi-mation (presented in chapter 3). In particular, we look for whether theintegral form resulting from a method of approximation for such operatorsis variationally consistent or variationally inconsistent.

(1) The non-self-adjoint differential operators are linear but not symmetric.For such operators the adjoint A∗ of the operator A is not the same.

(2) The properties of the integral forms (presented in chapter 3) resultingfrom various methods of approximation for non-self-adjoint differentialoperators are summarized in the following

(a) GM, PGM and WRM yield VIC integral forms.(b) GM/WF also yields VIC integral forms. However, in this approach

the contributions of the even order terms in the differential operatorto the coefficient matrix becomes symmetric due to integration byparts. When these terms dominate the behavior of the solution,GM/WF is beneficial compared to GM, PGM and WRM.

(c) LSM or LSP, when either used for GDEs containing highest orderderivatives of the dependent variables or for those cast as systemsof first order GDEs, yields VC integral forms.

(d) Thus, for non-self-adjoint operators only GM/WF and LSP are wor-thy of consideration.

(3) In GM/WF, the VIC integral forms remain VIC regardless of the reme-dies employed to alter them. In such cases, one must establish on aproblem-by-problem basis when the computations will remain stable,i.e. the restrictions on h, p and k and the dimensionless parameters inthe mathematical model.

363

364 NON-SELF-ADJOINT DIFFERENTIAL OPERATORS

(4) In finite element processes based on GM/WF one constructs an integralform based on fundamental lemma over the discretization ΩT . Weakform is constructed using integration by parts. Thus, if Aφ − f = 0 inΩ is the BVP and if ΩT = ∪eΩe is the discretization of Ω, then we canwrite

(Aφh− f, v)ΩT =∑e

(Aφeh− f, v

)Ωe

=∑e


)= 0 (6.1)

in which φh is the approximation of φ over ΩT and φeh is the local ap-proximation of φ over Ωe. For an element e with domain Ωe we constructthe weak form

(Aφeh − f, v)Ω = Be(φeh, v)− le(v), v = δφeh (6.2)

using the same procedure as described in chapter 5 for self-adjoint dif-ferential operators. In (6.2), we note that Be(·, ·) is bilinear but notsymmetric and le(·) is linear. We use the local approximation

φeh =n∑i=1

Ni δei = [N ]δe (6.3)

with v = Nj , j = 1, . . . , n (6.4)

to obtain

Be(φeh, v) = [Ke]δe (6.5)

le(v) = F e+ P e (6.6)

in which [Ke] is not symmetric, i.e. Kij 6= Kji ; i, j = 1, . . . , n. Using(6.5) and (6.6) in (6.1), the assembly of the element equations followsstandard procedure.

(5) In the finite element processes based on minimization of residual func-tional we construct the residual functional I(φh) over ΩT (noting thatA is linear). Let E = Aφh− f over ΩT and Ee = Aφeh− f over Ωe, then


Ie =∑e

(Ee, Ee)Ωe (6.7)

The necessary conditions are given by

δI = 2(E, δE)ΩT =∑e

δIe = 2∑e

(Ee, δEe)Ωe = 2∑e

ge = 2g = 0

(6.8)For an element e with domain Ωe, we construct

ge = (Ee, δEe)Ωe = Be(φeh, v)− le(v) (6.9)

6.2. 1D CONVECTION-DIFFUSION EQUATION 365

In the integral form (6.9) we note that Be(·, ·) is bilinear and symmetricand le(·) is linear. Upon substituting from (6.3) and (6.4) into (6.9) andnoting that δEe = ANj ; j = 1, 2, . . . , n, we obtain

Be(φeh, v) = [Ke]δe (6.10)

le(v) = F e (6.11)

in which

Keij = (ANi , ANj)Ωe , i, j = 1, . . . , n (6.12)

F ei = (fi , ANi)Ωe , i = 1, . . . , n (6.13)

Clearly, the coefficients Keij = Kji ; i, j = 1, . . . , n. Assembly of the

element equations (6.10) and (6.11) in (6.8) follows standard procedure.

In the following sections we consider specific model problems to derivefinite element processes based on GM/WF and LSP and present some nu-merical studies.

6.2 1D convection-diffusion equation

The steady-state one-dimensional convection-diffusion equation is rep-resentative of the energy equation encountered in more complex two- andthree-dimensional flows. The steady-state one-dimensional convection-diffusionequation in the dimensionless form can be written as

dφ

dx− 1

Pe

d2φ

dx2= 0 ∀x ∈ Ω = (0, 1) ⊂ R (6.14)

where φ is dimensionless temperature, Pe is the Peclet number defined asPe = uL/R = 1/k, u is the velocity, L is the length of the domain and k is thediffusion coefficient. We shall consider the following boundary conditions:

φ(0) = 1.0 and φ(1) = 0.0 (6.15)

6.2.1 Analytical solution

We note that

φ(x) = c1 + c2 exPe (6.16)

satisfies (6.14). The constants c1 and c2 are evaluated using φ(0) = 1.0 andφ(1) = 0 and we have

φ(x) =(ePe − exPe

ePe − 1

)(6.17)


It is instructive to examine the derivatives of φ(x) in (6.17) of order n

φ(n)(x) ≡ dnφ

dxn=−(Pe)n exPe

ePe − 1(6.18)

Therefore

φ(n)(1) =−(Pe)n ePe

ePe − 1=−(Pe)n

1− 1/ePe(6.19)

From (6.19), we note that

limPe→∞

φn(1) = −∞ (6.20)

However, for a finite value of Pe, the derivatives of all orders of φ(x) at x = 1and elsewhere remain bounded.

Remarks

(1) For very low Pe the convective term dφdx and diffusion term 1

Ped2φdx2

bothplay significant role over the entire domain Ω.

(2) As Pe increases the problem becomes convection dominated, i.e. withprogressively increasing Peclet number, the diffusion becomes more andmore isolated near x = 1.0. The solution φ remains one over the ma-jority of Ω except in the very small neighborhood near x = 1, generallyO(1/Pe) for large values of Pe over which φ changes from 1 to 0. Hence,at Pe = 106, φ remains 1 from x = 0.0 to x = 1−O(10−6) and changesfrom 1 to 0 over a length of O(10−6) near x = 1.0. Thus, in the physics ofthe problem, with increasing Pe the solution φ is described by the con-vection dominated process for the majority of the domain Ω. However,when attempting a numerical simulation of such a process, it is clearthat resolution of the isolated behavior of the diffusion near x = 1.0must be correctly incorporated in the numerical process.

(3) Based on the previous remark, we clearly see that accurate simulationof the isolated diffusion near x = 1.0 for high values of Pe is of criticalsignificance in solving this problem satisfactorily.

(4) Figure 6.1 shows plots of the analytical solution φ(x) vs x for Pecletnumbers of 10, 100 and 1000. Progressively increasing gradients of φnear x = 1.0 are rather obvious with increasing Peclet numbers.


0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

φ

x

Pe=10Pe=100Pe=1000

Figure 6.1: Theoretical solution φ versus x for Pe = 10, 100 and 1000

(5) From the analytical solution (6.17) it is clear that

φ(x) ∈ C∞(Ω) (6.21)

(6) Another significant point to note is to clearly understand the minimumrequirements on the approximation φh of φ in a numerical process toensure that the physics is not disturbed in the numerical process. From(6.14) we note that

(a) convection dφdx must at least be continuous over Ω which implies that

φ ∈ C1(ΩT ) to ensure such behavior.

(b) diffusion 1Pe

d2φhdx2

must also at least be continuous over Ω which ne-cessitates that φh must at least be of class C2 over ΩT .

From (a) and (b) we note that in order for convection and diffusion bothto be continuous over ΩT , φh must at least be of class C2 over ΩT . Thisis the minimum requirement to ensure that the physics of convectionand diffusion is not disturbed in the numerical process.

(7) These points discussed above will be helpful in understanding the na-ture of various computed solutions in the subsequent sections and therestrictions on the local approximations.

6.2.2 The Galerkin method with weak form (GM/WF)

We write (6.14) as

Aφ− f = 0 ∀x ∈ (0, 1) = Ω ⊂ R1 (6.22)


in which

A =d

dx− 1

Pe

d2

dx2, f = 0 (6.23)


φ(0) = 1 and φ(1) = 0 (6.24)

Let ΩT = ∪eΩe be the discretization of Ω and φh be the approximationof φ over ΩT and φeh, the local approximation of φ over Ωe = [xe, xe+1] →Ωξ = [−1, 1]. We consider


xe+1∫xe

(dφehdx− 1

Pe

d2φehdx2

)v dx, v = δφeh (6.25)

From (6.25) we obtain the weak form

(Aφeh−f, v)Ωe =

xe+1∫xe

(vdφehdx

+1

Pe

dv

dx

dφehdx

)dx−v(xe)P

e1−v(xe+1)P e2 (6.26)

where

P e1 = − 1

Pe

dφehdx

∣∣∣xe

(6.27)

P e2 =1

Pe

dφehdx

∣∣∣xe+1

(6.28)

or


in which

Be(φeh, v) =

xe+1∫xe

(vdφehdx

+1

Pe

dv

dx

dφehdx

)dx (6.30)

le(v) = v(xe)Pe1 − v(xe+1)P e2 (6.31)

Equations (6.29)–(6.31) are the weak form of (6.22). We note that

(a) Be(·, ·) is the bilinear but not symmetric.

(b) le(·) is linear.


Approximation space

The BVP contains up to second order derivative of the dependent variableφ but the weak form only contains up to first order derivatives of φeh and v.

(i) Admissibility of φh in Aφ−f = 0 in ΩT in the pointwise sense requiresφeh ∈ Vh ⊂ Hk,p(Ωe) ; k = 3 is minimally conforming. For this choicethe integrals in the following are Riemann.

(Aφh−f, v)ΩT =∑e

(Aφ−f, v)Ωe =∑e

(Be(φeh, v)−le(v)

)= 0 (6.32)

That is, if we choose φeh of class C2(Ωe), then all integrals are Riemannin all forms in (6.32), hence all these forms in (6.32) are equivalent.

(ii) Based on the weak form, if we choose φeh of class C1(Ωe), that is ifφeh ∈ Vh ⊂ H2,p(Ωe), then the following holds.

(Aφh − f, v)ΩT ⇔L

∑e

(Aφeh − f, v)Ωe ⇔L

∑e

Be(φeh, v)− le(v)︸︷︷︸R

(6.33)For this choice of local approximation

∑e(B

e(φeh, v) − le(v)) holds inthe Riemann sense, but all other integral forms in (6.33) hold only inthe Lebesgue sense.

(iii) If we choose φeh ∈ Vh ⊂ H1,p(Ωe), then∑

e(Be(φeh, v)−le(v)) only holds


Local approximation φeh

Let

φeh =

n∑i=1

Ni(ξ)φei = [N ]δe (6.34)

be the local approximation of φ over Ωe → Ωξ. Then

v = δφeh = Nj(ξ), j = 1, . . . , n (6.35)

in which Ni(ξ) are the local approximation functions (see chapter 8). Sub-stituting from (6.34) and (6.35) into the weak form (6.29)–(6.31)

Be(φeh, Nj) =

xe+1∫xe

(Nj

( n∑i=1

dNi

dxδei

)+dNj

dx

1

Pe

( n∑i=1

dNi

dxδei

))dx (6.36)

le(Nj) = Nj(xe)Pe1 −Nj(xe+1)P e2 , j = 1, 2, . . . , n (6.37)


i, j = 1, 2, . . . , n, which can be written as

Be(φeh, v) = [Ke]δe (6.38)

le(v) =

(P e1 )ξ=−1

0...0

(P e2 )ξ=+1

(6.39)

where

Keij =

xe+1∫xe

(NidNj

dx+

1

Pe

dNi

dx

dNj

dx

)dx, i, j = 1, 2, . . . , n (6.40)

Consider C0 local approximation with p = 1

That is, consider a two-node linear element. For this choice, we have

φeh(ξ) =(1− ξ

2

)φe1 +

(1 + ξ

2

)φe2 = N1 φ

e1 +N2 φ

e2

v = Nj(ξ), j = 1, 2(6.41)

and

x(ξ) =

2∑i=1

Ni(ξ)xi =(1− ξ

2

)x1 +

(1 + ξ

2

)x2

dx = J dξ =(dxdξ

)dξ =

he2dξ, he = xe+1 − xe

dv

dx=dNj

dx=

1

J

dNj

dξ=

2

he

dNj

dξ, j = 1, 2

(6.42)

Additionally,dN1

dξ= −1

2,dN2

dξ=

1

2(6.43)

First, we write the matrix [Ke] as

[Ke] = [K1e] + [K2e] (6.44)

in which

K1eij =

1∫−1

NidNj

dξdξ

K2eij =

1∫−1

1

Pe

dNi

dξ

dNj

dξ

2

hedξ =

2

Pehe

1∫−1

dNi

dξ

dNj

dξdξ

(6.45)


Therefore we have

[K1e] =

1∫−1

[N1

dN1dξ N1

dN2dξ

N2dN1dξ N2

dN2dξ

]dξ =

1

2

[−1 1−1 1

](6.46)

[K2e] =2

Pehe

1∫−1

[dN1dξ

dN1dξ

dN1dξ

dN2dξ

dN2dξ

dN1dξ

dN2dξ

dN2dξ

]dξ =

1

Pehe

[1 −1−1 1

](6.47)

and

P e =

P e1P e2

(6.48)

Remarks

(1) The matrix [K1e] is due to the convective term in the GDE. It is notsymmetric.

(2) The matrix [K2e] is purely due to the diffusion term in the GDE. It issymmetric, a benefit of the integration by parts for the terms with evenorder derivative, i.e. the diffusion term.

(3) The matrix [K1e] due to convection is independent of he, the elementlength, i.e. convection is simulated correctly in the computations re-gardless of he. In other words, convection is independent of he.

(4) The element matrix [K2e] due to diffusion is a function of he and Pe. Achange in Pe influences the physics of diffusion and he influences howthe physics of diffusion is simulated in the computations.

Numerical studies

Consider a four-element uniform discretization using two-node linear el-ements shown in Fig. 6.2. Using local node numbers 1 and 2 and (6.46) and(6.47), we can write the following for each of the four elements shown inFig. 6.2.

(Aφeh − f, v) =

(1

2

[−1 1−1 1

]+

1

Pehe

[1 −1−1 1

])φe1φe2

−P e1P e2

(6.49)

for e = 1, . . . , 4.

After imposing inter-element continuity conditions on the nodal valuesof φ at the element nodes and then the assembly of the element matricesand vectors P e gives the following:(

[K1] + [K2])δ = [K]δ = P (6.50)


31 42 4 5321

φ1 φ2 φ3 φ4 φ5

y

x

φ1 = 1.0 φ5 = 0.0

(BC)(BC) he = 14

= 0.25; e = 1, . . . , 4

Figure 6.2: Four-element uniform discretization with boundary conditions

or

−12

12 0 0 0

−12 0 1

2 0 0

0 −12 0 1

2 0

0 0 −12 0 1

2

0 0 0 −12

12

+1

Pehe

1 −1 0 0 0−1 2 −1 0 0

0 −1 2 −1 00 0 −1 2 −10 0 0 −1 1

φ1

φ2

φ3

φ4

φ5

=

P 1

1

P 12 + P 2

1

P 22 + P 3

1

P 32 + P 4

1

P 42

(6.51)

Essential BCs are given by

φ1 = 1.0, φ5 = 0.0 (6.52)

and the inter-element conditions on secondary variables are

P 11 = ? as φ1 = 1.0

P 12 + P 2

1 = 0

P 22 + P 3

1 = 0

P 32 + P 4

1 = 0

P 42 = ? as φ5 = 0.0

(6.53)

We impose conditions (6.52) and (6.53) in (6.51) by modifying the rows andcolumns of the assembled [K] and P:


1 0 0 0 0

− 12 −

1Pehe

2Pehe

12 −

1Pehe

0 0

0 −12 −

1Pehe

2Pehe

12 −

1Pehe

0

0 0 −12 −

1Pehe

2Pehe

12 −

1Pehe

0 0 0 0 1

φ1

φ2

φ3

φ4

φ5

=

10000

(6.54)

We note that the assembled matrix [K] in (6.54) is a tri-diagonal matrix.For nodes i− 1, i, and i+ 1 we can write(

− 1

2− 1

Pehe

)φi−1 +

2

Peheφi +

(1

2− 1

Pehe

)φi+1 = 0 (6.55)

Pre-multiply (6.55) by Pehe yields(Pehe2− 1)φi+1 + (2)φi −

(Pehe2

+ 1)φi−1 = 0 (6.56)

This is a second order difference equation that can be solved exactly byletting

φi = pi−1 (6.57)

Therefore, (6.55) becomes(Pehe2− 1)pi + (2) pi−1 +

(− Pehe

2− 1)pi−2 = 0 (6.58)

and dividing throughout by pi−2 yields

p2(Pehe

2− 1)

+ (2) p−(Pehe

2+ 1)

= 0 (6.59)

Equation (6.59) is quadratic in p and hence its roots p1 and p2 are

p =−2±

√(2)2 − 4

(Pehe

2 − 1) [−(Pehe

2 + 1)]

2(Pehe

2 − 1) (6.60)

or

p1 = 1, p2 =1 + Pehe

2

1− Pehe2

(6.61)

The solution φi consists of a linear combination of pi−11 and pi−1

2 and canbe written as

φi = C pi−11 +Dpi−1

2 (6.62)


Substituting for p1 and p2 in (6.62) from (6.61)

φi = C +D

(1 + Pehe

2

1− Pehe2

)i−1

(6.63)

in which, the constants C and D depend upon the boundary conditions, i.e.

at i = 1 : φi = φ1 = 1 ⇒ 1 = C +D

at i = N + 1 : φi = φN+1 = 0 ⇒ 0 = C +D

(1 + Pehe

2

1− Pehe2

)N(6.64)

Solving for C and D from (6.64)

C =

(1+Pe he

2

1−Pe he2

)N1−

(1+Pe he

2

1−Pe he2

)N , D =1

1−(

1+Pe he2

1−Pe he2

)N (6.65)

Therefore

φi =

(1+Pe he

2

1−Pe he2

)N1−

(1+Pe he

2

1−Pe he2

)N +

(1+Pe he

2

1−Pe he2

)i−1

1−(

1+Pe he2

1−Pe he2

)N (6.66)

From (6.66), we note that

(a) If Pehe2 < 1, then

(1+Pe he

2

1−Pe he2

)is positive and thus

(1+Pe he

2

1−Pe he2

)i−1

is positive

for all values of i.

(b) If Pehe2 > 1, then

(1+Pe he

2

1−Pe he2

)is negative and hence

(1+Pe he

2

1−Pe he2

)i−1

changes

sign with each successive i. This implies that the solution given by (6.66)oscillates from one node to the next, noting that C and D in (6.64) or(6.66) are constants.

The theoretical solution for Pe = 10, 100 and 1000 is shown in Fig. 6.1.The influence of the choice of Pehe

2 on the computational process and thecomputed solution can also be shown through the finite element computa-tions. Let us consider Pehe

2 = 0.5, 1.5 and 2.5. The numerical studies forthese three choices can be done in two ways: (i) We can choose a fixed Peand determine he values corresponding to Pehe

2 = 0.5, 1.5 and 2.5 for whichnumerical studies can be performed. (ii) We can also choose a fixed he valueand then determine Pe values corresponding to Pehe

2 = 0.5, 1.5 and 2.5. We


present details here for the second choice. Consider a uniform discretizationwith he = 0.01 (100 element uniform mesh) with local approximations ofclass C0 at p-level of one. For this value of he (fixed) and Pehe

2 = 0.5, 1, 5and 2.5 we obtain Pe = 100, 300 and 500 for which we compute numer-ical solutions using the formulation based on GM/WF. In all three cases[K]δ − F − P = 0 is satisfied by the computed solution confirmingthat assembled coefficient matrix is not poorly conditioned for these choices.For Pehe

2 = 0.5 (less than one), the computed solution is oscillation free (see

Fig. 6.3). For Pehe2 = 1.5, the computed solution has spurious oscillation in

the vicinity of x = 1.0. For Pehe2 = 2.5, the oscillations grow in magnitude

and spread over larger domain. If we continue to increase Pe, i.e. continueto increase Pehe/2, the oscillation will continue to increase in magnitudeand will continue to spread over larger domain until to a point at which thecomputations will cease due to near singular assembled coefficient matrix.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1

φh

Distance, x

C0(

—

Ωe ) ; p=1 , h

e=0.01

1/2 Peh

e=0.5 ; Pe=100

1/2 Peh

e=1.5 ; Pe=300

1/2 Peh

e=2.5 ; Pe=500

Figure 6.3: Numerical solution for different(Pe he

2

)(The Galerkin method with weak

form: two node linear element)

Remarks

(1) Consider a fixed discretization, p-level and k suitable for Pe = Pe1.If the Peclet number is increased form Pe1 to Pe2 > Pe1 and if thediscretization, p-level and k are kept fixed, then it is clear that morelocalized behavior of diffusion at Pe = Pe2 will no longer be simulatedcorrectly in the numerical process. Thus, what must happen in the nu-merical process is that the localized behavior of diffusion must diffuse


over a larger length. Progressively increasing Peclet number must con-tinue to yield progressively more diffused solutions until it cannot diffuseany more. Such a solution will be a straight line connecting φ = 1 at

x = 0 to φ = 0 at x = 1. This in fact is the solution of BVP: d2φdx2

= 0with φ(0) = 1 and φ(1) = 0 (φ(x) = (1 − x)). A consistent (i.e. VC)numerical process must behave in this manner.

0

0.2

0.4

0.6

0.8

1

0 0.25 0.5 0.75 1

φh

Distance, x

C1(

—

Ωe ) ; p=13 , h

e=0.2

Pe=100

Pe=200

Pe=300

Pe=400

Pe=500

Pe=1,000

Pe=10,000

Figure 6.4: Behavior of φh(x) with increasing Pe for a fixed uniform discretization withcharacteristic length he of 0.2. VC integral form: minimization of residual functional

This aspect of the VC integral form can be illustrated numerically. Con-sider LSP for this model problem (see previous sections and sections 6.1,6.2.3) without using auxiliary variable and auxiliary equation. This for-mulation yields VC integral form. We choose Pe = 100, a five-elementp-version uniform discretization with local approximation in H2,13(Ωe

x)space, i.e. local approximation of class C1(Ωx) at p-level of 13. For thischoice of he, p and k the numerically computed solution for Pe = 100is sufficiently close to the theoretical solution (see Fig. 6.4). Keepinghe, p and k constant, we increase Pe to 200, 300, 400, 500, 1,000 and10,000 and compute numerical solutions (shown in Fig. 6.4) using VCLS formulation. Solutions for all Pe numbers are smooth, i.e. oscillationfree. With progressively increasing Pe the computed solutions continueto diffuse and eventually become φh = (1 − x) for Pe = 10, 000 as ex-pected. Computations remain stable with positive definite coefficientmatrices (see reference [1]).


(2) Next, we examine the VIC integral forms from GM/WF constructedin this section. We note that the assembled equations after imposingboundary conditions would have the following form (assuming uniformdiscretization).

[K]δ =[[K1] + 1

Pehe [K2]]δ = f

nonzero due to BC

due to diffusion

due to convection

(6.67)

For a fixed value of he, progressively increasing Pe would result insmaller and smaller values of (1/Pe he) giving rise to more and moreisolated diffusion and as a consequence the correct simulation of diffu-sion is no longer present in the computational process. In the limit,when Pe→∞, the term 1

Pehe[K2] approaches a null matrix and (6.67)

reduces to

[K1]δ = f (6.68)

in which [K1] is a non-symmetric matrix with zeros on the diagonal,hence [K1] is a singular matrix. Thus, computation of δ is not possi-ble. We see that for a fixed discretization, VIC GM/WF progressivelydegrades with increasing Pe and eventually becomes a degenerate pro-cess in which computation of the solution δ ceases.

(3) From (6.68) it is quite clear that since [K1] is a non-symmetric ma-trix with zeros on the diagonal, a solution δ is possible from (6.67)if and only if 1

Pehe[K2] makes contribution to [K1] that is significant

enough to overcome the singular nature of [K1]. With increasing Pe,contribution of 1

Pehe[K2] to [K1] diminishes, i.e. [K] matrix which is

well-conditioned for some low Pe starts to become ill-conditioned andhence the appearance of spurious nodal oscillations in the solution andeventually becomes singular.

(4) Based on (1)–(3) it is clear that the root causes of nodal oscillations inthe solution are:

(a) VIC integral form in GM/WF resulting in non-symmetric [K1] in(6.67).

(b) The inability of a fixed discretization to simulate the physics ofdiffusion correctly for progressively increasing Pe.

(5) In the published literature this phenomenon described above is termedas negative diffusion in the Galerkin processes and hence the notionof addition of artificial diffusion through streamline upwinded Petrov-Galerkin techniques such as SUPG, SUPG/DC, SUPG/DC/LS, and soon [2–8]. Arguments presented in (1)–(4) based on the mathematics and


physics clearly support what is concluded in (4) and hence the notionof ‘negative diffusion’ in the Galerkin processes has no basis at all andtherefore all upwinding techniques to cure the spurious behaviors of thesolution as possible remedies have no basis either [2–8].

In other words, upwinding methods in whatever form are completelywithout mathematical foundation and are totally unjustifiable from thepoint of view of the mathematics as well as the physics (see Surana etal. [1, 9–13]).

(6) In conclusion, the Galerkin method for non-self-adjoint operators leadsto VIC finite element processes which would only work satisfactorily withC0 local approximations if appropriate discretization is considered inwhich physics of diffusion is simulated correctly in the numerical process.Such discretizations for the present model problem would result in thenumber of elements of the order of Pe in R1 (for higher Pe) which areimpractical (i.e. 106 elements for Pe = 106 in R1).

(7) It is natural to ask, whether the solutions of classes higher than C0

would result in any benefit in the GM/WF. This question is meritoriousand is a subject of study in a published paper by Surana et al. [1] andalso discussed in a later section.

6.2.3 Finite element formulation of convection-diffusionequation based on residual functional

In this section we present a finite element formulation of the 1D steadystate convection diffusion equation (6.14) based on the residual functional.Let φeh be the local approximation of φ over a typical element e with domainΩe of the discretization ΩT =

⋃Ωe of the spatial domain Ω = [0, 1]. Then

Ee =dφehdx− 1

Pe

d2φehdx2

∀x ∈ Ωe (6.69)

is the residual equation for an element e. The residual functional Ie for anelement e and the residual functional I for the discretization ΩT are givenby

Ie = (Ee, Ee)Ωe (6.70)

I =

M∑i=1

Ie =

M∑i=1

(Ee, Ee)Ωe (6.71)


Therefore

δI =M∑i=1

2(Ee, δEe)Ωe = 0 (6.72)

δ2I = 2

M∑i=1

(δEe, δEe)Ωe > 0 (6.73)

Equation (6.73) implies that a solution obtained from (6.72) minimizesI in (6.71). Equations (6.71)–(6.73) confirm that the integral form (6.72) inthe LSP is variationally consistent. Since (6.14) is a second order ODE, thelocal approximation φeh ∈ Hk,p(Ωe) ; k ≥ 3, p ≥ 2k− 1 in which k = 3 is theminimally conforming order of the approximation space. For this choice ofk all integrals over ΩT hold in the Riemann sense.

Details of element equations

Let Ωξ = [−1, 1] be the map of Ωe in the natural coordinate space ξ andlet

φeh(ξ) =n∑i=1

Ni(ξ)φei = [N ]δe (6.74)


Ee =d

dx

( n∑i=1

Ni(ξ)φei

)− 1

Pe

d2

dx2

( n∑i=1

Ni(ξ)φei

)(6.75)

which yields the following in matrix and vector notation:

Ee =([dN

dx

]− 1

Pe

[d2N

dx2

])δe (6.76)

δEe =dNdx

− 1

Pe

d2N

dx2

(6.77)

We note that (6.72) can be written as (since Ee is linear in φei )

M∑e−=1

[Ke]δe =

M∑e=1

[(δEe, δEe)Ωe

]δe =

[ M∑e=1

(δEe, δEe)Ωe

]δ = 0

(6.78)in which δ =

⋃eδe. Therefore

[Ke] =

∫Ωe

δEeδEeT dx (6.79)

is explicitly defined by using (6.77). Numerical values are computed usingGauss quadrature. Numerical studies using this formulation are presentedin later sections.


6.2.4 A finite element formulation of convection diffusionequation based on residual functional: first ordersystem of equations

A finite element formulation based on minimization of residual functionalof the convection diffusion equation (6.14) containing up to second derivativeof φ would require that the local approximation φeh of φ over an element atleast belong to the space H3,p(Ωe), i.e. φeh should be at least of class C2.

If we permit inter-element discontinuity of d2φdx2

, then φeh of the class C1 areadmissible in (6.14). However, if we wish to use φeh of the class C0 in the leastsquares processes then (6.14) must at least be recast as a system of first orderdifferential equations using auxiliary variable. This form is sometimes alsoreferred to as weak differential form or a first order system of the governingdifferential equation (6.14). If we let

τ =dφ

dx, τ being an auxiliary variable (6.80)

then, the GDE (6.14) can be written as

dφ

dx− 1

Pe

dτ

dx= 0

τ − dφ

dx= 0

∀x ∈ Ω = (0, 1) ⊂ R1 (6.81)

The first order differential equations in φ and τ in (6.81) will permit C0

local approximation for φ and τ over an element Ωe. The second equationin (6.81) is called auxiliary equation.

Let φeh and τ eh be local approximations of φ and τ over an element Ωe andlet φeh, τ

eh ∈ Vh ⊂ H1,p(Ωe), i.e. let φeh and τ eh be of class C0. Furthermore

choice of p = 1 (linear behavior of φeh and τ eh over Ωe) is also admissible in(6.81). Upon substituting φeh and τ eh in (6.81) we obtain, residuals functionsEe1 and Ee2 for an element e:

Ee1 =dφehdx− 1

Pe

dτ ehdx

Ee2 = τ eh −dφehdx

∀x ∈ Ωe (6.82)

The residual functional Ie for an element e can be constructed using

Ie = (Ee1, Ee1)Ωe + (Ee2, E

e2)Ωe (6.83)

and for the whole discretization with M elements we have

I =

M∑e=1

Ie =

M∑e=1

( 2∑i=1

(Eei , Eei )Ωe

)(6.84)


δI = 2

M∑e=1

( 2∑i=1

(Eei , δEei )Ωe

)= 0 (6.85)

δ2I = 2M∑e=1

2∑i=1

(δEei , δEei )Ωe > 0 (6.86)

Equations (6.86) imply that a solution δ from (6.85) minimizes I in(6.84). From (6.84)–(6.86) we confirm that the integral form (6.85) in theLSP is variationally consistent.


Consider equal order, equal degree local approximations for φ and τ overΩe. Let us choose a two-node linear element to illustrate the main steps inthe process. Let

φeh =2∑i=1

Ni φei = [N ]φe and τ eh =

2∑i=1

Ni τei = [N ]τ e (6.87)

in which

N1(ξ) =(1− ξ

2

)and N2(ξ) =

(1 + ξ

2

)x(ξ) =

(1− ξ2

)xe +

(1 + ξ

2

)xe+1

(6.88)

Therefore

J =dx

dξ=xe+1 − xe

2=he2

(6.89)

We note that (6.85) can be written as (since Eei are linear in φei and τ ei )

M∑e=1

[Ke]δe =

M∑e=1

[ 2∑i=1

(δEei , δEei )]δe =

[ M∑e=1

2∑i=1

(δEei , δEei )]δ = 0

(6.90)in which

δe =

φeτ e

=

φe1

φe2

τ e1

τ e2

, Nodal values of φ and τ for an element e

(6.91)


and

δEei =

∂Eei∂φe∂Eei∂τe

=

∂Eei∂φe1∂Eei∂φe2∂Eei∂τe1∂Eei∂τe2

, i = 1, 2 (6.92)

therefore

[Ke] =

2∑i=1


The element coefficient matrix requires details of δEei in (6.92). Using (6.82)and (6.87) we can write

Ee1 =d

dx

( 2∑i=1

Ni(ξ)φei

)− 1

Pe

d

dx

( 2∑i=1

Ni(ξ) τei

)Ee2 =

2∑i=1

Ni τei −

d

dx

( 2∑i=1

Ni(ξ)φei

) (6.94)

Details of δEei in (6.92) can be easily obtained using (6.94)

δEe1T =

[[dNdx

], − 1

Pe

[dNdx

]]δEe2T =

[−[dNdx

], [N ]

] (6.95)

in which

[N ] = [N1(ξ) , N2(ξ)] =[1− ξ

2,

1 + ξ

2

][dNdx

]=[dN1

dx,dN2

dx

]=

1

J

[dN1

dξ,dN2

dξ

]=

2

he

[− 1

2,

1

2

] (6.96)

Therefore

δEe1T =[− 1

he,

1

he,

1

Pehe, − 1

Pehe

]δEe2T =

[ 1

he, − 1

he,

1

2

(1− ξ

),

1

2

(1 + ξ

) ] (6.97)


and we can write the following:

(δEe1, δEe1)Ωe =

1∫−1

δEe1δEe1The2dξ

(δEe2, δEe2)Ωe =

1∫−1

δEe2δEe2The2dξ

(6.98)

which can be expressed as

− 1Pe

he2dξ

− 1Pe

1Pe

1Pe − 1

Pe

1Pe

− 1Pe

1Pe

1Pe2

− 1Pe2

− 1Pe2

1 −1

−1 1

1Pe2

(δEe1, δE

e1)Ωe =

1∫−1

1

he2

(6.99)

he2dξ

(1−ξ)2he

(1+ξ)2he

1he

2 − 1he

2

− 1he

2 − 1he

2

(1+ξ)2he

(1−ξ)2he

(1−ξ2)4

(1−ξ)24

−(1−ξ)2he

−(1+ξ)2he

(1−ξ2)4

(1+ξ)2

4

−(1+ξ)2he

−(1−ξ)2he

(δEe2, δE

e2)Ωe =

1∫−1

(6.100)

After performing integration in (6.99) and (6.100), we obtain the follow-ing element coefficient matrix [Ke]

-1

-1

0 0 -1 1

0 0 1

-111-1

1 -1 1

+1

Pe he

−12

[Ke] =

2he

−12

−12

he3

− 2he

he6

he3

−12

12

12

he6

2he

− 2he

12

12

(6.101)

and the element equations become

ge = [Ke]δe (6.102)

in which δe is defined by (6.91).

Remarks

(1) For a discretization with M elements the process of assembly and im-position of boundary conditions φ(0) = 1.0 and φ(1) = 0.0 follows theusual procedure.


(2) For a fixed discretization that works satisfactorily for a Pe = Pe1 using(6.101) as element matrix, if one progressively increases the Pe, then forvery large Peclet numbers, (6.101) would become

−12

2he

−12

−12

he3

− 2he

he6

he3

−12

12

12

he6

2he

− 2he

12

12

[Ke]Pe→∞ = (6.103)

The element matrix [Ke] in (6.103) is obviously due to pure convection.We note [Ke] in (6.103) is symmetric with nonzero elements on the diag-onal. Assembly of [Ke] in (6.103) and imposition of boundary conditionswould yield a symmetric and positive definite matrix from which δ canbe computed. Thus we see that LSP are non-degenerate processes. Thisproperty of the LSP is due to the variational consistency of the integralform.

Numerical solutions

In this section we consider some special cases and compute solutions forthem. Consider a two-element uniform mesh in which each element is a two-node linear element. This mesh is obviously deficient to simulate the solutionof convection diffusion equations for Pe → ∞. Nonetheless, we consider itto demonstrate the non-degenerate nature of LSP.

321 1 2

x1 = 0.0 x2 = 0.5 x3 = 1.0

φ1 φ2 φ3

τ1 τ2 τ3

φ3 = 0.0φ1 = 1.0consider Pe→∞

h1 = 12 h2 = 1

2

Figure 6.5: Two-element uniform discretization and boundary conditions

The element equations in this case would be (6.101) in which [Ke] isdefined by (6.103). Substituting h1 = h2 = 1

2 we obtain the following element


matrices:

[K1] = [K2] =

4 −4 0.5 0.5−4 4 −0.5 −0.50.5 −0.5 1

6112

0.5 −0.5 112

16

(6.104)

The degrees of freedom for the two elements are (after substituting inter-element continuity conditions on φ and τ)

δ1 =

φ1

φ2

τ1

τ2

and δ2 =

φ2

φ3

τ2

τ3

(6.105)

Assembled equations for the two element discretization are

τ2

φ2

τ1

φ3

φ1

τ3

−0.5+0.5

−0.5+0.5

0.5

φ3τ2φ2τ1φ1

1

6

0.5

1

12

0

00

0 0

0

0.5

-0.5

0.5

-4 -0.5

4

1

6

1

120.5

-4

1

12

-0.5

-0.5

0.5

1

12

-0.5

τ3

= 0

4 -4 0

-4

0

-0.5

4 + 4

16

+ 16

(6.106)

We note that assembled matrix [K] is symmetric (due to symmetry of[Ke]). After imposing boundary conditions φ1 = 1.0 and φ3 = 0.0, thereduced system of equations from (6.106) becomes

16 −0.5 1

12 0−0.5 8 0 0.5

112 0 1

3112

0 0.5 112

16

τ1

φ2

τ2

τ3

=

−0.5

4−0.5

0

(6.107)

A plot of the solution is shown in Fig. 6.6. We note that τ = dφdx = −1

at all three nodes. The solution is in agreement with the explanation pro-vided in previous remarks. Thus, we note that even with a ridiculous meshand hopeless local approximation, the formulation based on minimization ofresidual functional does not degenerate and allows us to compute a solutionsatisfying the boundary conditions at x = 0.0 and x = 1.0. This propertyof the formulation based on minimization of residual functional is due tovariational consistency of the integral form. This property also plays anextremely crucial role in designing adaptive processes.


x

1

0.5

10.5

φh

Figure 6.6: Plots of φh versus x, p = 1, 2 element mesh, first order system

Remarks

(1) It is rather obvious that a single element discretization would also yieldthe same solution as shown above.

(2) Even the use of higher degree C0 local approximations (C0 p-version)one could confirm the same behavior.

(3) Since in this case Pe = ∞, all discretizations, any number of degreesof freedom, and all orders of local approximation would yield the samesolution as shown in Fig. 6.6.

(4) The readers are reminded that the purpose of the above exercise is todemonstrate the non-degenerate nature of finite element processes basedon minimization of the residual functional due to variational consistencyof the integral form.

(5) It is perhaps instructive to examine the analytical solutions of the BVPwhen Pe =∞. In this case we have

dφ

dx= 0, 0 < x < 1 (6.108)

withφ(0) = 1 and φ(1) = 0 (6.109)

We note that (6.108) is a first order ordinary differential equation. Nonethe-less, we have two boundary conditions to be satisfied. Thus, this BVPhas no solution.

When attempting to compute a numerical solution of (6.108)–(6.109),one observes that the boundary conditions (6.109) must be satisfied(since they are imposed and thus no other choice). A solution φ(x)that satisfies both boundary conditions is of course

φ(x) = 1− x (6.110)


Obviously (6.110) is not the solution of (6.108)–(6.109), as it does notsatisfy (6.108). Nonetheless, finite element processes based on mini-mization of residual functional will compute such a solution due to thefact that it satisfies boundary conditions and the residual minimizationcriterion. Due to solution (6.110), one finds that

I =

∫Ω

(dφ

dx

)2

dΩ =

1∫0

(−1)2 dx = 1 (6.111)

which is what the least squares numerical process yields.

(6) In the finite element processes based on minimization of residual func-tional one could indeed start with a rather coarse mesh and invoke hpk-adaptive processes and arrive at an optimal combination (least dofs) toachieve the desired solution due to the fact that non-degenerate natureof the resulting algebraic system is always assured.

(7) On the other hand, in GM/WF due to VIC integral form, the coarsemeshes may result in degenerate computational processes and hencecomputations of the solution may not be possible and thus true adaptiveprocesses may not be possible either.

Numerical studies using local approximations in Hk,p(Ωe) spacesfor k ≥ 2, p ≥ 2k − 1

In this section we present some numerical studies for Pe = 100 and106 using graded discretizations and local approximations in Hk,p(Ωe); k ≥2, p ≥ 2k − 1 spaces [1]. In the finite element processes that are basedon minimization of residual functional I, examination of the behavior of Iversus degrees of freedom (dofs) is essential in determining its dependenceon h, p and k. We consider spaces Hk,p(Ωe) with k = 2, 3, . . ., and computesequences of solutions for increasing p-levels up to 19 in each space. Weconsider Pe = 100 and 106 and use discretizations:

Element lengths for five-element graded discretization for Pe = 100:

0.6, 0.28, 0.08, 0.02, 0.02

Element lengths for 10-element graded discretization for Pe = 106:

0.722215, 2.5e-1, 2.5e-2, 2.5e-3, 2.5e-4, 2.5e-5, 2.5e-6, 2.5e-6, 2.5e-6


1e-18

1e-16

1e-14

1e-12

1e-10

1e-08

1e-06

1e-04

1e-02

1e+00

10 100

Resid

ual

Fu

ncti

on

al

I

Degrees of Freedom

p=5

p=9

p=13

p=17

p-convergence

k-convergence

Hk,p

(—

Ωe ) Spaces (Pe=100)

k=2

k=3

k=4

k=5

k=6

k=7

Figure 6.7: Minimization of residual functional (LSP), residual functional I versus dofs:Pe = 100

1e-14

1e-12

1e-10

1e-08

1e-06

1e-04

1e-02

1e+00

10 100

Resid

ual

Fun

cti

onal

I

Degrees of Freedom

p=7

p=11

p=15

p-convergence

k-convergence

Hk,p

(—

Ωe ) Spaces (Pe=10

6)

k=2

k=3

k=4

k=5

k=6

k=7

Figure 6.8: Minimization of residual functional (LSP), residual functional I versus dofs:Pe = 106

Behavior of the residual functional

Figures 6.7 and 6.8 show graphs of I versus dofs for Pe = 100 andPe = 106, respectively. We make the following remarks.

(1) For a fixed k, the order of space and hence the global smoothness oforder k− 1, the rate of convergence of I increases as p-level is increased.

(2) As the order of the space k is increased from k to k + 1, lower valuesof I are achieved in the space of order k + 1 compared to the space of


order k, regardless of the values of k, indicating improved solution inthe space of order k + 1 compared to the space of order k.

(3) While the rate of convergence of I (i.e. slope of I versus dofs) increasesslightly with the change of the order of the space at lower p-levels, athigher p-levels (p ≥ 12), I versus dofs graphs are almost parallel to eachother for all values of k, indicating same rate of convergence for all valuesof k.

(4) For fixed p-level i.e., p = constant, and since the discretization is fixed,h is constant as well; one could study the k-process for the residualfunctional. In these cases h and p are fixed and only k, the order ofthe space Hk,p(Ωe) is changing, hence these represent k-process for thefunctional I. Dependence of I on the order of the space k, i.e. degreeof global smoothness is quite obvious. At lower p-levels (3 − 5), thedependence of I on k is not as strong as it is for p ≥ 5. With increasingp-levels, the dependence of I on k becomes even stronger. This confirmsthat the order of the space k in Hk,p(Ωe) or in other words, the degree ofglobal smoothness is undoubtedly an independent parameter in additionto h and p.

(5) The k and pk-convergence of the error functional I (and likewise otherquantities) is perhaps most illustrative of the influence of the order ofspace k on the convergence of I. The pk-convergence can be viewed in atleast two different ways. (i) For increasing k as well as increasing p butp = 2k−1, i.e. minimum p-level for the order of the space k. In this casewe observe much higher rate of convergence of pk-processes compared top-convergence of I in any of the spaces (i.e., any value of k in Hk,p(Ωe)).We also refer to this as the k-convergence. We note that increase inp with increasing k is inevitable, but in this case p is minimum. Inthe graphs this is represented as k-convergence. (ii) Perhaps, the mostdramatic is the pk-convergence of the error or residual functional I inwhich p and k both change in such a way that the total degrees offreedom for the discretization do not vary much. Almost vertical linesshow such behaviors for various combination of p and k in which totaldofs do not change significantly but the I values decrease significantly.We observe exceptionally high slopes of such lines, even for very lowtotal dofs. With increasing total dofs, slopes of these lines increaseindicating an increase in the convergence rate of I. At close to 100dofs, the pk-convergence graphs of I are almost vertical straight lines,indicating exponential convergence rate. It is worth noting that suchbehavior is the consequence of the fact that for fixed dofs, an increasein k would permit an increase in p-level, which in fact is responsible forthe improvement i.e. decrease in the value of I.


(6) We observe that I continues to decrease in progressively higher orderspaces for a given dofs, indicating improved performance of the LSP inhigher order spaces. That is, the best approximation property of theLSP in E-norm yields progressively improved solutions in higher orderspaces.

Using the five and ten element discretizations, we present some numericalstudies for Pe = 100 and 106 in H2,p(Ωe) and H3,p(Ωe) spaces. We presentφh versus x graphs for both values of Pe numbers.

Figures 6.9 (a) and (b) show plots of φh versus x in H2,p(Ωe) spacesfor p = 3, 5, 7, . . .. Figures 6.10 (a) and (b) show graphs of φh versus xin H3,p(Ωe); p = 5, 7, 9, . . . spaces. Due to graded mesh and low Pe (of100) even for low p-levels the computed solutions are sufficiently accurate.Figures 6.11 (a) and (b) show the plots of φh versus x in H2,p(Ωe) for p =3, 5, 7, 9, 18.

In Figs. 6.11 (a), we observe the linear behavior of φh in H2,3(Ωe) dueto the fact for this mesh at such low p-levels the solutions in H2,3(Ωe) areunable to account for the physics of diffusion in the numerical process. Asthe p-levels are increased, φh improves and at p = 18, φh is in excellentagreement with the theoretical solution. What appears to be a reasonablesolution at p = 7 in Fig. 6.11 (a), is in error upon closer examination asshown in Fig. 6.11 (b). However, for p ≥ 9 all computed solutions are inexcellent agreement with the theoretical solution (up to more than 5 or 6decimal places). Solutions in H3,p(Ωe); p = 5, 7, 9, 18 shown in Fig. 6.12(a) and Fig. 6.12 (b) exhibit similar behaviors as the solutions in H2,p(Ωe),i.e. for low values of p (≤ 7) the solutions are diffused but for p ≥ 9 sharpfronts are simulated accurately with excellent agreement with the theoreticalsolution. We remark that the computed solutions: (a) are always free ofspurious oscillations and that the computational process in each case is non-degenerate due to variational consistency of LSP, (b) are always physical ifjudged based on how well chosen h, p and k allow the physics of the BVP tobe incorporated in the numerical process, and (c) LSP permit much coarserdiscretizations than the variationally inconsistent Galerkin processes.

6.3 2D convection-diffusion equation

The steady state 2D convection-diffusion equation is the 2D form of theenergy equation in Eulerian description for an inviscid medium. The dimen-sionless form can be written as

u∂φ

∂x+ v

∂φ

∂y− ∂

∂x

(kxx

∂φ

∂x+ kxy

∂φ

∂y

)− ∂

∂y

(kyx

∂φ

∂x+ kyy

∂φ

∂y

)− f = 0

∀x, y ∈ Ω = Ωx × Ωy = (0, 1)× (0, 1) ⊂ R2 (6.112)


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

φh

x

p-convergence

H2,p

(—


p=3

p=4

p=5

p=7

p=19

(a) Function value φh over entire domain

0.7

0.75

0.8

0.85

0.9

0.95

1

0.7 0.75 0.8 0.85 0.9 0.95 1

φh

x

p-convergence

H2,p

(—


p=3

p=4

p=5

p=7

p=19

theoretical

(b) Function value φh: expanded view

Figure 6.9: Minimization of residual functional (LSP); φh versus dofs: Pe = 100, k = 2

in which u and v are known velocities in the x and y direction, φ is di-mensionless temperature and kij (i, j = x, y) are the thermal conductivitycoefficients. In the following, we consider more specific form of (6.112) forwhich analytical solution is possible.


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

φh

x

p-convergence

H3,p

(—


p=5

p=7

p=8

p=9

p=19


0.9

0.92

0.94

0.96

0.98

1

0.9 0.92 0.94 0.96 0.98 1

φh

x

p-convergence

H3,p

(—


p=5

p=7

p=9

p=19

theoretical



Theoretical solution of the simplified form

If we assume u = v = 1, kxx = kyy = 1Pe and kxy = kyx = 0 in addition

to f = 0 then (6.112) can be written as

∂φ

∂x+∂φ

∂y− 1

Pe

(∂2φ

∂x2+∂2φ

∂y2

)−f = 0, ∀x, y ∈ Ω = (0, 1)×(0, 1) ⊂ R2 (6.113)


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

φh

x

p-convergence

H2,p

(—

Ωe ) Spaces (Pe=10

6)

p=3

p=5

p=6

p=7

p=19


0.996

0.997

0.998

0.999

1

0.9999 0.99992 0.99994 0.99996 0.99998 1

φh

x

p-convergence

H2,p

(—

Ωe ) Spaces (Pe=10

6)

p=7

p=9

p=13

p=19

theoretical



We consider (6.113) with the following BCs (also shown in Fig. 6.13) andsimplification.

φ(1, y) = 0.0, 0 ≤ y ≤ 1

φ(x, 1) = 0.0, 0 ≤ x ≤ 1

φ(x, 0) =1− e(x−1)Pe

1− ePe, 0 ≤ x ≤ 1

φ(0, y) =1− e(y−1)Pe

1− ePe, 0 ≤ y ≤ 1

(6.114)


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

φh

x

p-convergence

H3,p

(—

Ωe ) Spaces (Pe=10

6)

p=5

p=7

p=8

p=9

p=19


0.996

0.997

0.998

0.999

1

0.9999 0.99992 0.99994 0.99996 0.99998 1

φh

x

p-convergence

H3,p

(—

Ωe ) Spaces (Pe=10

6)

p=7

p=9

p=13

p=19

theoretical



The theoretical solution of the BVP given by (6.113) with BCs (6.114)is

φ(x, y) =(1− e(x−1)Pe)(1− e(y−1)Pe)

(1− e−Pe)(1− e−Pe)(6.115)


The differential operator A in (6.114) is

∂

∂x+

∂

∂y− 1

Pe

( ∂2

∂x2+

∂2

∂y2

)(6.116)

hence (6.114) can be written as

Aφ = 0 as f = 0 (6.117)

The differential operator A is linear but not symmetric, hence A is a non-self-adjoint differential operator, thus the Galerkin method with weak formwould yield VIC integral form. In the following we consider least squaresfinite formulations of (6.113) based on the residual functional as these yieldVC integral forms.

y

x

Ω

φ(1, y) = 0

φ(x, 1) = 0

(0, 0) (1, 0)

(1, 1)(0, 1)

φ(0, y) = 1−e(y−1)Pe

1−e−Pe

φ(x, 0) = 1−e(x−1)Pe

1−e−Pe

Figure 6.13: Domain of definition of the BVP (6.113) and boundary conditions

6.3.1 Least squares finite element formulation based on theresidual functional

Consider the non-simplified form of 2D convection-diffusion equation i.e.equation (6.112). Let φeh(x, y) be the local approximation of φ over a typicalelement e with domain Ωe of the discretization ΩT =

⋃e Ωe of the spatial

domain Ω = (0, 1) × (0, 1). The residual equation Ee for an element e canbe written as

Ee = u∂φeh∂x

+ v∂φeh∂y− ∂

∂x

(kxx

∂φeh∂x

+ kxy∂φeh∂y

)− ∂

∂y

(kyx

∂φeh∂x

+ kyy∂φeh∂y

)− fe ∀x, y ∈ Ωe (6.118)


The residuals Ie and I for Ωe and ΩT can be written as

Ie = (Ee, Ee)Ωe (6.119)

I =M∑i=1

Ie =M∑e=1

(Ee, Ee)Ωe (6.120)

Therefore

δI = 2M∑e=1

(Ee, δEe)Ωe = 0 (6.121)

δ2I = 2

M∑e=1

(δEe, δEe)Ωe > 0 (6.122)

Since (6.113) is a second order ODE in φ, x and y, the local approxi-mation φeh(x, y) ∈ Hk,p(Ωe) ; k ≥ 3 and p = 2k − 1 in which k = 3 is theminimally conforming order of the approximation space. For this choice ofk, all integrals over ΩT hold in the Riemann sense.

Details element equations

Let Ωξη = (−1, 1)× (−1, 1) be the map of an element e with domain Ωe

in the natural coordinate space ξ, η and

φeh(ξ, η) =

n∑i=1

Ni(ξ, η)φei = [N ]δe (6.123)

be the local approximation of φ over Ωe (see chapter 8). Substituting (6.123)into (6.118)

Ee = u∂

∂x

( n∑i=1

Ni φei

)+ v

∂

∂y

( n∑i=1

Ni φei

)− ∂

∂x

(kxx

n∑i=1

∂Ni

∂xφei + kxy

n∑i=1

∂Ni

∂yφei

)− ∂

∂y

(kyx

n∑i=1

∂Ni

∂xφei + kyy

n∑i=1

∂Ni

∂yφei

)− fe

δEe = u∂Ni

∂x+ v

∂Ni

∂y− ∂

∂x

(kxx

∂Ni

∂x+ kxy

∂Ni

∂y

)− ∂

∂y

(kyx

∂Ni

∂x+ kyy

∂Ni

∂y

), i = 1, 2, . . . , n

(6.124)


We can also write these using the matrix and vector notation.

Ee =

u[∂N∂x

]+ v[∂N∂y

]− ∂

∂x

(kxx

[∂N∂x

]+ kxy

[∂N∂y

])− ∂

∂y

(kyx

[∂N∂x

]+ kyy

[∂N∂y

])δe − fe

δEe = u∂N∂x

+ v∂N∂y

− ∂

∂x

(kxx

∂N∂x

+ kxy

∂N∂y

)− ∂

∂y

(kyx

∂N∂x

+ kyy

∂N∂y

)(6.125)

Since Ee is linear in φei , then Ee = δEeT δe − fe, hence (6.121) canbe written as (for a discretization with M elements)

M∑e=1

[Ke]δe =

M∑e=1

[(δEe, δEe)Ωe

]δe

=[ M∑e=1

(δEe, δEe)Ωe

]δ =

M∑e=1

(δEe, fe)Ωe (6.126)

and therefore

[Ke] =

∫Ωe

δEeδEeT dΩ and F e =

∫Ωe

δEefe dΩ (6.127)

is explicitly defined by using (6.125). Numerical values of the coefficients of[Ke] are computed using Gauss quadrature.

6.3.2 Least squares finite element formulation of (6.113) byrecasting it as a system of first order PDEs

In (6.113), we introduce auxiliary variables α and β defined by the fol-lowing auxiliary equations

α = −(kxx

∂φ

∂x+ kxy

∂φ

∂y

)β = −

(kyx

∂φ

∂x+ kyy

∂φ

∂y

) (6.128)

Negative sign in (6.128) are introduced to be consistent with definitionsof heat vector [14]. Substituting from (6.128) into (6.113).

u∂φ

∂x+ v

∂φ

∂y+∂α

∂x+∂β

∂y− f = 0

α+ kxx∂φ

∂x+ kxy

∂φ

∂y= 0

β + kyx∂φ

∂x+ kyy

∂φ

∂y= 0

∀x, y ∈ Ω = (0, 1)× (0, 1) (6.129)


Let φeh, αeh, βeh be the local approximations of φ, α and β over Ωe, domainof a typical element e of the discretization ΩT . We consider φeh, α

eh, β

eh ∈

Vh ⊂ H1,p(Ωe), local approximations of class C0(Ωe) with equal degree ofapproximation functions. Upon substituting the local approximations in(6.129) we obtain the following residual equations.

Ee1 = u∂φeh∂x

+ v∂φeh∂y

+∂αeh∂x

+∂βeh∂y− fe

Ee2 = αeh + kxx∂φeh∂x

+ kxy∂φeh∂y

Ee3 = βeh + kyx∂φeh∂x

+ kyy∂φeh∂y

∀x, y ∈ Ωe (6.130)

The residual functional Ie and I over Ωe and ΩT can now be defined.

Ie =3∑i=1

(Eei , Eei )Ωe (6.131)

I =M∑e=1

Ie =M∑e=1

( 3∑i=1

(Eei , Eei )Ωe

)(6.132)

therefore

δI =

M∑e=1

δIe = 2

M∑e=1

( 3∑i=1

(Eei , δEei )Ωe

)= 0 (6.133)

δ2I =M∑e=1

δ2Ie = 2M∑e=1

( 3∑i=1

(δEei , δEei )Ωe

)> 0 (6.134)


Consider unequal degree C0 local approximation for φ, α and β over Ωe

for the sake of generality

φeh =

nφ∑i=1

Nφi (ξ, η)φei = [Nφ]φe

αeh =

nα∑i=1

Nαi (ξ, η)αei = [Nα]αe

βeh =

nβ∑i=1

Nβi (ξ, η)βei = [Nβ]βe

(6.135)



Ee1 = u∂

∂x

( nφ∑i=1

Nφ φei

)+ v

∂

∂y

( nφ∑i=1

Nφ φei

)+

∂

∂x

( nα∑i=1

Nα αei

)+

∂

∂y

( nβ∑i=1

Nβ βei

)− fe

Ee2 =

nα∑i=1

Nα αei + kxx

nφ∑i=1

∂Nφ

∂xφei + kxy

nφ∑i=1

∂Nφ

∂yφei

Ee3 =

nβ∑i=1

Nβ βei + kyx

nφ∑i=1

∂Nφ

∂xφei + kyy

nφ∑i=1

∂Nφ

∂yφei

(6.136)

which can be written as

Ee1 = u[∂Nφ

∂x

]φe+ u

[∂Nφ

∂y

]φe+

[∂Nα

∂x

]αe+

[∂Nβ

∂y

]βe − fe

Ee2 = [Nα]αe+ kxx

[∂Nφ

∂x

]φe+ kxy

[∂Nφ

∂y

]φe (6.137)

Ee3 = [Nβ]βe+ kyx

[∂Nφ

∂x

]φe+ kyy

[∂Nφ

∂y

]φe

Let us define

δe =

φeαeβe

and δ =⋃e

δe (6.138)

We can now consider δEe1, δEe2 and δEe3 which are given by

δEei =

∂Eei∂φe

∂Eei∂αe

∂Eei∂βe

, i = 1, 2, 3 (6.139)


and using (6.139) and (6.137), we obtain

δEe1 =

u∂Nφ

∂x

+ v

∂Nφ

∂y

∂Nα

∂x

∂Nβ

∂y

δEe2 =

kxx

∂Nφ

∂x

+ kxy

∂Nφ

∂y

Nα0

δEe3 =

kyx

∂Nφ

∂x

+ kyy

∂Nφ

∂y

0Nβ

(6.140)

We note that (6.133) can be written as

M∑e=1

([ 3∑i=1

(δEei , δEei )Ωe

]δe

)=[ M∑e=1

( 3∑i=1

(δEei , δEei )Ωe

)]δ =

M∑e=1

3∑i=1

(δEei , fe)Ωe

(6.141)

orM∑e=1

[Ke]δe = [K]δ =M∑i=1

F e = F (6.142)

Therefore

[Ke] =3∑i=1

(δEei , δEei )Ωe =

3∑i=1

∫Ωe

δEei δEei T dΩ

F e =

3∑i=1

∫Ωe

δEei fe dΩ

(6.143)

are explicitly defined by using (6.139). Numerical values of the coefficientsof [Ke] and F e are obtained using Gauss quadrature.

In the following we consider a number of model problems using a slightlysimplified form of (6.112). In all cases, numerical solutions are computedusing least squares finite element formulation based on residual functionalas presented above.


6.3.3 Convection dominated thermal flow (advection skewedto a square domain)

In this example [2,5–7,10] we consider advection skewed to a square do-main with φ, dimensionless temperature, specified on two adjacent bound-aries with ‖−→a ‖ =

√u2 + v2 = 1, that is, unidirectional flow skewed to the

boundaries by angle θ and θ = 30, 45 and 60 [see Fig. 6.14(a)]. Thediffusivity coefficients are chosen such that the diffusivity matrix [K] = k[I]with k = 10−6 (i.e. kxx = kyy = 10−6 and kxy = kyx = 0) which resultsin a Peclet number Pe = 1/k = 106. Thus, in this case, (6.112) reduces to(assuming f = 0)

u∂φ

∂x+ v

∂φ

∂y− 1

Pe

(∂2φ

∂x2+∂2φ

∂y2

)= 0 ∀x, y ∈ Ω = (0, 1)× (0, 1) ⊂ R2

(6.144)For such high value of Peclet number, the solution is essentially that of

the pure advection case. We compare the numerical solutions computed herewith those reported in the literature. The temperature φ is specified by thefollowing boundary conditions (for sides AB and AD):

φ = 100, x = 0, 0 ≤ y ≤ 1

φ = φ(x), y = 0, 0 ≤ x ≤ h1

φ = 50, y = 0, h1 ≤ x ≤ 1

(6.145)

where h1 is the length of the element along the x-axis located at A and φ(x)is a cubic function satisfying the following four conditions:

φ(0) = 100 , φ(h1) = 50 ,∂φ

∂x

∣∣∣x=0

= 0 ,∂φ

∂x

∣∣∣x=h1

= 0 (6.146)

In this study, we consider solutions of class C00(Ωe), hence we use themathematical model (6.129) consisting of a system of first order PDEs. First,we consider a coarse 4×4 uniform discretization [see Fig. 6.14(b)] consistingof nine node p-version elements with boundary conditions defined by (6.145)and (6.146). The p-levels in the ξ and η directions are increased uniformlybeginning with pξ = pη = 1.

Figure 6.15 shows p-convergence of the residual functional I (√I versus

degrees of freedom). At p-level of pξ = pη = 9, the value of√I is of the

order of O(10−3), i.e. the value of I is of the order of O(10−6) confirminggood accuracy of the computed solution. Figure 6.16(a) shows a contourplot and 3D plot of φ over the square domain at p-level of p = 8 in whichφ = 100 along AC and φ = 50 along EF . In the triangular region ACD, thetemperature φ has a constant value of 100 and likewise in the region EBF ,φ has a constant value of 50. Smooth and oscillation free computed solution


1

1

(a) Schematic of the BVP

direction

Flow

(b) Finite element discretization and boundary conditions

y

x

h1 φ = 50

φ = 100

100

50

φ

x

θ

A B

CD

on sides BC and CD

φ = 50

y

x

‖a‖ = 1

φ = 100

Zero flux boundary conditions

Figure 6.14: Schematic, boundary conditions, and finite element mesh for advectionskewed to a square domain

is quite obvious from the 3D plot. Transition from φ = 50 to φ = 100 occursin the region ACFE.

Remarks

(1) The first observation we make is that the computed solution (even forsuch a coarse discretization) is oscillation free. The present least squaresformulation based on residual functional requires no addition of stream-wise diffusion that is necessary in the Galerkin method with weak form.

(2) We note that lines of constant φ are parallel to each other, that is, ACis parallel to EF and so is the case for all other contour lines between


10-4

10-3

10-2

10-1

100

101

101

102

103

104

105

Sq

uare

roo

t of

resid

ual

functi

on

al,

√I

Degrees of freedom

p=1

p=1

p=1

p=9

p=8

p=8

Uniform discretization

4x4

10x10

20x20

Figure 6.15: Square root of the residual functional versus degrees of freedom for θ = 45,solutions of class C00(Ωe)

AC and EF confirming the absence of crosswind diffusion.

(3) The width h, given by h = h1 sin(θ), over which the function φ changesfrom 100 to 50 is a function of the element length h1 of the elementlocated at x = 0, y = 0 and angle θ and it is not, in any way, related tothe least squares formulation of the boundary value problem.

(4) To illustrate the point discussed in (3), we consider two more refineduniform discretizations: 10 × 10 and 20 × 20. For both discretizations,p-levels are increased uniformly (pξ = pη = p) beginning with p = 2 andending with p = 8. Figure 6.15 shows p-convergence of I (

√I versus

dofs) for all three discretizations (4 × 4, 10 × 10 and 20 × 20). Wenote that convergence rates (slopes of

√I versus dofs) are almost the

same for the three discretizations. However, the coarsest discretizationyields lowest values of I for a given number of degrees of freedom. Thisis in agreement with the expected behavior due to the smoothness ofthe solution φ, that is, for smooth solutions, coarse discretizations withhigher p-levels result in the best performance.

(5) Figures 6.16(b) and (c) show 3D and contour plots of φ for 10 × 10


0

1

0

1

40

50

60

70

80

90

100

110

φ

h1=0.25 x

y

φ

0

1

0 1x

y

A

B

CD

F

E

h1=0.25

(a) 4× 4 uniform discretization

0

1

0

1

40

50

60

70

80

90

100

110

φ

h1=0.10

x

y

φ

0

1

0 1x

y

A

B

CD

F

E

h1=0.10

(b) 10× 10 uniform discretization

0

1

0

1

40

50

60

70

80

90

100

110

φ

h1=0.05

x

y

φ

0

1

0 1x

y

A

B

CD

F

E

h1=0.05

(c) 20× 20 uniform discretization

Figure 6.16: Plots of φ for θ = 45: solutions of class C00(Ωe) at pξ = pη = p = 8

and 20 × 20 discretizations respectively at p = 8. We note that thesolution for 10×10 and 20×20 discretizations are also free of crosswinddiffusion. The length h1 decreases as the discretization is refined and asa consequence, the width h over which φ changes from 100 to 50 alsodecreases.


0

1

0

1

40

50

60

70

80

90

100

110

φ

h1=0.25 x

y

φ

0

1

0 1x

y

h1=0.25


0

1

0

1

40

50

60

70

80

90

100

110

φ

h1=0.10

x

y

φ

0

1

0 1x

y

h1=0.10


0

1

0

1

40

50

60

70

80

90

100

110

φ

h1=0.05

x

y

φ

0

1

0 1x

y

h1=0.05



(6) Numerical solutions are computed for θ = 30 and 60 with the threeuniform discretizations and uniform p-level. Figure 6.17 shows contourplots and 3D plots for θ = 30 for the three discretizations at p-level ofpξ = pη = 8. Figure 6.18 shows similar plots for θ = 60.

All three studies confirm that for a fixed value of h (given by h =


0

1

0

1

40

50

60

70

80

90

100

110

φ

h1=0.25 x

y

φ

0

1

0 1x

y

h1=0.25


0

1

0

1

40

50

60

70

80

90

100

110

φ

h1=0.10

x

y

φ

0

1

0 1x

y

h1=0.10


0

1

0

1

40

50

60

70

80

90

100

110

φ

h1=0.05

x

y

φ

0

1

0 1x

y

h1=0.05



h1 sin(θ)), the computed solutions are free of crosswind diffusion. The resultsfrom the least squares finite element formulation computed and presentedhere are compared with Grygiel and Tanguy [15] and those reported by manyresearchers using streamline upwinded Petrov-Galerkin method (SUPG) andthe method of characteristics. Discussions based on on these comparisons


are presented in the following.

Comparison with published works, discussion and remarks

(1) Brookes and Hughes [2] have presented results for this problem forθ = 22.50, 45 and 67.5 using 10 × 10 uniform mesh of bilinear el-ements employing four formulations: G (the Galerkin formulation), QU(quadrature method of adding diffusion, i.e., the convective terms areevaluated using one point quadrature), SU1 (one point quadrature forconvective term in SUPG) and SU2 (2× 2 quadrature for convective aswell as diffusion term in SUPG). We make the following observations

(a) The Galerkin formulation has wild oscillations in the computed so-lutions. The magnitude of the oscillations varies with angle θ.

(b) The QU method adds excessive diffusion and smears the solution.Sharp front is destroyed.

(c) The SU1 has oscillations similar to the Galerkin formulation forθ = 22.5 and 67.5. For the case of θ = 45, the solution is nodallyexact.

(d) The SU2 produces the best results out of the four techniques con-sidered but some oscillations are still present.

Hughes, Mallet and Mizukami [4] have also investigated the same prob-lem using: SUPG (2× 2 Gaussian quadrature rule for all terms for fournode bilinear element), DC1 (SUPG with discontinuity capturing op-erator (τ1 = τ and τ2 = τ11), using four node bilinear), DC2 (SUPGwith discontinuity capturing behavior (τ1 = τ and τ2 = max(0, τ11 − τ),using four node bilinear element) and MH (Mizukami and Hughes usetriangular elements with treatment similar to DC1 and DC2). Based onthe results, we make the following comments.

(a) The SUPG solution has oscillations.

(b) The DC1 smoothes the oscillations while still maintaining the rea-sonable sharp front.

(c) Discontinuity-capturing operator DC2 produces less diffusive solu-tion than DC1.

(d) In the case of the MH scheme, each triangular element is internallysubdivided into more than one triangle with the diagonals of thetriangles roughly oriented parallel to the internal layers. This schemeis monotonic and produces the sharpest fronts.

(2) The SUPG based methods have significant amount of crosswind diffusionThe amount of crosswind diffusion varies with the angle θ.


(3) Method of characteristics produces the best results for θ = 45. Themethod of characteristics also has significant diffusion but it is less thanthe SUPG method.

(4) The p-version LEFEF has no crosswind diffusion and solution is oscilla-tion free for all values of θ.

(5) Furthermore, the p-version least squares finite element formulation doesnot utilize upwinding, discontinuity capturing operator or any other suchmeans.

The p-version least squares finite element process is a straight forwardformulation that utilizes the fact that: the presence of artificial diffusionin the solution of the LEFEF is a consequence of inadequate representationof the function behavior over each element. If this argument is valid, thenadequate mesh with appropriate p-levels should be able to produce a goodsolution. The studies presented here confirm this.

6.3.4 Advection of a cosine hill in a rotating flow field

In this numerical study [2, 5–7, 10] we consider almost pure advection(Pe = 106) of a cosine hill in a rotating flow field over a unit square [seeFig. 6.19(a)]. The velocities u and v are given by the following.

u = −y, v = x (6.147)

We consider kxx = kyy = k = 10−6 and kxy = kyx = 0 which gives usPe = 1

k = 106 and (6.112) reduces to (6.144). Due to such small value ofdiffusivity k, this problem is almost the same pure advection, i.e.

−y∂φ∂x

+x∂φ

∂y− 1

Pe

(∂2φ

∂x2+∂2φ

∂y2

)= 0 ∀x, y ∈ Ω = (0, 1)×(0, 1) ⊂ R2 (6.148)

We set φ = 0 on all four boundaries of the domain of unit square. Along theline OA (internal boundary) φ is specified by a cosine hill.

φ =1

2

(cos(4πy + π) + 1

), −1

2≤ y ≤ 0 (6.149)

We consider a 4×4 uniform discretization of nine node p-version elementsshown in Fig. 6.19(b).

The p-levels are increased uniformly in ξ and η. Figure 6.20 shows agraph of

√I versus dofs (p-convergence of I). At p-level of 5, the value of I

is already of the order of O(10−5). Figure 6.21 shows the elevation of φ atp-level of 8 over the unit square computed using least squares finite elementformulation.


direction

Flow

1

1

(a) Schematic of the BVP and boundary conditions (b) An uniform discretization

φ = 0

φ = 0

x

y

A

O

φ = 0

φ = 0

φ

O A

Figure 6.19: Schematic, boundary conditions and finite element mesh for advection in arotating flow field

10-5

10-4

10-3

10-2

10-1

101

102

103

104

Squ

are

ro

ot

of

resid

ual

functi

onal,

√I

Degrees of freedom

p=1

p=2

p=4

p=6

p=8

Figure 6.20: Square root of the residual functional versus degrees of freedom, solutionsof class C00(Ωe)

Figure 6.22 shows plots of φ versus y and φ versus x at x = 0 and y = 0respectively (i.e. along the vertical and horizontal center lines of the unitsquare domain).

These results can be compared with those reported by Hughes and Mallet[3] using various upwinding formulations based on the Galerkin method withweak form.


-0.5

0.5

x

0.5

y

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

φ

Figure 6.21: 3D Plot of φ: 4x4 uniform discretization; solutions of class C00(Ωe), p-levelof 8

0

0.2

0.4

0.6

0.8

1

-0.5 -0.25 0 0.25 0.5

φ

y

0

0.2

0.4

0.6

0.8

1

-0.5 -0.25 0 0.25 0.5

φ

x

(a) φ versus y at x = 0 (b) φ versus x at y = 0

Figure 6.22: Plots of φ along x = 0 and y = 0: 4x4 mesh; solutions of class C00(Ωe),p-level of 8

Comparison with published works, discussion and remarks [10]

(1) For this problem, the Galerkin method with weak form produces rea-sonable results. Some oscillations are present in the computed solutionnear the boundaries of the domain.

(2) The QU formulation fails drastically due to the presence of crosswinddiffusion.

(3) Both SU1 and SU2 give good results.


(4) The p-version LSFEF produces excellent results for a rather coarse meshwithout the need of upwinding or any other adjustments. From Fig. 6.21,the remarkable accuracy of the computed solution φ is worth noting. Wenote that the p-version LSFEF has no dispersion or diffusion. The cosinehill distribution specified along OA is maintained as it gets advected inthe flow field.

6.3.5 Thermal boundary layer

In this model problem [2,5–7,10], we consider a rectangular domain 1×0.5with

φ = 1,

x = 0.0, 0 ≤ y ≤ 0.5y = 0.5, 0 ≤ x ≤ 1.0

φ = 0, y = 0, 0 ≤ x ≤ 1.0

(6.150)

as the boundary conditions. The velocity components u and v are given by

u = 2y, v = 0 (6.151)

No boundary conditions were imposed at the outflow (x = 1.0, 0 ≤ y ≤0.5). This model problem has been investigated by Franca et al. [8] fork = 7 × 10−4 which corresponds to Pe = 1428.57. In the present study weconsider Pe of 100, 1428.57, 10, 000 and 106. Figure 6.23 shows a schematic,and discretization using nine node p-version finite elements. We note that atx = y = 0, specification of φ is not unique. In the present study, we assumeφ = 1.0 at x = y = 0 and allow it to vary in a cubic manner from 1 to 0along the x-axis (0 ≤ x ≤ h1) for the element located at x = y = 0:

φ = φ(x),φ(0) = 1.0, φ(h1) = 0,

∂φ

∂x

∣∣∣x=0

= 0,∂φ

∂x

∣∣∣x=h1

= 0 (6.152)

0

0

0.5

1.0

Velocities:

0.41206250.41206250.150.0225

(a) Schematic and boundary conditions (b) Graded discretization (numbers indicate element lengths)

0.0016875

0.01125

0.075

0.20603125

0.20603125

h = 0.0033751

φ = 1

φ = 1

φ = 0

u = 2yv = 0

x

y

Figure 6.23: Schematic, boundary conditions and finite element mesh for thermal bound-ary layer problem


10-7

10-6

10-5

10-4

10-3

10-2

10-1

101

102

103

104

Sq

uare

ro

ot

of

resid

ual

fun

cti

on

al,

√I

Degrees of freedom

p=1

p=1

p=1

p=1

p=8

p=8

p=8

p=8

Pe = 100

Pe = 1,428.57

Pe = 10,000

Pe = 1,000,000

Figure 6.24: Square root of the residual functional versus degrees of freedom, solutionsof class C00(Ωe)

To accommodate φ in (6.152) we choose a rather small element at x =y = 0 as shown in Fig. 6.23(b). The p-levels in the ξ and η directions forall elements of the discretization are increased uniformly (pξ = pη = p).Figure 6.24 shows plots of

√I versus degrees of freedom (p-convergence of

the residual functional I for Peclet numbers of 100, 1428.57, 10, 000 and 106.At p-level of 8, the residual functional I has a magnitude lower than of theorder of O(10−4) confirming good accuracy of the computed solution. Plotsof the computed solution φ at p-level of 8 versus y at x = 0.003375, 0.025875,0.5 and 1.0 for Pe = 100, 1428.57, 10, 000 and 106 are shown in Fig. 6.25.

In addition, 3D plots of the computed solution φ over the rectangulardomain 1× 0.5 for all four values of Peclet numbers are shown in Fig. 6.26.

Remarks.

(1) A 5 × 5 graded coarse discretization works well for the wide range ofPeclet numbers considered (100–106).

(2) A small element size at x = y = 0 and the graded discretization isnecessitated due to: (1) the non-unique nature of φ at x = 0 and (2)higher Peclet numbers for which higher gradients of φ are present, i.e.sharper boundary layers.

6.4. SUMMARY 413

0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1

Dis

tance, y

Solution, φ

Pe:

At location, x=0.003375

100

1,428.57

10,000

106

0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1

Dis

tance, y

Solution, φ

Pe:

At location, x=0.025875

100

1,428.57

10,000

106

0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1

Dis

tance, y

Solution, φ

Pe:

At location, x=0.5

100

1,428.57

10,000

106

0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1

Dis

tance, y

Solution, φ

Pe:

At location, x=1.0

100

1,428.57

10,000

106

Figure 6.25: Plots of φ at different x locations: 5x5 mesh; solutions of class C00(Ωe),p-level of 8

(3) For Pe = 1428.57, the computed solutions are in good agreement withthe numerical solutions reported by Franca et al. [8].

(4) The numerical solution presented here are computed using a straight for-ward least squares finite element formulation based on a residual func-tional without any special treatments or use of upwinding methods. Themethod used here has excellent convergence characteristics.

6.4 Summary

Through simple model problems we are able to demonstrate the mostsignificant aspects of VIC and VC integral forms. In all applications withnon-self-adjoint differential operators, the findings reported for the 1D and2D convection-diffusion equations hold without exception. Another signif-icant aspect to note here is that all upwinding methods in whatever formhave no mathematical or physical basis and the solutions computed usingsuch methods are in fact not the true solutions of the BVP. Higher orderspaces play a significant role for such a BVP. The various issues and theresearch in this regard have been reported by Surana et al. [10]. Failure ofGM/WF due to VIC integral form in very simply model problems is more


0

1

0.5

0

0.2

0.4

0.6

0.8

1

1.2

φ

x

y

φ

0

1

0.5

0

0.2

0.4

0.6

0.8

1

1.2

φ

x

y

φ

(a) Plots of φ for Pe = 100 and 1, 428.57

0

1

0.5

0

0.2

0.4

0.6

0.8

1

1.2

φ

x

y

φ

0

1

0.5

0

0.2

0.4

0.6

0.8

1

1.2

φ

x

y

φ

(b) Plots of φ for Pe = 10, 000 and 106

Figure 6.26: 3D plots of φ for different values of Pe: 5x5 mesh; solutions of class C00(Ωe),p-level of 8

than sufficient to alert to refrain from using GM/WF for BVPs in two andthree dimensions in which the differential operators are non-self-adjoint. Im-portance of higher order spaces and higher degree local approximations isdemonstrated. Finite element formulations based on the residual functional(LSP) are shown to yield VC integral forms regardless of whether the mathe-matical model consists of a higher order system of differential equations or afirst order system of differential equations. VC integral forms yield uncondi-tionally stable finite element computational processes, hence are completelyfree of upwinding processes. Use of higher order spaces in differential modelscontaining the highest orders of the derivatives of the dependent variablesis most prudent choice as this choice results in the computational processeswith least number of degrees of freedom, hence most efficient computations.

Problems

6.1 Consider steady state one-dimensional convection-diffusion equation

dφ

dx− 1

Pe

d2φ

dx2= 0, 0 < x < 1 = Ω

φ(0) = 1, φ(1) = 0

This BVP represents dimensionless form of one-dimensional energy equation with unitvelocity field. Pe is called Peclet number and φ is dimensionless temperature.

PROBLEMS 415

(a) Construct weak form of the GDE over Ω using GM/WF. Give details of PV, SV, EBCand NBC as well as the nature of th resulting functionals. Establish VC or lack of itof the resulting integral form.

(b) Consider a two-node element Ωe with linear geometry as well as linear local approx-imation of φ over Ωe. Give details of the weak form over Ωe. Derive discretizedequations for an element with domain Ωe using natural coordinate system.

(c) Derive theoretical solution of the BVP.

(d) Perform numerical studies for Pe = 100 using element equations derived in (b). Con-sider progressively refined discretizations until you observe convergence of the numer-ical results.

(1) Tabulate nodal values of φ obtained from finite element calculations and thetheoretical values (at selected locations).

(2) Also tabulate dφdx

obtained from F.E. studies and the theoretical values (at selectedlocations).

(3) Plot graphs of φh versus x for all discretizations and compare with the theoreticalsolution φ (one single graph).

(4) Plot graphs of dφhdx

versus x for all discretizations and compare with the theoretical

solution dφdx

.(5) Calculate L2-norm of φh and dφh

dxfor all discretizations and compare with the

corresponding L2-norm obtained using the theoretical solution.

Discuss the F.E. results and their behavior with progressively refined meshes. Commenton the accuracy of the F.E. solution compared to theoretical solutions using the graphsand the L2-norms.

6.2 Consider one-dimensional steady state convection-diffusion equation in the dimen-sionless form.

dφ

dx− 1

Pe

d2φ

dx2= 0, 0 < x < 1 = Ω

φ(0) = 1, φ(1) = 0

(1)

The GDE (1) is referred to as the strong form of the GDE due to the fact that it containshighest order derivatives of the dependent variable φ.

(a) Construct a least squares formulation of (1) over Ω using φh as global approximationof φ over Ω. Do not substitute approximation φh in the integrals.

(b) Consider a discretization ΩT of Ω in which Ωe is a two-node element. Derive theequations for LSP over Ωe. Consider φeh of class C1(Ωe) for the two-node elementgiven by (for element map in ξ space: −1 ≤ ξ ≤ 1)

φeh(ξ) =( (1− ξ)

2+

(ξ3 − ξ)4

)φe1 +

( (1 + ξ)

2− (ξ3 − ξ)

4

)φe2

+( (ξ3 − ξ)

4− (ξ2 − 1)

4

) he2

(dφehdx

)1

+( (ξ3 − ξ)

4+

(ξ2 − 1)

4

) he2

(dφehdx

)2

(2)

in which φe1 and φe2 are nodal values of φ at nodes 1 and 2 and(dφehdx

)1

and(dφehdx

)2

are the nodal values of derivate of φeh at nodes 1 and 2. he is the element length forelement Ωe. Derive discretized equations of equilibrium for an element Ωe using localapproximation of class C1(Ωe) given by (2) and linear geometry mapping between xand ξ spaces.


(c) Perform numerical studies for Pe = 100 using element equations derived in (b). Con-sider five progressively refined discretizations.

(1) Tabulate φh, dφhdx

and d2φhdx2

for all discretizations and their theoretical values.(2) Plot graphs of (for all discretizations)

(i) φeh versus x and φ versus x(ii) dφh

dxversus x and dφ

dxversus x

(iii) d2φhdx2

versus x and d2φdx2

versus x

(3) Calculate L2-norms of φh and dφhdx

for all discretizations and compare them withtheir theoretical values. Plot graphs of the L2-norms versus dofs.

(4) Compute least squares functional I for each discretization and plot a graph of Iversus total dofs for each discretization on a single graph. What can you inferfrom these graphs?

Compare these solutions with GM/WF computed in 6.1. Provide a discussion ofresults and findings. [1, 9–13,16–18]


for non-self-adjoint operators in BVP. Int. J. Comp. Eng. Sci., 4(4):737–812, 2003.

[2] A. Brooks and T. J. R. Hughes. Streamline Upwinding/Petrov-Galerkin Formulationfor Convection Dominated Flows with Particular Emphasis on the IncompressibleNavier-Stokes Equations. Comp. Meth. in Appl. Mech. and Engg., 32:199–259, 1982.

[3] T. J. R. Hughes and M. Mallet. A new Finite Element Formulation for Computa-tional Fluid Dynamics; III. A Generalized Streamline Operator for MultidimensionalAdvection Diffusion Systems. Comp. Meth. in Appl. Mech. and Engg., 58:305–328,1986a.

[4] T. J. R. Hughes, M. Mallet, and A. Mizukami. A new Finite Element Formulationfor Computational Fluid Dynamics; II. Beyond SUPG. Comp. Meth. in Appl. Mech.and Engg., 54:341–355, 1986b.

[5] G. F. Carey and T. Plover. Variable Upwinding and Adaptive Mesh Refinement inConvection-Diffusion. Int. J. of Num. Meth. in Engg., 19:341–353, 1983.

[6] J. Christie, D. F. Griffiths, and A. R. Mitchell. Finite Element Method for SecondOrder Differential Equations with Significant First Derivatives. Int. J. of Num. Meth.in Engg., 10:1389–1396, 1976.

[7] J. Christie and A. R. Mitchell. Upwinding of Higher Order Galerkin Methods inConduction-Convection Problems. Int. J. of Num. Meth. in Engg., 12:1764–1771,1978.

[8] L. P. Franca, S. L. frey, and T. J. R. Hughes. Stabilized Finite Element Methods:I. Application to the Advenction-Diffusion Model. Comp. Meth. in Appl. Mech. andEngg., 95(3):253–276, 1992.

[9] D. Winterscheidt and K. S. Surana. p-Version Least Squares Finite Element Formula-tion for Convection-Diffusion Equation. International Journal of Numerical Methodsin Engineering, 36:111–133, 1993.

[10] K. S. Surana and J. S. Sandhu. Investigation of Diffusion in p-Version LSFE andSTFSFE formulations. Computational Mechanics, 16(3):151–169, 1995.

[11] K. S. Surana, O. Gupta, P. W. TenPas, and J. N. Reddy. h, p, k least squares finiteelement processes for 1d helmholtz equation. Int. J. Comp. Meth. in Eng. Sci. andMech., 7(4):263–291, 2006.


[12] Surana, K. S., Mahanti, R. K. and Reddy, J. N. Galerkin/Least Squares Finite Ele-ment Processes for BVPs in h, p, k Mathematical Framework. International Journalof Computational Engineering Sciences and Mechanics, 8:439–462, 2007.

[13] K. S. Surana, O. Gupta, and J. N. Reddy. Galerkin and least squares finite elementprocesses for 2d helmholtz equation in h, p, k framework. Int. J. Comp. Meth. in Eng.Sci. and Mech., 8:341–361, 2007.

[14] K. S. Surana. Advanced Mechanics of Continua. CRC/Taylor and Francis, 2015.

[15] J. M. Grygiel and P. A. Tanguy. Finite Element Solution for Advection DominatedThermal Flows. Comp. Meth. in Appl. Mech. and Engg., 93:277–289, 1991.




7

Non-Linear DifferentialOperators

7.1 Introduction

In this chapter we consider finite element processes for BVPs describedby non-linear differential operators. These BVPs could contain single ormulti-dependent variables and could be in single or multi-dimensional spaces.First, we review the properties of these differential operators as well as theproperties of the integral forms based on methods of approximation (pre-sented in Chapter 3).

(1) The non-linear differential operators are neither linear nor symmetric.

(2) The properties of the integral forms (presented in Chapter 3) resultingfrom various methods of approximation for non-linear differential oper-ators are summarized in the following.

(a) GM, PGM, WRM and GM/WF all yield variationally inconsistentintegral forms.

(b) However, in GM/WF the contribution of the even order terms in thedifferential operator to the total integral form becomes symmetricdue to transferring half of the order of differentiation from the de-pendent variable to the test function. When the even order terms inthe GDEs dominate the solution behavior, GM/WF is meritoriousover GM, PGM and WRM.

(3) The LSM or LSP when used for GDEs containing highest orders of thederivatives of the dependent variables or for those cast as a system offirst order PDEs using auxiliary equations can be made variationallyconsistent by:

(a) neglecting second variation of the residuals in the second variationof the residual functional

(b) solving the system of non-linear algebraic equations using Newton’slinear method (or Newton-Raphson method).

Justification for (a) and importance of (b) have been discussed in Chap-ter 3 in connection with classical methods of approximation.

419

420 NON-LINEAR DIFFERENTIAL OPERATORS

(4) In GM/WF one constructs and integral form based on FundamentalLemma of the calculus of variations, which is then converted to thediscretization. For each element of the discretization weak form is con-structed using integration by parts. Thus, if Aφ−f = 0 in Ω is the BVPand if ΩT =

⋃e Ωe is the discretization of Ω, then we can write




)(7.1)

in which φh is the approximation of φ over ΩT and φeh is the local approx-imation of φ over Ωe. For an element e with domain Ωe, we constructthe weak form

(Aφeh − f, v)Ωe = Be(φeh, v)− le(v), v = δφeh (7.2)

using the same procedure as described in Chapter 5 for self-adjoint op-erators. In (7.2), we note that

(a) Be(φeh, v) is linear in v but not linear in φeh due to the fact that thedifferential operator A is non-linear.

(b) Be(φeh, v) is obviously not symmetric (as bilinearity of Be(·, ·) isessential for symmetry).

(c) le(·) is linear.

We use the local approximation

φeh =n∑i=1

Ni δei = [N ]δe (7.3)

with v = δφeh = Nj , j = 1, . . . , n (7.4)

in which δei are nodal degrees of freedom and Ni are local approximationfunctions to obtain

Be(φeh) = [Ke]δe (7.5)

le(v) = P e+ F e (7.6)

In (7.5), [Ke] is a function of δe and Keij 6= Ke

ji, i.e. [Ke] is notsymmetric and P e is a vector of secondary variables resulting fromthe concomitant due to integration by parts.

(5) In least squares finite element processes we construct residual functionalI(φh) over ΩT . Let E = Aφh − f over ΩT and Ee = Aφeh − f over Ωe,then


Ie =∑e

(Ee, Ee)Ωe (7.7)

7.1. INTRODUCTION 421

The necessary conditions are given by

δI(φh) = (E, δE)ΩT =∑e

δIe = 2∑e

(Ee, δEe)Ωe

= 2∑e

ge = 2g = 0 (7.8)

whereg =

∑e

ge = 0 (7.9)

For an element e with domain Ωe, we construct ge = (Ee, δEe)Ωe andthen use (7.9) to obtain g. The necessary condition g = 0 mustbe used to calculate the nodal values δ =

⋃eδe for the discretiza-

tion ΩT . Since the differential operator A is non-linear, we must find aδ that satisfies (7.9) iteratively. Let δ0 be an assumed or startingsolution, then

g(δ0)6= 0 (7.10)

Letδ = δ0 + ∆δ (7.11)

be such that g(δ)

=g(δ0 + ∆δ)

= 0 (7.12)

Expandg(·)

in (7.12) in Taylor series about δ0 and retain only upto linear terms in ∆δ (Newton’s linear method or Newton–Raphsonmethod):

g(δ0 + ∆δ)

=g(δ0)

+∂g∂δ

∣∣∣δ0∆δ = 0 (7.13)

Therefore

∆δ = −[∂g∂δ

∣∣∣δ0

]−1g(δ0)

(7.14)

We note that

∂g∂δ

= δg =1

2δ2I = (δE, δE)ΩT + (E, δ2E)ΩT (7.15)

We approximate ∂g∂δ in (7.15) by (see Chapter 3)

1

2δ2I =

∂g∂δ

= δg ≈ (δE, δE)ΩT > 0 (7.16)

Hence, with this approximation we achieve variational consistency of theintegral form. Therefore, (7.14) becomes

∆δ = −[(δE, δE)ΩT

]−1

δ0

g(δ0)

(7.17)


and

(δE, δE)ΩT =∑e

(δEe, δEe)Ωe (7.18)

and we use line search to determine δ, i.e. δ = δ0 + α∆δ with0 < α ≤ 2 such that I(δ) ≤ I(δ0). We consider the iterative processconverged when |gi| ≤ ∆ ; i = 1, . . . where ∆ is a preset tolerance forzero.

Summary of steps:

1. Choose an assumed or starting solution δ02. I =

∑eIe =

∑e

(Ee, Ee)Ωe

3. g =∑ege =

∑e

(Ee, δEe)Ωe

4. (δE, δE)ΩT =∑e

(δEe, δEe)Ωe

5. ∆δ = −[(δE, δE)ΩT ]−1δ0g(δ0)

6. δ = δ0 + α∆δ; 0 ≤ α ≤ 2 such that I(δ) ≤ I(δ0)

7. If |gi| ≤ ∆, i = 1, . . . then the solution is converged. Otherwise setδ0 = δ and repeat steps 2 through 7.

In the following we consider a specific model problem to present detailsof finite element processes based on GM/WF and LSP.

7.2 One dimensional Burgers equation

The steady state 1D Burgers equation represents one dimensional form ofmomentum equation in viscous flows. The dimensionless form of 1D steadystate Burgers equation is given by

φdφ

dx− 1

Re

d2φ

dx2= 0 ∀x ∈ (0, 1) = Ω ⊂ R1 (7.19)

We consider the following boundary conditions:

φ(0) = 1, φ(1) = 0 (7.20)

in which Re = ρuLµ is the Reynolds number where ρ, u, L and µ are reference

density, velocity, length, and viscosity. Analytical solution of (7.19) and(7.20) is given by

φ = φ

(1− eφ Re(x−1)

1 + eφ Re(x−1)

)(7.21)

7.2. ONE DIMENSIONAL BURGERS EQUATION 423

in which φ is the solution of

φ− 1

φ+ 1= e−φ Re (7.22)

We note that for Re > 5, φ in (7.22) is very close to 1.0. For Re < 5, we needto solve (7.22) to find φ in (7.21). The behavior of φ and its derivatives issimilar to convection diffusion equation. Thus, for higher Re, the gradientsof φ become localized near x = 1.0 with progressively increasing values. Wenote that the analytical solution is of class C∞(Ω).

7.2.1 The Galerkin method with weak form

Let the Burgers equation be defined by Aφ − f = 0 in Ω, where A =φ ddx −

1Re

d2

dx2and f = 0. Let ΩT =

⋃e Ωe be the discretization of Ω such

that Ωe = [xe, xe+1] with he = xe+1 − xe, the element length. Let φh bethe approximation of φ in ΩT and φeh, the approximation of φ over Ωe withv = δφeh. Consider


∫Ωe

(Aφeh − f)v dx =

xe+1∫xe

(φehdφehdx− 1

Re

d2φehdx2

)v dx (7.23)

Using integration by parts for the second term in (7.23)


∫Ωe

(vφeh

dφehdx

+1

Re

dv

dx

dφehdx

)dx−

[v( 1

Re

dφehdx

)]xe+1

xe

(7.24)

Let

P e1 =(− 1

Re

dφehdx

)∣∣∣xe, P e2 =

( 1

Re

dφehdx

)∣∣∣xe+1

(7.25)

Therefore

(Aφeh−f, v)Ωe =

∫Ωe

(vφeh

dφehdx

+1

Re

dv

dx

dφehdx

)dx−v(xe)P

e1−v(xe+1)P e2 (7.26)

or(Aφeh − f, v)Ωe = Be(φeh, v)− le(v) (7.27)

where

Be(φeh, v) =

∫Ωe

(vφeh

dφehdx

+1

Re

dv

dx

dφehdx

)dx (7.28)

le(v) = v(xe)Pe1 + v(xe+1)P e2 (7.29)


Equation (7.26) is the weak form of (7.19), the BVP being considered.

(a) Be(φeh, v) is linear in v but is not linear in φeh.

(b) Be(·, ·) is not symmetric.

(c) le(v) is linear in v.

Approximation space

The BVP contains up to second order derivative of the dependent variableφ but, the weak form only contains up to the first order derivatives of φ andv.

(i) Admissibility of φh in Aφ−f = 0 in ΩT in the pointwise sense requiresφh ∈ Vh ⊂ Hk,p(ΩT ); k ≥ 3 for which k = 3 is minimally conforming.For this choice the integrals in the following are Riemann.

(Aφh−f, v)ΩT ⇔R

∑e

(Aφeh−f, v)Ωe ⇔R

∑e


)︸︷︷︸

R

(7.30)That is, if we choose φeh of class C2(Ωe) (or higher) then, all integrals in(7.30) are Riemann and hence, all three forms in (7.30) are equivalent.

(ii) Based on the weak form, if we choose φeh of class C1(Ωe), that is ifφeh ∈ Vh ⊂ H2,p(Ωe), then the following holds.

(Aφh − f, v)ΩTLL

∑e

(Aφeh − f, v)ΩeLL

∑e

(Be(φeh, v)− l(v)

)︸︷︷︸

R

(7.31)For this choice of local approximation φeh, the sum

∑e(B

e(·, ·)− le(·))holds in the Riemann sense but all other integral forms in (7.31) are inthe Lebesgue sense.

(iii) If we choose φeh ∈ Vh ⊂ H1,p(Ωe) then∑

e(Be(φeh, v)− l(v)) only holds


Local approximation φeh

Let

φeh =

n∑i=1

Ni(ξ) δei = [N ]δe (7.32)

be the local approximation of φ over Ωe → Ωξ, then

v = δφeh = Nj(ξ), j = 1, . . . , n (7.33)


in which Ni(ξ) are local approximation functions (see Chapter 8) and δei arenodal degrees of freedom. Substituting from (7.32) and (7.33) into the weakform (7.28) and (7.29) yields (with v = Nj)

Be(φeh, Nj) =

∫Ωe

[Nj

( n∑i=1

Ni δei

)( n∑i=1

dNi

dxδei

)+

1

Re

dNj

dx

( n∑i=1

dNi

dxδei

)]dx

(7.34)

le(Nj) = Nj(xe)Pe1 +Nj(xe+1 P

e2 (7.35)

or

Be(φeh, v) = [Ke]δe (7.36)

le(v) =

(P e1 )ξ=−1

0...0

(P e2 )ξ=+1

(7.37)

The coefficients Keij of [Ke] can be written in at least two possible ways.

That is, we can write (from (7.34))

Keij =

∫Ωe

(Ni φ

eh

dNj

dx+

1

Re

dNi

dx

dNj

dx

)dΩ, i, j = 1, . . . , n (7.38)

or

Keij =

∫Ωe

(NiNj

dφehdx

+1

Re

dNi

dx

dNj

dx

)dΩ, i, j = 1, . . . , n (7.39)

In (7.38), we see clearly that Keij 6= Ke

ij . In (7.39), it appears that Keij is

symmetric butdφehdx is a function of Ni. Thus, the integral of NiNj

dφehdx over

Ωe cannot be ensured to be positive.

C0 local approximations with p = 1

We consider a two-node linear element. For this choice we have

φeh =(1− ξ

2

)φe1 +

(1 + ξ

2

)φe2 =

2∑i=1

Ni φei = [N ]δe (7.40)

v = δφeh = Nj(ξ), j = 1, 2 (7.41)


and

x(ξ) =(1− ξ

2

)xe1 +

(1 + ξ

2

)xe2, J =

he2

= (xe2 − xe1)

dx = J dξ =he2dξ

(7.42)

It is instructive to consider (7.34) for j = 1 and 2. For j = 1 thenBe(φeh, v) becomes

Be(φeh, N1) =

1∫−1

[(1− ξ2

)((1− ξ2

)φe1 +

(1 + ξ

2

)φe2

)(− 1

2φe1 +

1

2φe2

)+

1

2Rehe

(φ1 − φ2

)]dξ (7.43)

and for j = 2 then Be(φeh, v) becomes

Be(φeh, N2) =

1∫−1

[(1 + ξ

2

)((1− ξ2

)φe1 +

(1 + ξ

2

)φe2

)(− 1

2φe1 +

1

2φe2

)+

1

2Rehe

(−φ1 + φ2

)]dξ (7.44)

which can be written as

Be(φeh, N1) = −1

3(φe1)2 +

1

3φe1 φ

e2 +

1

6(φe2)2 +

1

Rehe(φe1 − φe2)

Be(φeh, N2) = −1

6(φe1)2 − 1

3φe1 φ

e2 +

1

3(φe2)2 +

1

Rehe(−φe1 + φe2)

(7.45)

Writing (7.45) in the matrix form

Be(φeh, v) =

[− 1

3 φe1

13 φ

e1 + 1

6 φe2

− 16 φ

e1 − 1

3 φe2

13 φ

e2

]φe1

φe2

+

1

Rehe

[1 −1

− 1 1

]φe1

φe2

(7.46)

or

Be(φeh, v) =[[K1e(δe)

]+

1

Rehe

[K2e

]]δe = [Ke]δe (7.47)

If we consider a four-element uniform discretization with grid points 1-5 and function values at the grid points as φ1, . . . , φ5, then the assembled


equations would be

−1

6φ1 −

1

3φ2

−1

3φ1

0

0

0 0

0

0

1

3φ1 +

1

6φ2

1

3φ2 +

1

6φ3

1

3φ4 +

1

6φ5

−1

6φ4 −

1

3φ5

−1

6φ2 −

1

3φ3

−1

6φ3 −

1

3φ4

0

0

0

0

0

0

0

0

0

1

3φ5

1

3φ3 +

1

6φ4

φ4

φ3

φ2

φ5

φ1

φ4

φ1

φ3

φ2

φ5

P 12

+P 21

P 22

+P 31

P 32

+P 41

+1

Rehe

1

2

0

0

-1

0

-1

0

2-1

0

2

-1

-1

-1 1000

0 -1

=

P 42

P 110

0

0

-1

(7.48)

which can be written as[[K1(δ)

]+

1

Rehe

[K2]]δ = P (7.49)

Remarks

(1) From the BCs, we note that φ1 = 1 and φ5 = 0. Hence, after imposingBCs in (7.48), the reduced [K1] and [K2] would be (3 × 3) matrices inwhich [K1] would have zeros on the diagonals.

(2) [K1] is independent of he, that is, the non-linear convection part in [K]is not dependent on he, the discretization length.

(3) For a fixed discretization (i.e. fixed he), if one increases Reynolds numberRe then for very large Re, 1

Rehe[K2] → [0], a null matrix and as a

consequence [K] in (7.49) now only consists of [K1] and hence becomessingular.

(4) For a fixed discretization and for Reynolds numbers between 0 and ∞,the assembled [K] matrix progressively deteriorates for progressively in-creasing Reynolds number producing oscillations in the solution (in thesame fashion as in case of convection diffusion equation) and eventuallybecomes singular for large values of Re.

(5) Thus, we note that GM/WF leads to degeneracy of [K], a consequenceof variationally inconsistent integral form.


(6) The use of upwinding methods in this case can also be ruled out forthe same reasons as given in the case of convection-diffusion equation(chapter 6).

(7) Many numerical studies have been presented inHk,p(Ωe) spaces to demon-strate the various aspects discussed here (see references at the end of thechapter).

7.2.2 LSP based on residual functional

Let φeh be the local approximation of φ over typical element e with domainΩe of the discretization ΩT =

⋃e

Ωe of the spatial domain Ω = [0, 1], then

Ee = φehdφehdx− 1

Re

d2φehdx2

∀x ∈ Ωe (7.50)

is the residual equation for an element e. The residual functionals Ie and Ifor Ωe and ΩT are given by

Ie = (Ee, Ee)Ωe (7.51)

I =

M∑e=1

Ie =

M∑e=1

(Ee, Ee)Ωe (7.52)

and

δI = 2M∑e=1

(Ee, δEe)Ωe = 2∑e

ge = 2g = 0 (7.53)

δ2I ≈ 2M∑e=1

(δEe, δEe) = 2δg > 0 (7.54)

Since (7.50) is a second order ODE, the local approximation φeh ∈ Vh ⊂Hk,p(Ωe) ; k ≥ 3, p ≥ 2k − 1 in which k = 3 is the minimally conformingorder of the approximation space. For this choice of k all integrals over ΩT

hold in Riemann sense. Following the details of the LSP in the earlier sectionwe have

∆δ = −[(δE, δE)ΩT ]−1δ0g(δ0) (7.55)

We define[Ke] = (δEe, δEe)Ωe (7.56)

in which[Ke] = (δEe, δEe)Ωe (7.57)

The solution δ0 is an assumed or starting solution in the Newton’slinear method. A new updated solution δ is obtained using

δ = δ0+ α∆δ (7.58)

Details on how determining α have already been discussed.


Computational details

Let Ωξ = [−1, 1] be the map of Ωe in the natural coordinate space ξ andlet the local approximation be given by

φeh(ξ) =n∑i=1

Ni(ξ)φei = [N ]φe = [N ]δe (7.59)

Substituting (7.59) in (7.50)

Ee =( n∑i=1

Ni(ξ)φei

) ddx

( n∑j=1

Nj(ξ)φej

)− 1

Re

d2

dx2

( n∑i=1

Ni(ξ)φei

)(7.60)

or

Ee =([N ]δe

)([dNdx

]δe

)− 1

Re

[ d2

dx2

]δe

δEe = N([∂N

∂x

]δe

)+(

[N ]δe)dN

dx

− 1

Re

d2N

dx2

(7.61)

or

δEe = N∂φeh∂x

+ φeh

dNdx

− 1

Re

d2N

dx2

(7.62)

Thus [Ke] in (7.57) is defined. For an assumed solution δ0, φeh, anddφehdx

are known in (7.62), hence [Ke] is explicitly defined. Numerical values ofthe coefficients of [Ke] are calculated using Gauss quadrature.

7.2.3 LSP based on residual functional: first order system ofequations

Since the GDE for the BVP contains up to second order derivative ofthe dependent variable φ, the GDE must recast into a system of first orderODEs for C0 local approximation to be admissible. Let

τ =dφ

dx(7.63)

then the GDE (7.19) can be written as

φdφ

dx− 1

Re

dτ

dx= 0

τ − dφ

dx= 0

∀x ∈ (0, 1) = Ω ⊂ R1 (7.64)

with BCs: φ(0) = 1 and φ(1) = 0. Let φeh and τ eh be local approximations of φand τ over Ωe = [xe, xe+1]→ Ωξ = [−1, 1] an element e of the discretization


ΩT =⋃e Ωe. Then

Ee1 = φehdφehdx− 1

Re

dτ ehdx

Ee2 = τ eh −dφehdx

∀x ∈ (0, 1) = Ωe (7.65)

The residual functional Ie for an element e is given by

Ie = (Ee1, Ee1)Ωe + (Ee2, E

e2)Ωe (7.66)

For ΩT , we have

I =∑e

Ie =∑e

2∑i=1

(Eei , Eei )Ωe (7.67)

δI = 2∑e

2∑i=1

(Eei , δEei )Ωe = 2g = 2

(∑e

ge)

= 0 (7.68)

org =

∑e

ge = 0 (7.69)

Following the details of the LSP for nonlinear differential operators presentedin the earlier section we have

δ2I ∼= 2∑e

2∑i=1

(δEei , δEei )Ωe = 2δg > 0 (7.70)

and

∆δ = −1

2

[δ2I]−1

δ0g(δ0)

(7.71)

We define

[K] =1

2[δ2I] =

∑e

[Ke] (7.72)

in which

[Ke] =

2∑i=1



Consider equal order, equal degree local approximations. We choose two-node C0 linear local approximation for φeh and τ eh.

φeh =(1− ξ

2

)φe1 +

(1 + ξ

2

)φe2 = [N ]φe

τ eh =(1− ξ

2

)τ e1 +

(1 + ξ

2

)τ e2 = [N ]τ e

(7.74)


and

x(ξ) =(1− ξ

2

)xe +

(1 + ξ

2

)xe+1, he = xe+1 − xe, J =

he2

(7.75)

Let

δe =

φeτ e

=

φe1φe2τ e1τ e2

, nodal dofs for element e (7.76)

δEei =

∂Eei∂φe

∂Eei∂τe

=

∂Eei∂φe1∂Eei∂φe2

∂Eei∂τe1∂Eei∂τe2

, i = 1, 2 (7.77)

in which

Ee1 = φehdφehdx− 1

Re

dτ ehdx

, Ee2 = τ eh −dφehdx

(7.78)

Therefore

δEe1 = δφehdφehdx

+φehd(δφeh)

dx− 1

Re

d(δτ eh)

dx= N

dφehdx

+φeh

dNdx

− 1

Re

dNdx

δEe2 = δτ eh −

d(δφeh)

dx= N −

dNdx

(7.79)

Substituting for φeh, τ eh and δφeh = Nj ; j = 1, 2 and δτ eh = Nj ; j = 1, 2 in(7.79)

δEe1 =

N(dφehdx

)+ φeh

dNdx

− 1

Re

dNdx

, δEe2 =

dNdx

N

(7.80)

Therefore

[Ke] =

1∫−1

(δEe1δEe1T + δEe2δEe2T

) he2dξ (7.81)

We note that

[N ] =[

1−ξ2

1+ξ2

](7.82)[dN

dξ

]=[−1

212

](7.83)[dN

dx

]=

1

J

[dNdξ

]=

2

he

[−1

212

](7.84)

We remark that [Ke] is calculated at δ0, an assumed starting or guess

solution, at which (φeh)δ0 and(dφehdx

)δ0 appearing in (7.80) are known and

hence are constants as far as the integrals are concerned.


Remarks

(1) The formulation yields VC integral form with the approximation δ2I ∼=2(δE, δE)ΩT and the use of Newton’s linear method to find δ thatsatisfies the necessary condition g = 0.

(2) Computations of [Ke], and the solution procedure follows the standardprocedure.

Numerical studies using approximations in Hk,p(Ωe) spaces usingLSP

Since the differential operator is non-linear, GM/WF will yield VIC inte-gral form. Surana et al. [1–3] have presented numerical studies for GM/WFusing local approximations in Hk,p(Ωe) spaces.

A few important points to note: (1) LSP is variationally consistent andhence the resulting computational processes are always non-degenerate re-gardless of the choice of h, p and k, and the dimensionless parameters con-trolling the physics. (2) In this formulation we utilize auxiliary variable toreduce Burgers equation into a system of two first order differential equa-tions. (3) The minimally conforming space for LSP is H3,p(Ωe) for theoriginal differential equation without auxiliary variable, but if we permit

discontinuity of d2φdx2

at the inter-element boundaries, then it is possible touse H2,p(Ωe) spaces. Lack of convergence of such solutions will not be asurprise due to the fact that the local approximations in this space do notdescribe the physics of diffusion correctly in the computational process.

Consider a system of first order ODEs for Burgers equation

φdφ

dx− 1

Re

dτ

dx= 0, τ − dφ

dx= 0 (7.85)

Equation (7.85) is used in the majority of the published work. We makesome observations here.

(1) Consider a fixed discretization suitable for Re = Re1, if the Reynoldsnumber is increased from Re1 to Re2 > Re1 and if the discretization iskept fixed, then it is clear that more localized behavior of diffusion atRe = Re2 is no longer simulated correctly in the numerical process. Thusthe more localized behavior of diffusion at Re = Re2 must diffuse over alarger length in the computational process. Progressively increasing Remust continue to yield progressively diffused solutions until the solutioncannot diffuse anymore, one of the possible solutions will be a straightline connecting φ = 1 at x = 0 to φ = 0 at x = 1 which in fact is the

solution of the BVP d2φdx2

= 0 with φ(0) = 1 and φ(1) = 0.


(2) It is instructive to examine the analytical solution of the resulting differ-ential equation and its computed solution in case of 1D Burgers equationwhen Re =∞. In this case we have

φdφ

dx= 0, 0 < x < 1 (7.86)

φ(0) = 1, φ(1) = 0 (7.87)

We note that (7.86) is a first order non-linear differential equation,nonetheless we have two boundary conditions (7.87) to be satisfied by asolution of (7.86). Thus the BVP (7.86)–(7.87) has no unique solution.When attempting to compute a numerical solution of (7.86)–(7.87), oneobserved that boundary conditions (7.87) must be satisfied (since theyare imposed and thus no other choice). A solution φh of lowest degreethat satisfies the boundary conditions (7.87) is of course

φh = 1− x (7.88)

Obviously, φ in (7.88) is not the solution of (7.86) and (7.87) becauseit does not satisfy (7.86). However, LSP will compute such a solution(if the local approximation is linear) due to the fact that it satisfiesboundary conditions and LS minimization criterion, using (7.88) onefinds that

I(φh) =

∫Ω

(φhdφehdx

)2

dΩ =

1∫0

[(−1)(1− x)]2 dx =1

3(7.89)

which is in fact what the LSP yields. This nonzero I(φh) obviouslyconfirms that the GDE (7.86) is not satisfied by φh = 1 − x. Thetheoretical solution of 1D Burgers equation at Re→∞ is a step functionlocated at x = 1.0. Since (7.86) is a non-linear differential equation and(7.86) and (7.87) do not have a unique solution. For example

φh = 1−n∑i=1

xi

n, n = 2, 3, . . . (7.90)

satisfies the boundary conditions in (7.87). Each solution in (7.90), doesnot satisfy (7.86) and hence would yield a different value of the residualfunctional I. When solving (7.86) and (7.87) numerically, the nature ofφh obviously depends upon h, p, and k. Nonetheless, the computationalprocess in each case is non-degenerate.

(3) The motivation for the above exercise is to demonstrate the non-degeneratenature of LSP due to variational consistency of the integral form. Thecomputed solutions are always free of spurious oscillations and are al-ways physical within the limitations due to the choices of h, p, k, andRe (see [1] for numerical studies with varying h, p and k).


Graded discretizations: Re = 102 and Re = 106 (LSP)

In this section we consider numerical studies using the following gradeddiscretizations for Re of 102 and 106:

Element lengths for the 5-element graded discretization for Re = 100:0.779, 0.17, 0.017, 0.017, 0.017

Element lengths for the 13-element graded discretization for Re = 106:

0.4944443, 0.4944443, 10−2, 10−3, 5× 10−5, 5× 10−5,5× 10−6, 5× 10−6, 10−6, 10−7, 10−7, 10−7, 10−7

These discretizations by no means are optimal, but care has been taken

to ensure that these meshes permit resolution of the solution gradients. Weconsider solutions in H2,p(Ωe) and H3,p(Ωe) spaces. At Re = 102 the so-lution is relatively diffused; however, at Re = 106, the diffusion is isolatedover a length of O(10−6) near x = 1.0. First, we consider Re = 106. So-lutions are computed for p = 3 to p = 19 in both H2,p(Ωe) and H3,p(Ωe)spaces. Figures 7.1(a) and (b) show plots of computed solution φh versusx at different p-levels in H2,p(Ωe) space. The solutions in H2,p(Ωe) spaceprogressively approach the theoretical solution as p-level is increased. Atp = 19 the computed solution is in excellent agreement with the theoret-ical solution. Expanded plots of the solution shown in Figs. 7.1(b) revealthat the solutions are free of spurious oscillations. The solutions in H3,p(Ωe)space for various p-levels are shown in Figs. 7.2(a) and (b). We observe thesolutions to be diffused for p = 5, but for p ≥ 7 the computed solutionsare in good agreement with the theoretical solution. Expanded plots of φhversus x are shown in Fig. 7.2(b). We note that the solutions at all p-levelsare free of oscillations.

Results for Re = 102 (diφhdxi

versus x, i = 0, . . . , 3) H2,p(Ωe) and H3,p(Ωe)spaces are shown in Figs. 7.3(a) and (b) and 7.4(a) and (b). We observeminor oscillations in the solution at lower p-levels (< 6) near x = 1.0 in bothspaces, indicating inadequacy of the mesh in resolving solution gradientsat these p-levels. As the p-level is increased the solution and its gradientsconverge to the theoretical values. Oscillations in the expanded views com-pletely disappear at higher p-levels. The purpose of presenting these resultsis to demonstrate that even though the solution is quite diffused at Re = 102,but the inadequacy of the mesh and p-levels may results in oscillatory solu-tions which are eliminated in this case only by p-level increase. On the otherhand oscillatory solutions due to VIC integral form result due to degeneracyof the computational process.


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

φ

x

p-convergence

H2,p

(—

Ωe ) Spaces (Re=10

6)

p=3

p=5

k=6

k=7

k=9

k=19

(a) Function value over entire domain

1

1.005

1.01

1.015

0.99995 0.99996 0.99997 0.99998 0.99999 1

φ

x

p-convergence

H2,p

(—

Ωe ) Spaces (Re=10

6)

p=9

p=11

k=13

k=19

theoretical

(b) Function value: expanded view

Figure 7.1: LSP based on residual functional; 1D Burgers equation; φ versus x: Re = 106,k = 2

Dependence of the solution on order k of space Hk,p(Ωe) and k,pk-convergence

In this section we present numerical studies to demonstrate that the orderof space k in Hk,p(Ωe) is an independent parameter in all finite elementcomputations in addition to h and p. Dependence of the solution φh on


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

φ

x

p-convergence

H3,p

(—

Ωe ) Spaces (Re=10

6)

p=5

p=7

k=8

k=9

k=19


1

1.005

1.01

1.015

0.99995 0.99996 0.99997 0.99998 0.99999 1

φ

x

p-convergence

H3,p

(—

Ωe ) Spaces (Re=10

6)

p=9

p=11

k=13

k=19

theoretical



k in the Galerkin processes has already been demonstrated in the resultspresented by Surana et al. [1]. LSP are based on minimization of error orresidual functional I, hence examination of the behavior of I versus degreesof freedom is essential in determining its dependence on h, p, and k. Weconsider Hk,p(Ωe) ; k = 2, 3, . . . spaces and compute sequences of solutions


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

φ

x

p-convergence

H2,p

(—

Ωe ) Spaces (Re=100)

p=3

p=4

k=5

k=7

k=19


0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

0.8 0.85 0.9 0.95 1

φ

x

p-convergence

H2,p

(—


p=3

p=4

k=5

k=7

k=19

theoretical



for progressively increasing p-levels beginning with the lowest admissible p-level and increasing it up to 19 in each space. We consider Re = 102 andRe = 106 and use the discretizations given earlier.


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

φ

x

p-convergence

H3,p

(—


p=5

p=7

k=8

k=9

k=19


0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

0.8 0.85 0.9 0.95 1

φ

x

p-convergence

H3,p

(—


p=5

p=6

k=7

k=9

k=19

theoretical



Behavior residual functional I versus dof

Figures 7.7 and 7.8 show graph of residual functional I versus degrees offreedom for Re = 102 and Re = 106, respectively. We make the followingremarks.


1e-14

1e-12

1e-10

1e-08

1e-06

1e-04

1e-02

1e+00

10 100

Resid

ual

Fu

ncti

on

al,

I

Degrees of Freedom

p-convergence

k-convergence

Hk,p

(—


k=2

k=3

k=4

k=5

k=6

k=7

Figure 7.5: Residual functional I versus dof: Re = 100

1e-10

1e-08

1e-06

1e-04

1e-02

1e+00

10 100

Resid

ual

Fu

ncti

on

al,

I

Degrees of Freedom

p-convergence

k-convergence

Hk,p

(—

Ωe ) Spaces (Re=10

6)

k=2

k=3

k=4

k=5

k=6

k=7

Figure 7.6: Residual functional I versus dof: Re = 106

(1) For a fixed k, the order of space and hence the global smoothness, therate of convergence of I increases as p-level is increased.

(2) As the order of the space k is increased from k to k + 1 the computedvalues of I decrease in the space of order k + 1 compared to the space


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

φ

x

Re=10

Re=100

Re=1000

Figure 7.7: LSP based on residual functional; 1D Burgers equation; φ versus x: Re = 100

0.995

0.996

0.997

0.998

0.999

1

0.999 0.9992 0.9994 0.9996 0.9998 1

φ

x

Re=104

Re=105

Re=106

Figure 7.8: LSP based on residual functional; 1D Burgers equation; φ versus x: Re = 106

of order k, regardless of the values of k, indicating improved solution inthe space of order k + 1 compared to the space of order k.

(3) While the rate of convergence of I (i.e., slope of I versus dof graph) isincreased slightly with the change in the order of the space at lower p-


levels, at higher p-levels (p ≥ 12), I versus dof curves are almost parallelto each other for all values of k, indicating same rate of convergence forall values of k.

(4) For fixed p-level (i.e., p = constant) and for the fixed discretization (i.e.constant h) we could study the k-convergence of the residual functionalI (lines with dots). Along these lines, h and p are fixed and only k,the order of the space Hk,p(Ωe) is changing, hence these lines representk-convergence of the functional I for LSP. Dependence of I on the orderof the space k, that is, degree of global smoothness is quite obvious.At lower p-levels (3 − 5), the dependence of I on k is not as strongas it is for p > 5. With increasing p-levels, the dependence of I onk becomes even stronger. This confirms that the order of the spacek in Hk,p(Ωe) or in other words, the degree of global smoothness isundoubtedly an independent parameter in addition to h and p. Weobserve similar dependence of all quantities on k (not shown here).

(5) pk-convergence of the error functional I (and likewise other quantities)is perhaps most illustrative of the influence of the order of space k onthe convergence rate of I. pk-convergence can be viewed in at leasttwo different ways: (a) for increasing k as well as increasing p but p =2k − 1, i.e., minimum p-level for the order of the space k. In this casewe observe much higher rate of convergence of pk-processes compared top-convergence of I in any of the spaces (i.e, any value of k in Hk,p(Ωe))(b) Perhaps, the most dramatic is the pk-convergence of the residualfunctional I in which p and k both change in such a way that the totaldegrees of freedom for the discretization do not vary much. Almostvertical lines show such behaviors for various combination of p and k inwhich total dof do not change significantly but I decreases substantially.We observe exceptionally high slopes of such lines, even for very low totaldofs. With increasing total dofs, slopes of these lines increase showingan increase in the convergence rate of I. At close to 100 dofs, thepk-convergence graphs of I are almost vertical straight lines, indicatingexponential convergence rate. It is worth noting that such behavior is theconsequence of the fact that for fixed dofs, an increase in k would permitan increase in p-level, which in fact is responsible for the improvementin the value of I.

(6) We observe that I continues to decrease in progressively higher orderspaces for a given dofs, indicating improved performance of the LSP inhigher order spaces. That is, the best approximation property of theLSP in E-norm gets progressively better in higher order spaces.


7.3 Fully developed flow of Giesekus fluid betweenparallel plates (polymer flow)

In this model problem we consider fully developed flow of an incompress-ible Giesekus fluid [4] between parallel plates. Figure 7.9 shows a schematicof the flow. The plates are separated by a distance of 2H. The originof the xy-coordinate is located at the center of the plates and the posi-tive x-direction is the direction of the flow. The flow is pressure driven i.e.∂p∂x (negative) is specified. We assume the fluid to be incompressible. Weconsider contravariant Cauchy stress tensor and Almansi strain tensors asconjugate measures of the stress and strain tensors in Eulerian description.This yields upper convected Giesekus constitutive model. The mathemati-cal model describing the flow physics (for incompressible case with isother-mal flow assumption) consists of x-momentum equation and the constitutiveequations. The continuity equation in this case is satisfied identically. If wedecompose the contravariant Cauchy stress tensor in equilibrium stress anddeviatoric contravariant Cauchy stress tensor, then the equilibrium stress ismechanical pressure p and the deviatoric contravariant Cauchy stress tensorbecomes a dependent variable in the constitutive theory.

Flow direction

center line

(UCG1)

(UCG2)

B

BCs at B:

u = 0 , p = 0

x

H

y

A

BCs at A: ∂u

∂y= 0

τxy = 0

τ pxy = 0

Figure 7.9: Schematic of 1D fully developed flow between parallel plates (half domain)

We begin with all quantities with their usual dimensions (units) in thedevelopment of the mathematical model and then non-dimensionalize themusing the following. The quantities with the subscript zero are the reference

7.3. FLOW OF GIESEKUS FLUID BETWEEN PARALLEL PLATES 443

quantities.

x =x

L0), η = η =

η

η0), ηs =

ηsη0, ηp =

ηpη0

, ρ =ηsρ0, u =

u

u0axial velocity

p =p

p0, τ =

τ

τ0and p0 = τ0 =

ρ0y

20 ; Characteristic kinetic energy

orµ0u0L0

, Characteristic kinetic stress

(7.91)

We choose the large of the two for p0 (and τ0). This results in dimensionlessform of the mathematical model given in the following:

Momentum equations:

In the absence of body forces( p0

ρ0u20

)∂p∂x−( τ0

ρ0u20

)∂τxy∂y

= 0 (7.92)( p0

ρ0u20

)∂p∂y−( τ0

ρ0u20

)∂τyy∂y

= 0 (7.93)

Giesekus constitutive model:

We consider the upper convected Giesekus constitutive model derived[4] in deviatoric Cauchy stress tensor τττ . In this model, the first convectedtime derivative of τττ , the deviatoric contravariant Cauchy stress tensor, isa dependent variable in the constitutive theory. Dimensionless form of theconstitutive model is given by

τxx − 2De τxy∂u

∂y− αDe

η

(L0τ0

u0η0

)((τ2xx + (τxy)

2)

= 0

τyy − αDe

η

(L0τ0

u0η0

)((τ2yy + (τxy)

2)

= 0

τxy − 2De τyy∂u

∂y− αDe

η

(L0τ0

u0η0τxy

)(τxx + τyy) = η

L0τ0

u0η0τxy

∂u

∂y

(7.94)

Equations (7.92)–(7.94) constitute the complete mathematical model in de-pendent variables u, p, τxx, τyy and τxy for fully developed flow between

parallel plates. The flow is assumed to be pressure driven i.e. ∂p∂x is specified

as input data. Since p0 = τ0 and ∂p∂x is constant, from (7.92) we can deter-

mine τxy by integrating (7.92) with respect to y and using the BC: τxy = 0at y = 0 (due to symmetry).

τxy =(∂p∂x

)y (7.95)


A theoretical solution for the remaining dependent variables is not possibledue to complexity of the constitutive model. We consider LSP for (7.92) -(7.94) in Hk,p(Ωe) spaces; k ≥ 2, p ≥ 2k − 1. Theoretical solution for τxygiven by (7.95) will serve as one of the checks on the validity and accuracy ofcomputations. We consider equal order, equal degree local approximationsfor all dependent variables: u, p, τxx, τyy, τxy. With the choice p0 = τ0, thequantities in the brackets in (7.92) and (7.93) can be factored and eliminated,but we do not do so in the following in order to maintain generality.

Consider a discretization ΩT =⋃e

Ωe of Ω = (0, 1). Let ueh, peh, (τyy)eh

and (τxy)eh be local approximations of u, p, τxx, τyy and τxy. By substituting

the local approximations in (7.92) - (7.94) and using the notations

c1 =p0

ρ0u20

, c2 =τ0

ρ0u20

, c1 =L0τ0

u0η0and

1

c3=u0η0

L0τ0

we obtain the residual equations for an element e with domain Ωe.

Ee1 = c1∂peh∂x− c2

∂(τxy)eh

∂y

Ee2 = c1∂peh∂y− c2

∂(τyy)eh

∂y

Ee3 = (τxx)eh − 2De(τxy)eh

∂ueh∂y− αDe

ηc3

(((τxx

eh

)2+((τxy

)2)(7.96)

Ee4 = (τyy)eh − α

De

ηc3

(((τyy)

eh

)2+((τxy)

eh

)2)Ee5 = (τxy)

eh − 2De(τyy)

eh

∂ueh∂y− αDe

ηc3(τxy)

eh

((τxx)eh + (τyy)

eh

)− η

c3

∂ueh∂y

The residual functional Ie for an element e is given by

Ie =

5∑i=1

(Eei , Eei )Ωe (7.97)

The functional I for the discretization ΩT can be written as

I =∑e

Ie =∑e

( 5∑i=1

(Eei , Eei )Ωe

)(7.98)

Therefore

δI =∑e

δIe = 2∑e

( 5∑i=1

(Eei , δEei )Ωe

)= 2

∑e

ge = 2g = 0 (7.99)

δI ≈∑e

δIe = 2∑e

( 5∑i=1

(δEei , δEei )Ωe

)= 2δg > 0 (7.100)


and

∆δ = −[δ2I]−1δ0g(δ0)

(7.101)

We define

[K] = [δ2I] =∑e

[Ke] (7.102)

in which

[Ke] =

5∑i=1



Let Ωξ = [−1, 1] be the map of Ωe in the natural coordinate space ξ andlet

ueh =n∑i=1

Ni uei = [N ]ue

peh =n∑i=1

Ni pei = [N ]pe

(τxx)eh =n∑i=1

Ni (τxx)ei = [N ](τxx)e

(τyy)eh =

n∑i=1

Ni (τyy)ei = [N ](τyy)e

(τxy)eh =

n∑i=1

Ni (τxy)ei = [N ](τxy)e

(7.104)

in which Ni = Ni(ξ); i = 1, 2, . . . , n. Substituting (7.104) in the residualequations and noting that ∂p

∂x is given i.e. known, we obtain

Ee1 = c1∂p

∂x− c2

([∂N∂y

](τxy)

e)Ee2 = c1

([∂N∂y

]pe

)− c2

([∂N∂y

](τyy)

e)Ee3 =

([N ]

(τxx)e)− 2De

([N ]

(τxy)e)([∂N

∂y

]ue

)− αDe

ηc3

(([N ]

(τxx)e)2

+(

[N ]

(τxy)e)2)

(7.105)


Ee4 =(

[N ]

(τyy)e)− αDe

ηc3

(([N ]

(τyy)e)2

+(

[N ]

(τxy)e)2)

Ee5 =(

[N ]

(τxy)e)− 2De

([N ]

(τyy)e)([N ]ue

)− αDe

ηc3

([N ]

(τxx)e

+ [N ]

(τyy)e)− η

c3

([N ]ue

)Let

δeT =[ueT , peT

(τxx)e

T (τyy)

eT

(τxy)eT ]

(7.106)

be the nodal dofs for an element e, then

δEei T =

∂Eei∂δe

=

[∂Eei∂ue

T,

∂Eei∂pe

T,

∂Eei

∂(τxx)e

T,

∂Eei

∂(τyy)e

T,

∂Eei

∂(τxy)e

T], i = 1, 2, . . . , 5 (7.107)

Using (7.105)–(7.107) we can obtain∂Ee1∂δe

=

[0T , 0T , 0T , 0T , −c2

[∂N∂y

]]∂Ee2∂δe

=

[0T , c1

[∂N∂y

], 0T , −c2

[∂N∂y

], −c2

[∂N∂y

], 0T

]∂Ee3∂δe

=

[−2De(τxy)

eh

[∂N∂y

], 0T ,

[N ]− 2αDe

η(τxx)eh[N ]− 2Deueh[N ]− 2

αDe

αc3(τxy)

eh[N ], 0T

]∂Ee4∂δe

=

[0T, 0T, 0T, −2αDe

ηc3(τyy)

eh[N ], [N ]−2

αDe

αc3(τxy)

eh[N ]

]∂Ee5∂δe

=

[−2De(τyy)

eh[N ]− η

c3[N ], 0T , −αDe

ηc3(τxy)

eh[N ],

− 2Deueh[N ]− αDe

αc3(τxy)

eh[N ], [N ]− αDe

ηc3

((τxx)eh + (τxx)eh

)[N ]

](7.108)

Therefore

[Ke] =

∫Ωe

δEei δEei T dΩ (7.109)

Numerical values of the coefficients of [Ke] are calculated using Gauss quadra-ture with δ0, a known starting solution which i assumed zero in the fol-lowing.


Numerical studies

We consider fully developed flow of an incompressible Giesekus fluid [4]with the following material coefficients that are constant.

ρ = 800kg

m3, ηs = 0.002 Pa.s, ηp = 1.426 Pa.s, λ = 0.006 s, α = 0.15

We choose

H = L0 = 0.003175 m, ρ0 = ρ = 800kg

m3, η0 = η = 1.426 Pa.s

which gives

H = L0 = 1, p0 = τ0 = ρu2o = 800u2

o, Re =ρL0u0

η0= 1.7812u0,

De =λu0

L0= 18.89764u0

and we also choose u0 = 0.5 m/sec for which Re = 0.8906 and De = 9.45. Agood discretization of the spatial domain 0 ≤ y ≤ 1 is important in ensur-ing satisfactory convergence of the Newton’s linear method for the system ofnon-linear algebraic equations and good accuracy of the computed solutions.With progressively increasing ∂p

∂x , we expect a constant (approximately) ve-locity core or plug at the center of the flow. This suggests a highly biasedfiner discretization towards the walls. A four-element graded mesh with el-ement length of 0.1, 0.15, 0.225, and 0.525 starting from the wall is used inthe present study. The local approximations are p-version (3-node elements)in higher order spaces. Initial p-convergence studies with this discretizationsuggest p = 11 and k = 2 (local approximations of class C1(Ωe

x) to be suffi-cient for good accuracy of results. For this choice of mesh, p-level and ordero the space (k = 2), the residual or least squares functional values remainO(10−9) to O(10−31) indicating that the PDEs are satisfied accurately inthe pointwise sense as the integrals are Riemann when the local approxima-tions for u, p, τxx, τyy, and τxy are of class C1(Ωe). Newton’s linear methodused for solving the non-linear algebraic equations converges in less that 10iterations for all results presented in this section.

In numerical studies we begin with ∂p∂x = −0.1 for which a converged

solution is obtained and then progressively increase it up to ∂p∂x = −0.275

using a continuation procedure in which converged solutions at lower ∂p∂x

are used as initial (or starting) solution in the Newton’s linear method.Figure 7.10 shows graphs of velocity u versus y for different values of ∂p

∂x .

Graphs of velocity gradient ∂u∂y versus y for different values of ∂p

∂x are shown

in Fig. 7.11. As expected with progressively increasing ∂p∂x (corresponding


1

1.2

1.4

1.6

1.8

2

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4

Dis

tan

ce, y

Velocity, u

∂p/∂x values:

C1(

—

Ωe

x) ; p=9

-0.1

-0.2

-0.25

-0.265

-0.275

Figure 7.10: Velocity u versus distance y

1

1.2

1.4

1.6

1.8

2

-30 -28 -26 -24 -22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0

Dis

tan

ce, y

Velocity gradient, du/dy

∂p/∂x values:

C1(

—

Ωe

x) ; p=9

-0.1

-0.2

-0.25

-0.265

-0.275

Figure 7.11: Velocity gradient dudy

versus distance y

to increasing flow rate) almost a constant velocity core is observed in thecenter of the flow. I values of O(10−9) or lower, and use of C1(Ωe) localapproximations ensure that the computed solutions satisfy GDEs accuratelyin the pointwise sense. Plots of τxx, τyy and τxy versus y for different values

of ∂p∂x are shown in Figs. 7.12 to 7.14. Computed τyy is in perfect agreement


with the theoretical solution (7.95). This model problem demonstrates thesignificance of VC integral forms in which regardless of the complexity ofthe mathematical model, the computations can be performed in a routinemanner with good convergence characteristics and good accuracy.

1

1.2

1.4

1.6

1.8

2

0 1 2 3 4 5 6 7 8 9 10

Dis

tan

ce, y

Stress, τxx

∂p/∂x values:

C1(

—

Ωe

x) ; p=9

-0.1

-0.2

-0.25

-0.265

-0.275

Figure 7.12: Stress component τxx versus distance y

1

1.2

1.4

1.6

1.8

2

-0.12 -0.11 -0.1 -0.09 -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0

Dis

tan

ce, y

Stress, τyy

∂p/∂x values:

C1(

—

Ωe

x) ; p=9

-0.1

-0.2

-0.25

-0.265

-0.275

Figure 7.13: Stress component τyy versus distance y


1

1.2

1.4

1.6

1.8

2

-0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0

Dis

tan

ce, y

Stress, τxy

∂p/∂x values:

C1(

—

Ωe

x) ; p=9

-0.1

-0.2

-0.25

-0.265

-0.275

Figure 7.14: Stress component τxy versus distance y

7.4 2D steady-state Navier–Stokes equations

In this section we consider example problems that require numerical sim-ulation of two-dimensional Navier-Stokes equations for isothermal, incom-pressible, Newtonian fluids. The dimensionless form of the two-dimensionalsteady state Navier–Stokes equations in xy frame (Eulerian description) con-sisting of continuity equation, momentum equations and the constitutiveequations for isothermal, incompressible Newtonian fluids are given by (inthe absence of sources and sinks) [5]

ρ

(∂u

∂x+∂v

∂y

)= 0

ρ

(u∂u

∂x+ v

∂u

∂y

)+

p0

ρ0u20

∂p

∂x− τ0

ρ0u20

(∂τxx∂x

+∂τxy∂y

)= 0

ρ

(u∂v

∂x+ v

∂v

∂y

)+

p0

ρ0u20

∂p

∂y− τ0

ρ0u20

(∂τxy∂x

+∂τyy∂y

)= 0

(7.110)

where

τxx =

(η0u0

L0τ0

)2η∂u

∂x

τyy =

(η0u0

L0τ0

)2η∂v

∂y

τxy =

(η0u0

L0τ0

)η

(∂u

∂y+∂v

∂x

) ∀x, y ∈ Ω ⊂ R2 (7.111)

7.4. 2D STEADY-STATE NAVIER-STOKES EQUATIONS 451

in which u, v are velocity in x and y directions, p is mechanical pressure,τxx, τyy, τxy are components of deviatoric Cauchy stress tensor, ρ is densityand η is viscosity (assumed constant), all are dimensionless including x, y.We have used

x =x

L0, y =

y

L0, ρ =

ρ

ρ0, η =

η

η0, τij =

τijτ0, p =

p

p0(7.112)

in which all quantities with ˆ (hat) have their usual dimensions and thequantities with subscript zero are reference quantities. We note that p andτij both have units of F

L2 (F is force and L is length), hence τ0 and p0 mustnot be chosen independent of each other. In fact, once we choose F0 andL0, τ0 and p0 both may be defined as F0

L20. We use the following: τ0 = p0 =

max(ρ0u20,ηu0L0

). ρu20 is characteristic kinetic energy and η0u0

L0is characteristic

viscous stress. For inertia dominated flows, generally ρ0u20 >

η0u0L0

will hold.We note that when

(1) p0 = τ0 = ρ0u20,

τ0

ρ0u20

=p0

ρu20

= 1 andη0u0

L0τ0=

1

Re

(2) p0 = τ0 =η0u0

L0,

τ0

ρ0u20

=p0

ρu20

=1

Reand

η0u0

L0τ0= 1

(7.113)

where Re is Reynolds number.

Remarks

(1) Equations (7.111) is a first order system of non-linear PDEs in depen-dent variables u, v, p, τxxτyy, τxy that naturally result in this form di-rectly from the conservation laws and the constitutive theory. Equations(7.111) is a system of first order non-linear PDEs in u, v, p, τxxτyy, τxy.

(2) In this particular case it is possible to substitute in stresses in the mo-mentum equations. The resulting terms can be simplified using thefollowing relations that can be obtained by differentiating the continuityequation with respect to x and y.

∂2v

∂x∂y= −∂

2u

∂x2

∂2u

∂y∂x= −∂

2v

∂y2

(7.114)

The resulting continuity (unchanged) and momentum equations can bewritten as


ρ

(∂u

∂x+∂v

∂y

)= 0

ρ

(u∂u

∂x+∂u

∂y

)+

(p0

ρ0u20

)∂p

∂x−( η

Re

)(∂2u

∂x2+∂2u

∂y2

)= 0

ρ

(u∂v

∂x+∂v

∂y

)+

(p0

ρ0u20

)∂p

∂y−( η

Re

)(∂2v

∂x2+∂2v

∂y2

)= 0

(7.115)

for all x, y ∈ Ω ⊂ R2. Equations (7.115) are also a system of non-linearpartial differential equations that contain first order derivatives of themechanical pressure but up to second order derivatives of the velocitiesu and v.

(3) Since both systems of PDEs ((7.111) and (7.115)) are systems of non-linear PDEs, we only consider least squares finite element processes forboth them as only the LSP in this case yields VIC integral forms.

7.4.1 LSP based on residual functional: first order system ofPDEs

Let ueh, veh, p

eh, (τxx)eh, (τyy)

eh, (τxy)

eh be the local approximations of u, v,

p, τxx, τyy, τxy over an element Ωe of the discretization ΩT =⋃e Ωe of the

domain Ω ⊂ R2. For generality, we consider unequal order unequal degreelocal approximations.

ueh =nu∑i=1

Nui u

ei = [Nu] ue

veh =nv∑i=1

Nui v

ei = [Nv] ve

peh =

np∑i=1

Npi p

ei = [Np] pe

(τxx)eh =

nτxx∑i=1

N τxxi (τxx)ei = [N τxx ] (τxx)e

(τyy)eh =

nτyy∑i=1

Nτyyi (τyy)

ei = [N τyy ] (τyy)e

(τxy)eh =

nτxy∑i=1

Nτxyi (τxy)

ei = [N τxy ] (τxy)e

(7.116)


By replacing u, v, p, τxx, τyy, and τxy with their local approximation in(7.111) we obtain the residual equations.

Ee1 = ρ

(∂ueh∂x

+∂veh∂y

)Ee2 = ρ

(ueh∂ueh∂x

+ veh∂ueh∂y

)+

p0

ρ0u20

− τ0

ρ0u20

(∂(τxx)eh∂x

+∂(τxy)

eh

∂y

)Ee3 = ρ

(ueh∂veh∂x

+ veh∂veh∂y

)+

p0

ρ0u20

− τ0

ρ0u20

(∂(τxy)

eh

∂x+∂(τyy)

eh

∂y

)Ee4 = (τxx)eh −

(η0u0

L0τ0

)2η∂ueh∂x

Ee5 = (τyy)eh −

(η0u0

L0τ0

)2η∂veh∂y

Ee6 = (τxy)eh −

(η0u0

L0τ0

)η

(∂ueh∂y

+∂veh∂x

)

(7.117)

Upon substituting for local approximations from (7.116) in (7.117) we canobtain explicit forms of the residual equations. Let us define nodal degreesof freedom δe for an element e as

δeT =[ueT , veT , peT , (τxx)eT , (τyy)eT , (τxy)eT

](7.118)

Then

δEei T =

[∂Eei∂ue

T,

∂Eei∂ve

T,

∂Eei∂pe

T,

∂Eei

∂(τxx)e

T,

∂Eei∂(τyy)e

T,

∂Eei

∂(τxy)e

T], i = 1, 2, . . . , 6 (7.119)

and the element matrix [Ke] is given by

[Ke] =

6∑i=1

∫Ωe

∂Eei∂δe

∂Eei∂δe

TdΩ (7.120)

Remaining details follow the standard procedure presented earlier in con-nection with other model problems and hence are omitted here. In this caseleast admissible order k of the approximation space is 2 if the integrals overΩT are to be in Riemann sense. However, k = 1 i.e. local approxima-tion of class C0(Ωe) for all dependent variables are permissible if we acceptthe integrals in Lebesgue sense over ΩT . In applications generally local ap-proximations of class C0(Ωe) with same p-levels are used for all dependentvariables.


7.4.2 LSP based on residual functional: higher ordersystems of PDEs

Let ueh, veh, p

eh be the local approximations of u, v, and p over an element

Ωe of the discretization ΩT =⋃e Ωe of Ω ⊂ R2. In this case also we consider

unequal order, unequal degree local approximations for u, u and p.

ueh =nu∑i=1

Nui u

ei = [Nu] ue

veh =nv∑i=1

Nui v

ei = [Nv] ve

peh =

np∑i=1

Npi p

ei = [Np] pe

(7.121)

By replacing u, v and p with their approximations in (7.115) we obtainresidual equations

Ee1 = ρ

(∂ueh∂x

+∂veh∂y

)Ee2 = ρ

(ueh∂ueh∂x

+ veh∂ueh∂y

)+

p0

ρ0u0

∂peh∂x− τ0

L0ρ0u0

(∂2ueh∂x2

+∂2ueh∂y2

)Ee3 = ρ

(ueh∂veh∂x

+ veh∂veh∂y

)+

p0

ρ0u0

∂peh∂y− τ0

L0ρ0u0

(∂2veh∂x2

+∂2veh∂y2

) (7.122)

for all x, y ∈ Ω ⊂ R2. Upon substituting for local approximation from (7.121)in (7.122) we can obtain explicit forms of the residual equations. Let us definenodal degrees of freedom δe for an element e as

δeT =[ueT , veT , peT

](7.123)

Then

δEei T =

[∂Eei∂ue

T,

∂Eei∂ve

T,

∂Eei∂pe

T], i = 1, 2, 3 (7.124)

and the element coefficient matrix [Ke] is given by

[Ke] =3∑i=1

∫Ωe

∂Eei∂δe

∂Eei∂δe

TdΩ (7.125)

Remaining details follow the procedure presented in earlier examples. Inthis case k − 2 and k = 3 are minimum orders of approximation spaces for


peh and ueh, veh for k ≥ 3 for all dependent variables would be admissible as

well. Generally in applications in which the solutions are smooth k = 2 maysuffice. For this choice of k the integrals over ΩT are in Lebesgue sense.

In the following we present three numerical examples using the LS formu-lations presented in Sections 7.4.1 and 7.4.2. We consider numerical studiesfor the following model BVPs using Navier Stokes equations for incompress-ible isothermal flows.

(a) Section 7.4.3: slider bearing, flow of a viscous lubricant

(b) Section 7.4.4: a square lid-driven cavity

(c) Section 7.4.5: asymmetric backward facing step

(d) Section 7.4.6: flow past a circular cylinder

7.4.3 Slider bearing; flow of a viscous lubricant

The problem has been studies in references [6, 7]. In reference [7], anapproximated mathematical model for which the theoretical solution as wellas finite element solutions are presented. Figure 7.15 shows a schematic ofthe slider bearing. The bearing consists of a sliding pad moving at a velocityu0 relative to the stationary pad inclined at a small angle to the moving pad.The page between the pads is filled with lubricant. The two open ends areassumed atmospheric pressure p0 (we assume this to be zero in the numericalstudies). We consider the following properties of the fluid.

η = 8× 10−4 lb ft sec ft−2, ρ = 50.0lb

ft3

and choose the following reference quantities

ρ0 = ρ = 50.0lb

ft3 , η0 = η = 8× 10−4 lb ft sec ft−2,

L0 = 0.001 ft, v0 = 1.0ft

sec

With these choices of reference quantities

η =η

η0= 1, h1 =

h1

L0= 0.8, h2 =

h2

L0= 0.4, L =

L

L0=

0.36

0.001= 360

We choose ρ0 = τ0 = ρ0v20 = 50 based on characteristic kinetic energy

(CKE). With this choice

p0

ρ0 v20

=τ0

ρ0 v20

= 1 andη0 v0

L0τ0=

η0 v0

L0 ρ0 v20

=η0

L0 ρ0 v0=

1

Re=

1

62.5


u = u0, v = 0

u = 0, v = 0

y

x

p = p0 = 0

L

h1

h2

p = p0 = 0

h(x)

L = 0.36 ft, h1 = 8× 10−4 ft, h2 = h12

= 4× 10−4 ft, u0 = 30 ftsec

Figure 7.15: Schematic of slider bearing

Approximate theoretical solution

Following references [6, 7], the flow can be approximated by (using thenon-dimensionality form of the Navier–Stokes equations)

1

Reη∂2u

∂y2=∂p

∂x(7.126)

where

∂p

∂x=

1

Re

6η u0

h2

(1− H

h

), h(x) = h1 +

(h2 − h1

L

)x, H =

2h1 h2

h1 + h2(7.127)

and

u =(u0 −

h2

2ηRe

∂p

∂x

y

h

)(1− y

h

)(7.128)

p(x) =1

Re

6η u0 L(H − 1− h)(h− h2)

h2(h21 − h2

2)(7.129)

τxy =η

Re

∂u

∂y=∂p

∂x

(y − h

2

)− η u0

Reh(7.130)

Finite element solution

We present finite element solution of the Navier–Stokes equations us-ing least squares finite element formulation presented in section 7.4.1 for asystem of first order equations. Figure 7.16 shows a 5×3 uniform discretiza-tion of the dimensionless domain using nine-node p-version two-dimensionalelements (see chapter 8 for details). We consider local approximations ofclass C0(Ωe) with equal degree interpolations. Even though the integrals


for this choice of approximation space are in Lebesgue sense but due to thesmoothness of the solution, we expect accurate computed solutions uponconvergence. Computations are performed using C0(Ωe) equal degree inter-polations with p-level of 1, 2, 3, . . ..

u = 0, v = 0

y

x

p = 0

0.8

0.4

u0 = 30, v = 0

360

p = 0

Figure 7.16: Computational domain and 5 × 3 uniform mesh of nine-node p-versionelements

Figure 7.17 shows a plot of the square root of the residual functional Iversus degrees of freedom for progressively increasing p-levels. With increas-ing p-levels progressively reduce values of I confirm improved accuracy ofthe solution. Figure 7.18 shows plots of pressure p versus x at y = 0 (atthe moving plate) for various p-levels and comparison with the theoreticalsolution of (7.126) shows similar plots for shear stress τxy at y = 0. Wenote that the theoretical solution of the approximate mathematical modelunderestimates pressure distribution but overestimates the shear stress whencompared with the converged solution of the actual Navier–Stokes equationswithout approximation presented here.

7.4.4 A square lid-driven cavity

In this model problem we consider a dimensionless unit square lid-drivencavity, a schematic of which is shown in Figure 7.20(a). The boundary con-ditions for dimensionless velocities u, v and pressure p are also shown inFig. 7.20(a). Figures 7.20(b) and (c) two graded finite element discretiza-tions using 36 nine-node p-version finite elements. The main difference inthe two discretizations is the size of the elements adjacent to the four bound-aries of the cavity. In discretization A the elements are 0.1 units where indiscretization B they are 0.05 units. The physical size of the cavity is 3 cm× 3 cm and the fluid properties used are:

ρ = 998.2 kg/m3, η = 1.002× 10−3 Pa


10-4

10-3

10-2

10-1

100

100 1000 10000

Square

root

of

resid

ual

functi

onal,

√I

Degrees of freedom

C0(

—

Ωe )

p=1

p=3

p=5

p=7

p=9

Figure 7.17: Square root of the residual functional versus dofs

0

20

40

60

80

100

120

140

160

180

200

220

240

260

280

300

0 40 80 120 160 200 240 280 320 360

Pre

ssu

re,

p

Distance, x

C0(

—

Ωe ) ; LSP

p=1

p=2

p=3

p=4,5,...,9

Approx. Theoretical

Figure 7.18: Pressure solution along the moving plate (y = 0)

We consider the following reference values:

ρ0 = ρ =998.2 kg/m3, η0 = η = 1.002× 10−3 Pa,

L0 = 0.03 m, υ0 = 0.03346 m/s


-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0 40 80 120 160 200 240 280 320 360

Str

ess, τ x

y

Distance, x

C0(

—

Ωe ) ; LSP

p=1

p=2

p=3

p=4,5,...,9

Approx. Theoretical

Figure 7.19: Shear stress distribution along the moving plate (y = 0)

0.15

0.05

0.30

0.30

0.15

0.05

(c) Mesh B

0.30

0.30

(b) Mesh A

0.15

0.15

0.15

0.15

(a) Schematic

x

y

x

yhd

x

v = 0

u = 1

u = 0v = 0

v = 0

u = 0

p = 0A

B C

D

u = 0v = 0

y hd

Figure 7.20: Driven cavity problem schematic and the finite element discretizations

With this choice of reference values, we have

ρ = 1, η = 1, Re =ρ0 L0 υ0

η0= 1000

The velocity of the lid is assumed to vary from zero at the vertical walls toone in a continuous and differentiable manner over a length of hd representingthe characteristic lengths of the elements at the top two corners. Using theconditions

u = 0,∂u

∂x= 0 at xA = 0 and x = xB = hd

we can obtain a cubic distribution of u over 0 ≤ x ≤ hd that is continuousand differentiable. Likewise, using

u = 0,∂u

∂x= 0 at x = xC = 1− hd and x = xD = 1


we can obtain another cubic distribution of u over 1 − hd ≤ x ≤ 1 that iscontinuous and differentiable.

In the computations of the numerical solutions we consider the mathe-matical model consisting of a first order system of PDEs (7.111) as well asthe higher order system of equations (7.115). We consider local approxi-mations of classes C0(Ωe) and C1(Ωe) with equal degree interpolations forall variables. p-levels in ξ and η directions are chosen same (pξ = pη = p)and are uniformly increased from 3 to 9. Numerical solutions are computedfor both mesh A and mesh B. Newton’s linear method with line search isconsidered converged when |gi| ≤ ∆ ; i = 1, 2, . . . in which ∆ is a presenttolerance. Figure 7.21 shows graphs of

√I versus degrees of freedom for

both mathematical models with mesh A and mesh B when τ0 = p0 = ρ0υ20

(CKE) is used to non-dimensionalize pressure and stresses. Similar graphsfor po = τ0 = η0υ0/L0 (CVS) are shown in Fig. 7.22. From Figs. 7.21 and7.22 we note that C1 solutions yield lower values of I compared to C0 so-lutions for a given dofs confirming better accuracy of C1 solutions.

√I in

Fig. 7.21 is lower than in Fig. 7.22 for the same degrees of freedom suggestingthat CKE approach is superior to CVS approach for non-dimensionalizingpressure and stresses. Mesh B yields lower

√I compared to mesh A (for the

same dofs), hence more accurate computed solutions.

10-4

10-3

10-2

10-1

100

100 1000 10000 100000

Sq

uare

ro

ot

of

resid

ual

fun

cti

on

al,

√I

Degrees of freedom

p=3

p=3

p=9

p=9

Mesh A (Higher order PDEs) : C1(

—

Ωe )

Mesh B (Higher order PDEs) : C1(

—

Ωe )

Mesh A : C0(

—

Ωe )

Mesh B : C0(

—

Ωe )

Figure 7.21: Square root of the residual functional versus degrees of freedom(p0 = τ0 = ρ0υ

20); CKE

Convergence rates of√I vs dof for solutions of class C0 and C1 are

comparable. From these graphs of√I in Figs. 7.21 and 7.22, the most


10-3

10-2

10-1

100

100 1000 10000 100000

Sq

uare

ro

ot

of

resid

ual

fun

cti

on

al,

√I

Degrees of freedom

p=3 p=3

p=9p=9

Mesh A (Higher order PDEs) : C1(

—

Ωe )

Mesh B (Higher order PDEs) : C1(

—

Ωe )

Mesh A : C0(

—

Ωe )

Mesh B : C0(

—

Ωe )

Figure 7.22: Square root of the residual functional versus degrees of freedom(p0 = τ0 = η0υ0

L0); CVS

accurate results are obtained using mesh B for the mathematical modelconsisting of the first order system of PDEs with local approximation of classC0(Ωe) and using CKE to non-dimensionalize pressure and stresses, henceis used here to present the computed results. These are also compared withmesh A. Figures 7.23 and 7.24 have plots of velocities u and v versus distanceat vertical and horizontal centerlines of the cavity for different p-levels usingmesh A.

Similar plots for mesh B are shown in Figs. 7.25 and 7.26. At lower p-levels (p = 3), computed solutions using meshes A and B differ as expecteddue to coarse meshes and low p-levels, but beyond p = 3 both meshes yieldalmost same results. At p = 5 the computed solutions are almost converged.The solutions for these meshes are also reported in references [8, 9] and arecompared with the solutions of reference [10] with good agreement.

7.4.5 Asymmetric backward facing step

In this model problem we consider 2:3 asymmetric backward facing stepshown in Fig. 7.27(a) in dimensionless form. The fluid is water with thefollowing properties:

ρ = 998.2kg

m3, η = 1.002× 10−3 Pa


0

0.25

0.5

0.75

1

-0.5 0 0.5 1

Dis

tan

ce,

y

Velocity, u at x=0.5

Mesh A : C0(

—

Ωe )

p=3

p=5

p=7

p=9

Figure 7.23: Velocity u at x = 0.5 for mesh A using local approximation of class C0(Ωe)(CKE as reference pressure and stress)

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

0 0.25 0.5 0.75 1

Velo

cit

y,

v a

t y=

0.5

Distance, x

Mesh A : C0(

—

Ωe )

p=3

p=5

p=7

p=9

Figure 7.24: Velocity v at y = 0.5 for mesh A using local approximation of class C0(Ωe)(CKE as reference pressure and stress)

We choose the following reference values:

ρ0 = ρ = 998.2kg

m3, η0 = η = 1.002× 10−3 Pa, L0 = 0.015 m

Then

ρ = 1, n = 1, Re =ρ0 L0 u0

η0= 14943.1137u0


0

0.25

0.5

0.75

1

-0.5 0 0.5 1

Dis

tan

ce,

y

Velocity, u at x=0.5

Mesh B : C0(

—

Ωe )

p=3

p=5

p=7

p=9

Figure 7.25: Velocity u at x = 0.5 for mesh A using local approximation of class C0(Ωe)(CKE as reference pressure and stress)

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

0 0.25 0.5 0.75 1

Velo

cit

y,

v a

t y=

0.5

Distance, x

Mesh B : C0(

—

Ωe )

p=3

p=5

p=7

p=9

Figure 7.26: Velocity v at y = 0.5 for mesh A using local approximation of class C0(Ωe)(CKE as reference pressure and stress)

For u0 = 0.004885 we have Re = 73 and for u0 = 0.00153247 we obtainRe = 229. The experimental measurements for this model problem for thesetwo Reynolds numbers have been reported by Denham and Patrick [11].

The numerical simulations for Re = 73 and Re = 229 of the experimentin [11] have also been given in references [9] using the discretizations shownin Figs. 7.27(b) and (c). In the numerical simulations presented in refer-


(a) Schematic

(b) A 7 element discretization

(c) A 20 element discretization

(d) A 32 element discretization

y

y

x

τxx = 0v = 0

u = 0 , v = 0

u = 0 , v = 0

u = 0 , v = 0

x = 4 x = 8

x = 4 x = 8

x = 4 x = 8

y

x

x

x

y

0.2

0.2

0.20.8

0.8

0.2

0.2

4/3 28

2

Figure 7.27: Backward facing step problem and finite element discretizations

ences [9] using LSP based on residual functional using first order system ofPDEs, the experimental inlet velocities of reference [9] were used so that thenumerically computed solutions could be compared with experimental mea-surements. Published numerical solutions in references [9] show extremelygood agreement with the experimental measurements even for such coarsediscretizations of Figs. 7.27(b) and (c). Reference [9] shows that even thecoarsest possible mesh of Fig. 7.27(b) produces reasonable computed solu-tions. In reference [9] numerically computed solutions for discretization ofFig. 7.27(c) are compared with the experimental results of reference [11].

In the numerical solutions presented here we employ a slightly more re-fined discretization of 32 nine-node C0(Ωe) p-version hierarchical elementsshown in Fig. 7.27(d). We use LSP based on residual functional for the firstorder system of PDEs (7.111). Same p-levels are considered for all depen-


10-4

10-3

10-2

10-1

1000 10000 100000

Square

root

of

resid

ual

functi

onal,

√I

Degrees of freedom

p=3

p=3

p=9

p=9

C0(

—

Ωe )

CKE Re=73

CVS Re=73

CKE Re=229

CVS Re=229

Figure 7.28: Square root of the residual functional versus degrees of freedom

dent variable in ξ and η direction. p-levels are uniformly increased for allelements of the discretization from 3 to 9. Fully developed parabolic u ve-locity (dimensionless) with peak value of 1.5 is specified at the inlet for bothReynolds numbers. Figure 7.28 shows graphs of

√I versus dof for Re = 73

and Re = 229 when CKE and CVS are used to non-dimensionalize pressuresand deviatoric Cauchy stresses. As seen in Fig. 7.28, I is O(10−6) or lowerat p = 9 confirming extremely good accuracy of the computed solutions.Newton’s method with line search converges in less than 10 iterations forall computations using |gi| ≤ O(10−6) ; i = 1, 2, . . . as convergence criteria.Lower values of

√I for a given dofs for CKE compared to CVS indicate

better accuracy of the computed solutions when using CKE to nondimen-sionalize pressure and deviatoric stresses. Thus, in the following we presentthe computed solutions using CKE at p-levels of nine as these are convergedsolutions.

Plots of velocity u vs y at x = 0.0, 0.8, 2.0 and 28.0 or Reynolds numbersof 73 and 229 at p = 9 are shown in Figs. 7.29–7.30. As expected at x = 0,onset of expansion, the axial velocity u is parabolic for both Reynolds num-bers (same as at the inlet). Negative u velocity in vicinity of lower boundaryfor y < −0.25 (approximately) confirms recirculation in the expansion cor-ner.

At x = 28.0 we observe perfect fully-developed parabolic velocity profilesfor both Reynolds numbers.


-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Dis

tan

ce,

y

Velocity, u

C0(

—

Ωe ) ; p=9

CKE, Re=73

CKE, Re=229

Figure 7.29: Velocity u at x = 0.0 (step location) using local approximation of classC0(Ωe)

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Dis

tan

ce,

y

Velocity, u

C0(

—

Ωe ) ; p=9

CKE, Re=73

CKE, Re=229

Figure 7.30: Velocity u at x = 0.8 using local approximation of class C0(Ωe)

Figures 7.33 and 7.34 show contour plots of velocity u for Re = 73 andRe = 229. Contour plots of velocity v are shown in Figs. 7.35 and 7.36.We clearly observe larger recirculation zone for Re = 229, as expected.The contour plots confirm that the length of 28 dimensionless units beyondexpansion point is sufficient for the flow to become fully developed.


-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Dis

tan

ce,

y

Velocity, u

C0(

—

Ωe ) ; p=9

CKE, Re=73

CKE, Re=229

Figure 7.31: Velocity u at x = 2 (step location) using local approximation of class C0(Ωe)

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Dis

tan

ce,

y

Velocity, u

C0(

—

Ωe ) ; p=9

CKE, Re=73

CKE, Re=229

Figure 7.32: Velocity u at x = 28 using local approximation of class C0(Ωe)

7.4.6 Flow past a circular cylinder

Figure 7.37 shows a schematic of the flow domain. Figure 7.38 showsboundary conditions on the four boundaries of the domain and the boundaryof the cylinder.

Since L0 = 1 and the dimensionless radius of the cylinder is 1 we have


Figure 7.33: Velocity u for Re = 73 (CKE), C0(Ωe) ; p = 9

Figure 7.34: Velocity u for Re = 229 (CKE), C0(Ωe) ; p = 9

Figure 7.35: Velocity v for Re = 73 (CKE), C0(Ωe) ; p = 9

r = 1 meter. Flow domain consists of length L, height h and the cylinderis symmetrically located in the flow domain. Figure 7.39 (a) shows a 1300element graded discretization for (x×y) = (−30.5, 30.5)×(−21.5, 21.5). Thecylinder is located at the center of the domain. Figure 7.39(b) shows anotherdiscretization in which the discretization of Fig. 7.39(a) remains unchangedbut additional length has been added to the right of the cylinder. Thisdiscretization has 1400 elements.


Figure 7.36: Velocity v for Re = 229 (CKE), C0(Ωe) ; p = 9

ufs rh

L

Figure 7.37: Schematic of Flow over circular cylinder

-1.4

-1.2

-1

-0.8

-0.6

-0.4

0 40 80 120 160 200 240 280 320 360

Str

ess,

τ xy

Distance, x

p=1

p=2

p=3

p=4,5,...,9

Approx. Theoretical

v = 0

∂v/∂x = 0

∂v/∂y = 0

∂T/∂x = 0

∂u/∂x = 0

T = 1

v = 0

ρ = 1

ρ = 1, u = ufs, v = 0, T = 1

ρ = 1, u = ufs, v = 0, T = 1

u = 0, v = 0, ∂T/∂n = 0

ufs = 1r

(b) Boundary Conditions for incompressible flow past a circular cylinder

(a) Boundary Conditions for compressible flow past a circular cylinder

r

∂u/∂x = 0

v = 0

u = 0, v = 0, ∂T/∂n = 0

ufs = 1

u = ufs, v = 0

u = ufs, v = 0

∂v/∂y = 0

∂v/∂x = 0

v = 0

Figure 1: Boundary conditions for flow past a circular cylinder (compressible flow)

2

Figure 7.38: Boundary conditions for flow past a circular cylinder (incompressible flow)

We consider water at NTP with the following properties:

ρ0 = ρ = 997.78 kg/m3, µ0 = µ = 9.774× 10−4 Pa · s, L0 = 1 m

Therefore

ρ = 1, µ = 1, L = L and Re =997.78

9, 774× 10−4u0


−40 −30 −20 −10 0 10 20 30 40−25

−20

−15

−10

−5

0

5

10

15

20

25

−40 −20 0 20 40 60 80−25

−20

−15

−10

−5

0

5

10

15

20

25

(b) Mesh of 1400 elements

(a) Mesh of 1300 elements

Figure 7.39: Discretizations for the two different domain lengths for flow past a cylinder

We choose solution of class C22 at p-level of 5 for ρ, u, v in the mathe-matical model given by equations (7.115), higher order system. We chooseu0 such that Re =10, 20, 30, 40, 60, 80, 100, 200 and 500 are obtainedand ufs = 1 in all cases. For Re = 10 no initial or starting solution isused (other than null), thus Stokes flow becomes the starting solution atthe end for first iteration in Newton’s linear method. The convergence ofthe Newton’s method is achieved in 5 iterations with |gi|max = 0.332× 10−6

and I = 0.213 × 10−4. This solution at Re = 10 is used as initial solutionfor Re = 20 (new u0) to obtain a converged solution for Re = 20. Thiscontinuation process is continued until Re = 500 is reached for both 30.5and 60.5 lengths of the domain past the cylinder. Table 7.1 gives details ofRe, I, |gi|max and number of iterations.

Contour maps of u velocity for Re = 10, 100 and 500 for lengths of 30.5and 60.5 past the cylinder are shown in Figs. 7.40–7.42. We observe the

7.5. 2D COMPRESSIBLE NEWTONIAN FLUID FLOW 471

Table 7.1: Values of Re, I, g for incompressible flow calculations

Re I |gi|max no. of iterations

10 0.213× 10−4 0.332× 10−6 520 0.332× 10−4 0.645× 10−6 540 0.687× 10−4 0.159× 10−7 660 0.141× 10−3 0.196× 10−6 680 0.206× 10−3 0.598× 10−6 7100 0.254× 10−3 0.208× 10−6 8200 0.749× 10−3 0.510× 10−6 9500 0.204× 10−2 0.932× 10−6 25

following.

(a) For low Re (up to 200), the lengths of 30.5 and 60.5 yield same resultsdue to the lack of influence of outflow boundary on the flow physics inthe near field past the cylinder.

(b) At Re = 500, the length of 30.5 is obviously not sufficient whereas lengthof 60.5 is still sufficient to capture the flow physics behind the cylinder.

Figure 7.40: Comparison of velocity u for Re = 10 of lengths of 30.5 and 60.5 behindthe cylinder (incompressible flow)

7.5 2D compressible Newtonian fluid flow

Following reference [5], the conservation and balance laws: conservationof mass, balance of momenta and the first law of thermodynamics yieldcontinuity, momenta and energy equations. For stationary process (BVP)the dimensionless form of conservation and balance laws can be written as




(in the absence of sources and sinks and using ideal gas law):

∂(ρu)

∂x+∂(ρv)

∂y= 0 (7.131)

ρu∂u

∂x+ ρv

∂u

∂y+( p0

ρ0v20

)∂p∂x−( τ0

ρ0v20

)(τxx∂x

+τxy∂y

)= 0 (7.132)

ρu∂v

∂x+ ρv

∂v

∂y+( p0

ρ0v20

)∂p∂y−( τ0

ρ0v20

)((7.133)

ρcvEc

(u∂T

∂x+ v

∂T

∂y

)+

1

ReBr

(∂qx∂x

+∂qy∂y

)−( p0

ρ0 v20

)p(∂u∂x

+∂v

∂y

)− 1

Re

(2ηDij

∂vi∂xj

+ λ(Dkk)2)

= 0 (7.134)

τij =( η0 v0

L0 τ0

)(2ηDij + λδijDkk) , i, j = 1, 2 (7.135)

(7.136)


where

qx = −k∂T∂x

, qy = −k∂T∂y

(7.137)

Dij =1

2

( ∂vi∂xj

+∂vj∂xi

), i, j = 1, 2 (7.138)

p(ρ, T ) =(R0 p0 T0

p0

)RρT (7.139)

In deriving (7.131)–(7.139), we have used the following definition of thedimensionless variables

L =L

L0, vi =

viv0

, ρ =ρ

ρ0, η =

η

η0, λ =

λ

η0

τij =τijτ0

, p =p

p0, T =

T

T0, k =

k

k0

t =t

t0, t0 =

L0

v0, cv =

cvcv0

, p0 = τ0 = ρv20 (CKE)

(7.140)

In (7.140), the quantities with hat notation are with their usual dimen-sions and those with subscript zero are their reference values. Re, Br andEc are defined as

Re =v0 ρ0 L0

η0, Br =

η0 v20

k0 T0, Ec =

v20

cv0 T0

In addition we have used the following for specific internal energy e in theenergy equation

e = cv T (7.141)

in which we assume constant cv. Equations (7.131)–(7.137) is a system oftime first order PDEs in nine dependent variables ρ, u, v, τij , qx, qy, T .By substituting the Cauchy deviatoric stresses τij from (7.135) and the heatvectors qx, qy from (7.137) into (7.134), these can be reduced into a systemof four PDEs in four dependent variables ρ, u, v, T .

∂(ρvi)

∂xi= 0 (7.142)

ρvj∂vi∂xj

+( p0

ρ0 v20

) ∂p∂xi− 1

Re

∂

∂xj(2ηDIJ + λδijDkk) = 0 (i = 1, 2) (7.143)

ρ cvEc

(u∂T

∂x+ v

∂T

∂y

)− k

ReBr

(∂2T

∂x2+∂2T

∂u2

)−( p0

ρ0 v20

)(∂u∂x

+∂v

∂y

)− 1

Re

(2ηDij

∂vi∂xj

+ λ(D2kk

)= 0 (7.144)

∂p

∂xi− ∂p

∂ρ

( ∂p∂xi

)+∂p

∂T

( ∂T∂xi

)= 0 (7.145)


In (7.142)–(7.145), x1 and x2 refer to x and y, likewise v1 and v2 imply uand v, velocities in x and y directions. In (7.145), ∂p∂ρ and ∂ρ

∂T are deterministicfrom the equation of state (ideal gas law (7.139)). Equations (7.142)–(7.144)is a system of four PDEs in four dependent variables ρ, u, v, T . These containup to second order derivatives of velocities u, v and temperature T .

A variationally consistent least squares finite element formulation of(7.142)–(7.144) is constructed using residual functional (similar to one pre-sented in section 7.4. Details are straight forward. Since the PDEs containup to second order derivatives of u, v and T , the approximation space Vfor this must satisfy V ⊂ Hk,p(Ωe), k ≥ 3 for the integrals to be Riemannbut V ⊂ Hk,p(Ωe), k = 2 when the integrals are Lebesgue i.e. solutions ofclass C2 correspond to minimally conforming spaces but solutions of classC1 are admissible if the integrals can be accepted in the Lebesgue sense.Using equal order equal degree interpolation for all dependent variables wecan consider ρ, u, v, T to be of class C1 or C2.

In the following we present numerical solutions of two boundary valueproblems using the VC least squares finite element formulation based onresidual functional using PDEs (7.142)–(7.144).

(a) Carter’s plate: Laminar flow over a flat plate at Mach 1,3 and 5 isconsidered. The medium is air and the transport properties are assumedto be constant with ideal gas law.

(b) Flow over a circular cylinder: Flow over a circular cylinder with varyingtrailing lengths is considered. Medium is air with constant transportproperties at NTP and ideal gas law.

For both model problems we consider solutions for class C1(Ωe) as wellas solutions of class C2(Ωe) at various p-levels. These two model problemshave also been investigated in references [12,13].

7.5.1 Carter’s plate

In this study we consider flow of compressible fluid (air) over a stationaryimpermeable plate at various mach numbers. Plate is two meters in length.At the left end of the plate a constant velocity field is imposed. The objectiveis to compute the flow features in a 2 m × 0.01 m domain over the plate.Figure 7.43 shows a schematic of the Carter’s plate. Boundary conditionsand non-dimensional computational domain L × h are shown in Fig. 7.44.The following properties are used for air at NTP.

ρ0 = 1.12254kg

m3, η0 = 0.1983× 10−4 Pa · s, T0 = 410.52 K

cv0 = 717.0J

kg ·K, R = 286.9965

J

kg ·K, k0 = 0.028854

N

kg ·K


ufs

L = 2m

h = 0.01m

Figure 7.43: Schematic of Carter’s plate

L = 2

y

x

ρ = 1, T = 1, u = ufs, v = 0

A B

CD

u = 0, v = 0, ∂T/∂y = 0

h = 0.01

∂T/∂x = 0

ρ = 1 , ∂ρ/∂x = 0 , T = 1 , ∂T/∂x = 0 , ∂T/∂y = 0

ufs

∆y

Figure 7.44: Carter’s plate Boundary conditions

We choose the following values for the reference quantities:

L0 = 1.0m, ρ0 = ρ, µ0 = µ, k0 = k, cv0 = cv, v0 = 343.25m

s

T0 = 410.52 K, R0 = 286.99J

kg ·K, τ0 = p0 = ρo v

20 = 0.14438× 106 kg

m · sWith these reference quantities we have the following dimensionless quan-

tities and dimensionless parameters:

ρ = 1.0, µ = 1.0, k = 1.0, cv = 1.0, R = 1.0, γ = 1.4


Re = 0.212× 106, Br = 0.19724, Ec = 0.40027

We choose L = 2 (L = 2 m) and h = 0.01 (h = 0.01 m). Specificationof the dependent variables on the boundaries is crucial to ensure that theBVP is well posed but not overly constrained. That is, we only define whatis permissible based on physics and that to only basic quantities and thusavoiding redundant descriptions through BCs.

At the inlet(no heat transfer along and across inlet AD):

ρ = 1, T = 1 ⇒ ∂T

∂y= 0,

∂T

∂x= 0

At the plate (no heat transfer normal to the plate):

u = 0, v = 0,∂T

∂y= 0

At the top boundary CD (no heat transfer normal to the boundary CD):

ρ = 1 ⇒ ∂p

∂x= 0; T = 1 ⇒ ∂T

∂x= 0;

∂T

∂y= 0

At the outflow boundary BC (no heat transfer across the boundary BC):

∂T

∂x= 0

Fully developed flow conditions on BC are inappropriate to use if thelength L is not sufficient. To avoid over specification of BCs on CD, weavoid defining u and v and/or their gradients. We present numerical studiesfor free stream velocity ufs of Mach 1, 2, 3 and 5. The domain (2× 0.01) isdiscretized using a (70 × 30) graded mesh in which the element size at thelower left corner is 10−5. We choose us to be the speed of sound. Solutionsfor all Mach numbers are of class C11 at p = 3 in space and time.

7.5.1.1 Mach 1 flow

If we choose u0 = us then ufs = 1. The other reference quantities arechosen as shown above. This choice of u0 gives rise to ReM1, EcM1, BrM1,i.e. Re, Ec and Br at Mach 1. The choice of initial solutions is criticalas well. We choose ρ = 1, T = 1, v = 0 and u = 1 as initial solution.At the leading edge, the velocity distribution of 0 to ufs is applied over adistance of ∆y in the element located at the leading edge in a continuousand differentiable manner (shown in Fig. 7.44). Newton’s linear methodwith line search converged in 35 iterations with |gi|max = 0.862× 10−6 and


I = 0.325× 10−2. The largest value of I is from the element located at theleading edge. Other than the single element at the leading edge, I valueselsewhere are much smaller than O(10−2).

Figures 7.45–7.47 show contour plots of density, velocity and temperatureover (x, y) domain of (0 , 2)× (0 , 0.005). From the figures we observe thatboundary layer may not be fully developed, hence the reason for not imposingfully developed flow conditions at the outflow. Progressive development ofthe boundary layer is clearly observed. Temperature rise due to viscousdissipation at the plate with progressively higher values for increasing x areclearly seen in Fig. 7.47. Increase in density beyond 1 (initial state) indicatesexistence of mild shock.

Den

sity

,ρ

Figure 7.45: Contour of the density ρ for Mach 1.0 flow

Vel

oci

ty,

u

Figure 7.46: Contour of the velocity u for Mach 1.0 flow


Tem

per

ature

,T

Figure 7.47: Contour of the temperature T for Mach 1.0 flow

7.5.1.2 General consideration for higher Machnumber flows

For calculating flow at Mach 2, 3 and 5 a direct calculation similar tothat described for Mach 1 resulted in lack of convergence of the iterativesolution method for solving non-linear algebraic equations. This is largelydue to inadequate starting or initial solution for Newton’s linear method forsolving non-linear algebraic equations i.e., the choice of free stream valuesas initial solution (as used for Mach 1 flow) is not in close proximity of theactual solution sought from Newton’s linear method upon convergence.

Thus, a continuation procedure becomes essential to use. For examplewe could use Mach 1 solution as a starting or initial solution for Mach 2.For illustration purposes consider Mach 2 flow. For Mach 2 flow, u0 = 2usgives ufs = 1 (same as in case of Mach 1). But (Re)M2, (Ec)M2, (Br)M2

will be different compared to Mach 1 flow. With this choice, the boundaryand free stream values of u, ρ, T and v in Mach 2 solution are same as thosefor Mach 1 flow but new Re,Br and Ec for Mach 2 flow are characteristicdimensionless parameters of the flow at Mach 2. Thus we can proceed asfollows:

Mach 2 flow u0 = 2us, ufs = 1All other reference values same as those for Mach 1Starting solution from Mach 1 calculations

Mach 3 flow u0 = 3us, ufs = 1

(a) Starting solution from Mach 1 flow(b) Starting solution from Mach 2 flow

Mach 5 flow u0 = 5us, ufs = 1


(a) Starting solution from Mach 1 flow

(b) Starting solution from Mach 2 flow

(c) Starting solution from Mach 3 flow

7.5.1.3 Mach 2 flow

Using the continuation from Mach 1 described above with u0 = 2us,ufs = 1 (same as Mach 1 flow), the Newton’s linear method converges in 17iteration with |gi|max = 0.883 × 10−6 and I = 0.105 × 10−2. The contourplots of ρ, u, T are shown in Figs. 7.48–7.50 over the same (x, y) domain asused for Mach 1 flow. Thinner boundary layer and increased temperaturevalues at the plate are clearly observed compared to Mach 1 flow. Largerincrease in density (1.05) compared to Mach 1 (1.02) is also observed.

Den

sity

,ρ


Vel

oci

ty,

u



Tem

per

ature

,T


7.5.1.4 Mach 3 flow

In this case Mach 1 solution as well as Mach 2 solutions were used asstarting or initial solutions in two separate studies. The final convergedsolutions at Mach 3 were identical from either of these two as initial solutions.Using Mach 1 solution as initial solutions, Newton’s linear method convergesin 17 iterations with |gi|max = 0.254× 10−6 and I = 0.716× 10−3.

Contour plots of ρ, u, T are shown in Figs. 7.51–7.53. Thinner boundarylayer and increased temperature compared to Mach 2 flow are quite clear.

Den

sity

,ρ



Vel

oci

ty,

u


Tem

per

ature

,T


7.5.1.5 Mach 5 flow

For Mach 5 flow, the initial or starting solutions were used from Mach 1as well as Mach 2 and Mach 3 for three different studies (with u0 = 5us andufs = 1 ). All three studies converged to the identically same solution. Wepresent results obtained using the Mach 1 solution as the starting or initialsolution. Contour of ρ, u and T are shown in Figs. 7.54–7.56.

The Newton’s linear method converges in 29 iterations with |gi|max =0.473 × 10−6 and I = 0.425 × 10−3. Contour plots of ρ, u, T are shown inFigs. 7.54–7.56. We observe similar trend of diminishing boundary layerthickness with increasing temperature values at the plate. Density increasein this case is largest (1.1), higher than Mach 1, 2 and 3 flows indicatingslightly stronger shock.


Den

sity

,ρ


Vel

oci

ty,

u


Tem

per

ature

,T


Comparison of results for Mach 1, 2, 3 and 5 flows

A comparison of the results for mach 1, 2, 3, and 5 is presented inFigs. 7.57–7.61. Figure 7.57 shows plots of velocity u at the outflow bound-


ary for Mach 1, 2, 3 and 5 flows. An exploded view of these is also shownin Fig. 7.58. We clearly see thinning boundary layer for increasing Machnumbers. Free stream value of u = ufs = 1 for y > 0.004 for all Mach num-bers shows that h-value of 0.01 chosen in numerical studies is sufficientlylarge. Plots of ρ, T and v at the outflow for Mach 1, 2, 3 and 5 are shownin Figs. 7.59–7.61. Progressively decreasing density at the plate for progres-sively increasing Mach number flows is clearly observed. Free stream densityof one is clearly seen for y > 0.004, conforming adequate choice of h. FromFig. 7.60 we observe progressively increasing temperature values at the platefor progressively increasing Mach number flows and a free stream tempera-ture of 1 for y > 0.004. Plots of v at the outflow boundary confirm that v ofthe order of 10−5 implies very close to fully developed flow. Slightly largervalues of density beyond 1.0 indicate existence of milder shocks.

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0 0.2 0.4 0.6 0.8 1 1.2

Velocity, u

Dis

tance

,y

Mach 2Mach 3

Mach 1

Mach 5

Figure 7.57: Velocity u at the outflow for Mach 1, 2, 3 and 5 flows

7.5.2 Mach 1 flow past a circular cylinder

We consider same geometry, discretizations as used for example 7.4.6 incase of incompressible isothermal flow of a Newtonian fluid considered here.We consider flow of air (at NTP) past a circular cylinder at Mach 1 with thefollowing properties of the medium.

ρ = 1.12254kg

m3, µ = 0.1983× 10−4 Pa · s, T0 = 410.52 K

cv = 717.0J

kg ·K, R = 286.9965

J

kg ·K, k = 0.028854

N

kg ·Kand we choose the following reference quantities and their values.

L0 = 1.0m, ρ0 = ρ, µ0 = µ, k0 = k, cv0 = cv


0.0006

0.0008

0.001

0.0012

0.0014

0.0016

0.0018

0.002

0.8 0.85 0.9 0.95 1 1.05

Velocity, u

Dis

tance

,y

Mach 2Mach 3

Mach 1

Mach 5

Figure 7.58: Exploded view of velocity u for Mach 1, 2, 3 and 5 flows

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Density, ρ

Dis

tance

,y

Mach 2Mach 3

Mach 1

Mach 5

Figure 7.59: Density ρ at the outflow for Mach 1, 2, 3 and 5 flows

u0 = 343.25m

s, T0 = 410.52 K, R0 = 286.99

J

kg ·K

τ0 = p0 = ρ0 u20 = 0.14438× 106 kg

m · s2

With these reference quantities we have the following dimensionless quan-tities and the dimensional parameters.

ρ = 1.0, µ = 0, k = 1.0, cv = 1.0, R = 1.0

Re = 0.212× 106, Br = 0.19724, Ec = 0.40027


0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0.5 1 1.5 2 2.5 3 3.5 4

Temperature, T

Dis

tance

,y

Mach 2Mach 3

Mach 1

Mach 5

Figure 7.60: Temperature T at the outflow for Mach 1, 2, 3 and 5 flows

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

-0.0001 -8e-05 -6e-05 -4e-05 -2e-05 0 2e-05

Velocity, v

Dis

tance

,y

Mach 2Mach 3

Mach 1

Mach 5

Figure 7.61: Velocity v at the outflow for Mach 1, 2, 3 and 5 flows

A schematic with BCs is shown in Fig. 7.62. We choose r = 1 m, hencer = r/L0 = 1. The length from the inlet to the center of the cylinder is 6units.

From the studies performed for Carter’s plate we find that a good startingor initial solution is quite essential when solving BVPs for compressible flow.We choose the incompressible flow solution (for same geometry and samediscretization) as starting solution for compressible flow. When using alength of 60.5 behind the cylinder, the use of incompressible flow results innon-converged Newton’s linear method. Experiments with length of 30.5behind the cylinder show convergence of the Newton’s linear method. We


0 40 80 120 160 200 240 280 320 360

Distance, x

v = 0

∂v/∂x = 0

∂v/∂y = 0

∂T/∂x = 0

∂u/∂x = 0

T = 1

v = 0

ρ = 1

ρ = 1, u = ufs, v = 0, T = 1

ρ = 1, u = ufs, v = 0, T = 1

u = 0, v = 0, ∂T/∂n = 0

ufs = 1r

(b) Boundary Conditions for incompressible flow past a circular cylinder

(a) Boundary Conditions for compressible flow past a circular cylinder

r

∂u/∂x = 0

v = 0

u = 0, v = 0, ∂T/∂n = 0

ufs = 1

u = ufs, v = 0

u = ufs, v = 0

∂v/∂y = 0

∂v/∂x = 0

v = 0

Figure 1: Boundary conditions for flow past a circular cylinder (compressible flow)

2

Figure 7.62: Boundary conditions for flow past a circular cylinder (compressible flow)

realize that this length is not sufficient for the flow to be fully developed atthe outflow boundary. We present results to demonstrate that the VCLSPhas no difficulty in computing numerical solution. We consider solutionsof class C22(Ωe) with p-level of 5. Using the converged solution from theincompressible flow at Re = 500, the Newton’s linear method converges in50 iterations with |gi|max = 0.613× 10−4 and I = 0.206× 10−2 and only fewelements around.

Contour plots of u, v, ρ and T are shown in Figs. 7.63–7.66. High valuesof I indicate the GDEs are not satisfied accurately by the computed solutiondue to inadequate h and p and the length past the cylinder. Nonethelessthe computational process works well in the sense that with continuationin Re, for the given h and p, the length past the cylinder and the outflowboundary conditions a solution satisfying discretized form of the GDEs isfound (|gi| ≥ 10−6). With h, p refinement, adequate length past the cylinderand appropriate outflow conditions we see no issues in resolving the correctphysics with the laminar flow assumption.

7.6 Summary

Numerical solutions of a variety of model problems in R1 and R2 contain-ing nonlinear differential operators have been considered. It has been shownin chapter 3 that for such operators, all methods of approximations leadingto integral forms yield VIC integral forms. It was established that the inte-gral forms resulting from the residual functional (LSP) can be made VC bysmall adjustments in the solution process for nonlinear systems of algebraicequations. It is established that by neglecting the second variation of the

7.6. SUMMARY 487

Vel

oci

ty,

u

Figure 7.63: Contour plot of velocity u for compressible flow past a cylinder of lengthsof 30.5

Vel

oci

ty,

vFigure 7.64: Contour plot of velocity v for compressible flow past a cylinder of lengthsof 30.5

Den

sity

,ρ

Figure 7.65: Contour plot of density ρ for compressible flow past a cylinder of lengths of30.5

residuals in the second variation of the residual functional and by employingNewton’s linear method for obtaining numerical solutions of the nonlinearalgebraic equations, the integral forms become VC, hence the resulting co-efficient matrices in the algebraic systems remain unconditionally positivedefinite. The rationale and validity of this approach was also establishedin chapter 3. All numerical studies presented in this chapter utilize the ap-


Tem

per

atu

re,

T

Figure 7.66: Contour plot of temperature T for compressible flow past a cylinder oflengths of 30.5

proach (see chapter 3 for more details). Hence, regardless of the choice of p,p and k and the parameters in the mathematical models (like Pe, Re, Deetc.), the computational process used here remains unconditionally stable.

As discussed in chapter 3, the modified least squares formulation uti-lized here is free of upwinding methods. The residual functions and theresidual functionals used in the formulation correspond to actual nonlineardifferential operators. With appropriate choices of p and k, this methodcan incorporate desired features of the theoretical solution in the compu-tational process. Adaptive computations based on residual functional andh, p refinements provide a mechanism to achieve true convergence to thetheoretical solutions. By maintaining integrals in the Riemann sense (ap-propriate choice of k) and by ensuring that the residual functionals for thewhole discretization approach zero, we ensure that the computed numericalsolutions indeed satisfy the governing differential equations in the pointwisesense, hence these solutions are indeed same as the theoretical solutions.[1–3,7–39]

Problems

7.1 Consider the one-dimensional steady state Burgers equation:

φ∂φ

dx− 1

Re

d2φ

dx2= 0 0 < x < 1

φ(0) = 1, φ(1) = 0

A least squares finite element formulation of this BVP has been implemented in Finesse[40] using strong form of the GDE in SS03, ktype= 4, kifo= 15. This formulation allowssolutions of classes Ci ; i = 1, . . . , 6 with p-levels up to 15. The objective of this assignmentto perform numerical studies for solutions of classes Ci ; i = 1, . . . , 6 using highly gradedmesh. Mesh design strategy in which the element with the highest (Ie/he) value is split intotwo elements, leads to inefficient mesh in this case due to highly localized solution gradientnear x = 1.0. Numerical solutions for this problem can also be obtained using any other


computational platform or by developing a computational code using the requirementsdescribed here.

A more efficient mesh design strategy in this case is one in which the element length in thehigh gradient area is chosen adequately and then increased progressively in the directionaway from the localized gradient area. Consider Re = 10, 000. For this Re, the shock widthis roughly O(10−5) at x = 1.0. Thus, begin with an element size of 0.00001 at x = 1.0 andprogressively increase the element length by a factor of 5 until x = 0.0. Consider threenode p-version elements (kifo= 15 as indicated above) in the discretizations. For this fixedmesh consider the following numerical studies.

(i) Solutions of class C1 ; p = 3, 5, 7, 9, 11, 13, 15

(ii) Solutions of class C2 ; p = 5, 7, 9, 11, 13, 15

(iii) Solutions of class C3 ; p = 7, 9, 11, 13, 15

(iv) Solutions of class C4 ; p = 9, 11, 13, 15

(v) Solutions of class C5 ; p = 11, 13, 15

(vi) Solutions of class C6 ; p = 13, 15

Use (2p+ 1) quadrature rule for all calculations

(a) Plot graphs of least squares functional I versus dofs on log-log scale for all solutionson a single sheet. Comment on:

(i) accuracy(ii) convergence rates(iii) other features

(b) Plot graphs of diφdxi

, (i = 0, 1, 2, . . . , 7) for each classes of solutions to observe theirconvergence with increasing p-levels

(c) Compare numerically computed results with the theoretical solution.(d) Observe the convergence behavior of Newton’s method with line search with increasing

p-levels and increasing order of the space. Provide a clear but concise discussion ofthe results and the findings.


for non-linear operators in BVP. Int. J. Comp. Eng. Sci., 5(1):133–207, 2004.

[2] K. S. Surana, O. Gupta, P. W. TenPas, and J. N. Reddy. h, p, k least squares finiteelement processes for 1d helmholtz equation. Int. J. Comp. Meth. in Eng. Sci. andMech., 7(4):263–291, 2006.

[3] Surana, K. S., Mahanti, R. K. and Reddy, J. N. Galerkin/Least Squares Finite Ele-ment Processes for BVPs in h, p, k Mathematical Framework. International Journalof Computational Engineering Sciences and Mechanics, 8:439–462, 2007.

[4] K. S. Surana, D. Nunez and J. N. Reddy. Giesekus Constitutive Model for Thermovis-coelastic Fluids based on Ordered Rate Constitutive Theories. Journal Of ResearchUpdates in Polymer Science, 2:232–260, 2013.



[7] H. Schlinchting. Boundary Layer Theory. McGraw-Hill, New York, 3rd edition, 1960.


[8] B. C. Bell. p-Version Least Squares Finite Element Method for Steady and UnsteadyFluid Flow. PhD thesis, The University of Kansas, Mechanical Engineering Depart-ment, 1993.

[9] D. Winterscheidt. p-Version Least Squares Finite Element Method for Fluid Dynam-ics. PhD thesis, The University of Kansas, Mechanical Engineering Department,1992.

[10] U. Ghia, K. N. Ghia, and C. T. Shin. High-Re Solutions for Incompressible Flow usingthe Navier-Stokes Equations and a Multigrid Method. J. Comp. Phys., 48:387–411,1982.

[11] M K. Denham and M. A. Patrick. Laminar Flow over a Downstream-Facing Stepin a Two-Dimensional Channel. Trans. Institute of Chemical Engineers, 52:361–367,1974.

[12] S. Allu, K. S. Surana, A. Romkes, and J. N. Reddy. Numerical Solutions of BVPsin 2D Viscous Compressible Flows using h, p, k Framework. Int. J. Comp. Meth. inEng. Sci. and Mechanics, 10(1):158–171, 2009.

[13] S. Allu. Computations of Viscous Compressible Flows in h, p, k Finite Element Frame-work with Variationally Consistent Integral Forms. PhD thesis, The University ofKansas, Mechanical Engineering Department, 2008.

[14] D. Winterscheidt and K. S. Surana. p-Version Least Squares Finite Element Formula-tion for Convection-Diffusion Equation. International Journal of Numerical Methodsin Engineering, 36:111–133, 1993.

[15] D. Winterscheidt and K. S. Surana. p-Version Least Squares Finite Element Formu-lation for Burgers Equation. International Journal of Numerical Methods in Engi-neering, 36:3629–3646, 1993.

[16] D. Winterscheidt and K. S. Surana. p-Version Least Squares Finite Element For-mulation for Two Dimensional Incompressible Fluid Flow. International Journal ofNumerical Methods in Fluids, 18:43–69, 1994.

[17] B. Bell and K. S. Surana. p-Version Least Squares Finite Element Formulation for TwoDimensional Incompressible Non-Newtonian Isothermal and Non-Isothermal FluidFlow. International Journal of Numerical Methods in Fluids, 18:127–167, 1994.

[18] N. B. Edgar and K. S. Surana. p-Version Least Squares Finite Element Formulationfor axisymmetric, Incompressible, Non-Newtonian Fluid Flow. Comp. Meth. in Appl.Mech. and Engg., 113:271–300, 1994.

[19] K. S. Surana and J. S. Sandhu. Investigation of Diffusion in p-Version LSFE andSTFSFE formulations. Computational Mechanics, 16(3):151–169, 1995.

[20] M. Bagheri and K. S. Surana. p-Version LSFEF for Extended κ-ε Model of Turbulencefor Fully Developed Flow. Communications in Numerical Methods in Engineering,16:83–95, 2000.

[21] M. Bagheri and K. S. Surana. p-Version LSFEF for Steady State Two DimensionalTurbulent Flow using κ-ε Model of Turbulence. Communications in Numerical Meth-ods in Engineering, 16:97–120, 2000.

[22] K. S. Surana and M. Bona. Non-Weak/Strong Solutions of Linear and Non-linearHyperbolic and Parabolic Equations Resulting from a Single Conservation Law. In-ternational Journal of Computational Engineering Science, 1(2):299–330, 2000.

[23] K. S. Surana and D. G. Van Dyne. Non-Weak/Strong Solutions in Gas Dynamics:A C11 p-Version STLSFEF in Eulerian Frame of Reference using ρ, u, p PrimitiveVariables. International Journal of Numerical Methods in Engineering, 53:1051–1099,2002.


[24] K. S. Surana and D. G. Van Dyne. Non-Weak/Strong Solutions in Gas Dynamics:A C11 p-Version STLSFEF in Lagrangian Frame of Reference using ρ, u, p PrimiteVariables. International Journal of Numerical Methods in Engineering, 53:1025–1050,2002.

[25] K. S. Surana and D. G. Van Dyne. Non-Weak/Strong Solutions in Gas Dynamics:A C11 p-Version STLSFEF in Lagrangian Frame of Reference using ρ, u, T PrimiteVariables. International Journal of Computational Engineering Science, 2(3):357–382,2001.



[28] K. S. Surana and J. N. Reddy. An Accurate and Robust Computational Methodologyfor Structural Dynamics Problems. Proceedings of the 2nd International Conferenceon Structural Stability and Dynamics, Singapore, Dec. 16-18, 2002.

[29] K. S. Surana and J. N. Reddy. The k-Version of Finite Element Method: A NewComputational Methodology for Boundary Value Problems. Proceedings of the ISSS-SPIE, International Conference on Smart Materials, Structures and Systems, IndianInstitute of Science, Bangalore, India, July 17-19, 2002.

[30] K. S. Surana and A. Mohammed. k-Version of Finite Element Method in 2D PolymerFlows: Oldroyd-B Constitutive Model. International Journal of Numerical Methodsin Fluids, 52:119–162, 2006.

[31] K. S. Surana, S. Allu, P. W. TenPas, and J. N. Reddy. k-Version of Finite ElementMethod in Gas Dynamics: Higher Order Global Differentiability Numerical Solutions.International Journal of Numerical Methods in Engineering, 69:1109–1157, 2006.

[32] K. S. Surana. A New Mathematical and Computational Framework for BVPs andIVPs. Proceeding of the 1st International Conference on Enhancement and Promo-tion of Comp. Meth. in Eng. Sci. and Mechanics (CMESM), Aug 10-12, Changchun,China, pages 21–29, 2006.

[33] K. S. Surana, M. K. EngelKemier, J. N. Reddy, and P. W. TenPas. k-Version LeastSquares Finite Element Processes for 2D Generalized Newtonian Fluid Flows. Inter-national Journal of Comp. Meth. in Eng. Sci. and Mechanics, 8:243–261, 2007.

[34] K. S. Surana, L. R. Anthoni, S. Allu, and J. N. Reddy. Strong and weak formof governing differential equations in least squares finite element processes in hpkframework. Int. J. Comp. Meth. in Eng. Sci. and Mech., 2008.

[35] K. S. Surana, S. Allu, J. N. Reddy, and P. W. TenPas. Least Squares Finite Ele-ment Processes in hpk Mathematical Framework for Non-Linear Conservation Law.International Journal Num. Meth. in Fluids, 57(10):1545–1568, 2008.

[36] K. S. Surana, S. Bhola, J. N. Reddy, and P. W. TenPas. k-Version of Finite ElementMethod in @D Polymer Flows: Upper Convected Maxwell Model. Int. J. Comp. andStructures, 86(17-18):1782–1808, 2008.

[37] K. S. Surana, K. M. Deshpande, A. Romkes, and J. N. Reddy. Computations ofNumerical Solutions in Polymer Flows using Giesekus Constitutive Model in h, p, kFramework with Variationally Consistent Integral Forms. Int. J. Comp. Meth. inEng. Sci. and Mechanics, 10:217–344, 2009.

[38] K. S. Surana, K. M. Deshpande, A. Romkes, and J. N. Reddy. Numerical Simulationsof BVPs and IVPs in Fiber Spinning using Giesekus Constitutive Model in h, p, kFramework. Int. J. Comp. Meth. in Eng. Sci. and Mechanics, 10(2):143–157, 2009.


[39] J. N. Reddy. An Introduction to Continuum Mechanics. Cambridge University Press,2nd edition, 2013.

[40] K.S. Surana. FINESSE, Finite Element System. Computational Mechanics Labora-tory, Department of Mechanical Engineering, University of Kansas, 2015.

8

Basic Elements of Mappingand Interpolation Theory

In this chapter we present basic definitions, concepts and details of inter-polation theory and mapping. This material is an integral component of thefinite element method and allows one to map irregular element shapes intostandard shapes to evaluate the integrals as well as permits construction oflocal approximation functions. We consider 1D, 2D, and 3D cases.

8.1 Mapping in one dimension

In one dimensional space, say x, the geometry of interest is a line seg-ment of length he with the position coordinates of the two ends defined byxe and xe+1. This line segment may very well represent a finite elementfor solving a BVP involving only one independent variable x and the is-sue at hand is obviously to define the behavior of a dependent variable,say φ(x) ∀x ∈ [xe, xe+1] = Ωe. While it is more appealing to consider thephysical domain Ωe, mathematically from the point of view of establishingconcepts of interpolation theory it is certainly more advantageous to mapΩe into another domain, say Ωξ : [xe, xe+1] = Ωe → Ωξ = [−1, 1], in whichthe ξ coordinate is often referred to as the natural coordinate. In this ap-proach, all line segments Ωe = [xe, xe+1], e = 1, . . . in the physical domainx are mapped into the same domain Ωξ = [−1, 1] in the natural coordinatespace ξ. Thus, the interpolation theory for dependent variable φ only needto be developed in the natural coordinate ξ in which the geometry of anyand every line segment or element is fixed: Ωξ = [−1, 1] and thus the ap-proximation φ must be described over Ωξ instead of Ωe. This process possesno special problem as the mapping x → ξ would permit us to get to thephysical domain x when we wish to do so. Details of mapping and varioustransformations are discussed in the following.

Consider a line segment Ωe = [xe, xe+1] in the physical domain x. LetΩξ = [−1, 1] be its map in the natural coordinate space ξ, as shown inFig. 8.1. Mapping of [xe, xe+1] into [−1, 1] can be accomplished by usinga stretch transformation or contraction mapping depending upon whetherhe < 2 or he > 2 and when he = 2 the mapping is identity. That is, if he < 2

493

494 BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

2

1−1

211he

y

xe xe+1

x ξ

η

Ωe Ωξ

Figure 8.1: Change of coordinates: physical space to natural coordinate space

the mapping simply consists of stretching he uniformly such that he = 2 inξ coordinate space. On the other hand, when he > 2 we uniformly contracthe such that he = 2 in ξ coordinate space. The mapping so described isobviously linear. Other possibilities exist as well (discussed later).

8.1.1 Mapping of points

Regardless of the precise nature of the mapping, in the abstract sense wecould describe it by

x = x(ξ) (8.1)

Such mapping can be easily constructed by using, for example, functions inξ space and position coordinates in x space. If we consider the line segmentbeing defined by the position coordinates of its two ends then we could write

x = x(ξ) =(1− ξ

2

)xe +

(1 + ξ

2

)xe+1. (8.2)

One could verify that such a mapping is a linear stretch and that(1−ξ

2

)and

(1+ξ2

)indeed are functions corresponding to points 1 and 2 in ξ space

which are a map of xe and xe+1 in x-space. Other possibilities exist, butfor now it suffices to consider this abstract form of the mapping definedby (8.1) for developing the general theory. Form (8.1) we note that such amapping only maps points, i.e. given a location ξ∗ ∈ Ωξ we could determinethe corresponding x∗ = x(ξ∗) ∈ Ωe. Mapping (8.1) must be one-to-one andonto, i.e. the inverse of the mapping ξ = ξ(x) must exist and must be unique.For our purpose here, (8.1) suffices.

8.1.2 Mapping of lengths

Elemental length dx, dξ in x, ξ spaces can be related using

dx =dx

dξdξ (8.3)

in which we define J = dxdξ , called the Jacobian of mapping or transformation.

Obviously we have

he =

xe+1∫xe

dx =

1∫−1

dx

dξdξ =

1∫−1

J dξ (8.4)

8.2. ELEMENTS OF INTERPOLATION THEORY OVER Ωξ = [−1, 1] 495

8.1.3 Behavior of dependent variable φ over Ωe

Our interest of course is to describe the behavior of φ over Ωe. Instead ofdoing so, we can also describe the behavior of φ over Ωξ using interpolationtheory,

φ = φ(ξ) (8.5)

Since for a ξ, say ξ∗, the corresponding x, say x∗, is given by x(ξ∗) using(8.1) and using (8.5) we in fact have φ(ξ∗) and thus the behavior of φ overΩe. This approach is advantageous in that the interpolation theory onlyneeds to be developed for Ωξ and not for each Ωe (e = 1, 2, . . . , N).

Obtaining diφdxi

; i = 1, . . .

Since φ = φ(ξ) and x = x(ξ) we have

dφ(ξ)

dξ=dφ(ξ)

dx

dx

dξ=dφ(ξ)

dxJ (8.6)

Thereforedφ(ξ)

dx=

1

J

dφ(ξ)

dξ(8.7)

It is straight forward to show that

diφ(ξ)

dxi=

1

(J)idiφ(ξ)

dξi(8.8)

For example, when the mapping is defined by (8.2) we have

J =dx

dξ=

1

2(xe+1 − xe) =

he2

Other forms of mapping in one dimension will be discussed in later sections.In any case, once x = x(ξ) is defined, everything else is defined.

8.2 Elements of interpolation theory over Ωξ = [−1, 1]

Consider the element map Ωξ = [−1, 1] in the natural coordinate spaceξ. In this section we consider various alternatives to establish an analyticalexpression for the behavior of a function f = f(ξ), ξ ∈ [−1, 1].

8.2.1 A polynomial approximation in one dimension

Let f(ξ) have the values f(ξ0) = f0, f(ξ1) = f1, . . . , f(ξn) = fn at n+ 1points ξ0, ξ1, . . . , ξn ∈ [−1, 1]. then we can describe f(ξ) as a polynomial ofdegree n that passes through these n+ 1 points

f(ξ) = a0 + a1ξ + a2ξ2 + · · ·+ anξ

n (8.9)


or

f(ξ) = [1 ξ ξ2 · · · ξn]

a0

a1...an

(8.10)

Upon substituting f(ξ) = fi and ξ = ξi; i = 0, 1, . . . , n in (a), we obtainthe following set of linear simultaneous algebraic equations among ai (i =0, 1, . . . , n) from which ai (i = 0, 1, . . . , n) can be calculated:

1 ξ0 ξ20 . . . ξ

n0

1 ξ1 ξ21 . . . ξ

n1

......

.... . .

...1 ξn ξ

2n . . . ξ

nn

a0

a1...an

=

f0

f1...fn

(8.11)

or[C]a = f (8.12)

Thereforea = [C]−1f (8.13)

Substituting from (8.13) into (8.10) for a we obtain

f(ξ) = [1 ξ ξ2 · · · ξn][C]−1

f0

f1...fn

(8.14)

or

f(ξ) = [L0(ξ) L1(ξ) · · · Ln(ξ)]

f0

f1...fn

(8.15)

or

f(ξ) =n∑i=0

Li(ξ)fi (8.16)

in which Li(ξ) are basis functions or local approximation functions corre-sponding to locations ξi; i = 0, 1, . . . , n and f(ξ) is called local approxi-mation of f over Ωξ. Local approximation is a linear combination of thebasis functions and the function values fi (constants) at ξi; i = 0, 1, . . . , ngenerally called degrees of freedom.


Remarks.

(1) In order for a and, hence, f(ξ) to be unique, it is obvious that thematrix [C] should not be singular, i.e. det [C] 6= 0.

(2) If ξi (i = 0, 1, . . . , n) are distinct then det [C] 6= 0 is assured and, hence,the uniqueness of a is assured as well.

(3) Such approach is error prone due to matrix inversion and is computa-tionally not very efficient. As the number of points increases, we need toinvert increasingly large matrices. Choices of ξi are obviously crucial aswell. This process described in the above section can be accomplishedmore efficiently using Lagrange interpolating polynomials (described inthe following in section 8.2.2) in which the inverse of the matrices isavoided.

8.2.2 Lagrange interpolating polynomials in one dimension

Let Ωξ = [−1, 1] be the domain (i.e. map of an element in the naturalcoordinate system ξ). Let (ξi, f(ξi)) (i = 0, 1, . . . , n) be the pair of points

Theorem 8.1. There exists a unique polynomial φ(ξ) or degree not exceedingn called the Lagrange interpolating polynomial such that

φ(ξi) = f(ξi), i = 0, 1, . . . , n (8.17)

Proof. The existence of the polynomial φ can be proved if we can establishthe existence of polynomials Lk; k = 0, 1, . . . , n with the properties

(i) Each polynomial Lk is a polynomial of degree at most n.

(ii) Li(ξj) =

1 j = i

0 j 6= i, i = 0, 1, . . . , n

Assuming the existence of these polynomials, we can write

φ(ξ) =

n∑k=0

fkLk(ξ) (8.18)

where fk = f(ξk).

(iii) We also note thatn∑i=0

Li(ξ) = 1 (8.19)


Remarks.

(1) Lk(ξ) are polynomials of degree at most n.

(2) φ(ξ) is a linear combination of Lk(ξ) and fk, hence φ(ξ) is a polynomialof degree at most n.

(3) φ(ξm) = fm = f(ξm), since Lk(ξm) = 0 for k 6= m and Lk(ξm) = 1 fork = m .

(4) Lk(ξ) are called Lagrange interpolating polynomials.

The Lagrange interpolating polynomials Lk(ξ) can be obtained using

Lk(ξ) =

n∏m=0m6=k

( ξ − ξmξk − ξm

), k = 0, 1, . . . , n (8.20)

The polynomials Lk(ξ) have the desired properties as described earlier and,hence, we can write

f(ξ) = φ(ξ) =

n∑i=0

fiLi(ξ), fi = f(ξi) (8.21)

Example 8.1 (Quadratic Lagrange polynomials over Ωe = [−1, 1]). Con-sider the points (ξ1, f(ξ1)),(ξ2, f(ξ2)) and (ξ3, f(ξ3)) where (ξ1, ξ2, ξ3) are(−1, 0, 1). Then we can express f(ξ) by φ(ξ) given by

f(ξ) = φ(ξ) = L1(ξ)f1 + L2(ξ)f2 + L3(ξ)f3 ∀ξ ∈ [ξ1, ξ3] = [−1, 1]

So all we need to do is to establish L1(ξ), L2(ξ) and L3(ξ). We do so in thefollowing. Let

Lk(ξ) =3∏

m=1m6=k


), k = 1, 2, 3

Thus, we have

k = 1 : L1(ξ) =

3∏m=1m6=1

( ξ − ξmξ1 − ξm

)=

(ξ − ξ2)(ξ − ξ3)

(ξ1 − ξ2)(ξ1 − ξ3)=ξ(ξ − 1)

2

k = 2 : L2(ξ) =3∏

m=1m6=2


)=

(ξ − ξ1)(ξ − ξ3)

(ξ2 − ξ1)(ξ2 − ξ3)= (1− ξ2)

k = 3 : L3(ξ) =3∏

m=1m6=3


)=

(ξ − ξ1)(ξ − ξ2)

(ξ3 − ξ1)(ξ3 − ξ2)=ξ(ξ + 1)

2


Remarks.

(1) Lk(ξm) =

1 m = k

0 m 6= k, k = 1, 2, 3.

(2)∑3

k=1 Lk(ξ) = ξ(ξ−1)2 + (1− ξ2) + ξ(ξ+1)

2 = 1

(3) Plots of Lk(ξ) as a function of ξ for ξ ∈ [−1, 1] are shown in Fig. 8.2.

21 3ξ

ξ = −1 ξ = 0 ξ = 1

1 1

L1(ξ) L2(ξ) L3(ξ)

Figure 8.2: Plots of Li(ξ) (i = 1, 2, 3) for ξ ∈ [−1, 1]

(4) We have

f(ξ) = φ(ξ) = L1(ξ)f(ξ1) + L2(ξ)f(ξ2) + L3(ξ)f(ξ3)

or

f(ξ) = φ(ξ) =

3∑i=1

Li(ξ)f(ξi) =

3∑i=1

Li(ξ)fi

where f1, f2, f3 are the values of the function at the nodes or points1, 2, 3 located at ξ = −1, 0, 1. In this case, the element would be a threenode element with values of the function at the nodes as unknowns andφ(ξ) is quadratic over [−1, 1].

(5) We note that Li(ξ) (i = 1, 2, 3) are quadratic basis functions correspond-ing to nodes 1, 2, 3.

(6) Henceforth, we refer to f(ξ) = φ(ξ) interpolation as local approximationof the dependent variable φ over Ωξ. This is due to the fact that if Ωξ

is the map of an element Ωe, then φ(ξ) is purely local to this elementwithout regard to the neighboring elements.

Example 8.2 (Cubic Lagrange polynomials over Ωe = [−1, 1]). Considerthe points (ξi, f(ξi)), (i = 1, . . . , 4) where ξ1, . . . , ξ3) are (−1,−1

3 ,13 , 1). Then

we can express f(ξ) by φ(ξ) given by

f(ξ) = φ(ξ) =

4∑i=1

Li(ξ)f(ξi) =

4∑i=1

Li(ξ)fi ∀ξ ∈ [ξ1, ξ4] = [−1, 1]


In this case, also, we need to establish Lk(ξ). Recall that

Lk(ξ) =4∏

m=1m6=k


), k = 1, 2, 3, 4

or

L1(ξ) =4∏

m=1m6=1


)=

(ξ − ξ2)(ξ − ξ3)(ξ − ξ4)

(ξ1 − ξ2)(ξ1 − ξ3)(ξ1 − ξ4)

= − 9

16(1− ξ)(1

3+ ξ)(

1

3− ξ)

L2(ξ) =

4∏m=1m6=2


)=

(ξ − ξ1)(ξ − ξ3)(ξ − ξ4)

(ξ2 − ξ1)(ξ2 − ξ3)(ξ2 − ξ4)

=27

16(1 + ξ)(1− ξ)(1

3− ξ)

L3(ξ) =4∏

m=1m6=3


)=

(ξ − ξ1)(ξ − ξ2)(ξ − ξ4)

(ξ3 − ξ1)(ξ3 − ξ2)(ξ3 − ξ4)

=27

16(1 + ξ)(1− ξ)(1

3+ ξ)

L4(ξ) =4∏

m=1m6=4


)=

(ξ − ξ1)(ξ − ξ2)(ξ − ξ3)

(ξ4 − ξ1)(ξ4 − ξ2)(ξ4 − ξ3)

= − 9

16(1

3+ ξ)(

1

3− ξ)(1 + ξ)

Remarks.

(1) Lk(ξm) =

1 m = k

0 m 6= k, k = 1, 2, 3, 4.

(2)∑4

k=1 Lk(ξ) = 1

Plots of Lk(ξ) as a function of ξ for ξ ∈ [−1, 1] are shown in Fig. 8.3.

(3) As the degree of the polynomial increases over the domain Ωξ = [−1, 1]so do the number of nodes or points, that is, a linear polynomials needstwo nodes or points, a quadratic behavior requires three nodes or points,and likewise a cubic polynomial over Ωξ would need four nodes or pointsbetween [−1, 1] in the ξ coordinate space and so on.

(4) These domains Ωξ with 2, 3, 4, . . . nodes in the ξ coordinate system couldbe viewed as maps of the 1D finite elements from the physical space xto the natural coordinate space ξ.


(5) At each node only the function value is required to establish the ana-lytical behavior of the function over Ωξ. Thus, when such elements areused in the discretization, we only have inter-subdomain continuity ofthe function, i.e. all such interpolation may consist of higher degree butare of class C0 and, hence, will exhibit inter-subdomain discontinuity ofthe first derivatives of the function and the derivatives of order higherthan one are not defined at the inter-element node points.

(6) In summary, we can list 1D C0 interpolation of f over Ωξ as follows

f(ξ) =

n∑i=1

Li(ξ) fi

where n = 1, 2, 3, . . . for the linear, quadratic, and cubic cases. Li(ξ)are called Lagrange functions, local approximation functions, or basisfunctions (see Fig. 8.4).

1

1 2 3 4

1

1 2 3 4

1

1 2 3 4

1

1 2 3 4

L3(ξ)

ξ

−1 −13

113

L1(ξ)

ξ

−1 −13

113

L4(ξ)

ξ

−1 −13

113

L2(ξ)

ξ

−1 −13

113

Figure 8.3: Plots of Li(ξ); i = 1, 2, 3 for ξ ∈ [−1, 1]

8.2.3 p-version hierarchical functions in one dimension

Consider Ωξ = [−1, 1] as the domain of interpolation (referred to as el-ement or finite element map from the physical coordinate space x to ξ).Behavior of a function f over Ωξ is referred to as element interpolation orlocal approximation as it is local to the element domain Ωξ. The subjectof element interpolation or local approximation is crucial as it establisheshow the function or dependent variable behaves over the element domain.


C0

Basis functionsNodal configuration

L1(ξ) = 1−ξ2

L2(ξ) = 1+ξ2

1 2

-1 1ξ

Type

1 4

-1 1ξ

2

13

3

−13

L2(ξ) = 2716(1 + ξ)(1− ξ)(1

3 − ξ)L3(ξ) = 27

16(1 + ξ)(1− ξ)(13 + ξ)

L4(ξ) = − 916(1

3 + ξ)(13 − ξ)(1 + ξ)

L1(ξ) = − 916(1

3 + ξ)(13 − ξ)(1− ξ)

C0

L2(ξ) = 1− ξ2

L3(ξ) = ξ(ξ+1)2

L1(ξ) = ξ(ξ−1)2

1 3

-1 1ξ

2

0

C0

Figure 8.4: Linear, quadratic, and cubic Lagrange interpolation functions in terms of thenatural coordinate

It is well established that higher degree local interpolations as opposed tolower degree yield higher convergence rates and have many other beneficialproperties. The simplest manner in which the higher degree local inter-polations can be constructed is by using Lagrange interpolation functions.Consider the one dimensional case in which Ωξ = [−1, 1]. Figure 8.5 shows atwo-node linear, three-node quadratic and four-node cubic Lagrange config-urations in the natural coordinate space ξ for which the basis functions (orlocal approximation functions or Lagrange interpolation functions) can beeasily constructed using Lagrange interpolation (as shown in section 8.2.2).

Remarks.

(1) From Figure 8.5, we note that for linear local approximations or La-grange functions the degree of the polynomials is one (p = 1). Forquadratic local approximations the degree of the polynomial is two (p =2) and for cubic local approximations the degree of the polynomial isthree (p = 3). Thus, this process does provide a mechanism for increas-ing the degree of local approximation, i.e. p-level.

(2) As we increase the p-level (i.e. degree of local approximation) additionalnodes need to be added to the existing elements (domain Ωe), which isan undesirable feature.

(3) If we examine the basis functions (Lagrange interpolation functions) forthe linear and quadratic case, we note that Lagrange functions L1(ξ) =1−ξ

2 and L2(ξ) = 1+ξ2 for linear approximation do not explicitly appear


C0

Basis functionsNodal configurations

L1(ξ) = 1−ξ2

L2(ξ) = 1+ξ2

1 2

-1 1ξ

Type

1 4

-1 1ξ

2

13

3

−13

L2(ξ) = 2716(1 + ξ)(1− ξ)(1

3 − ξ)L3(ξ) = 27

16(1 + ξ)(1− ξ)(13 + ξ)

L4(ξ) = − 916(1

3 + ξ)(13 − ξ)(1 + ξ)

L1(ξ) = − 916(1

3 + ξ)(13 − ξ)(1− ξ)

C0

L2(ξ) = 1− ξ2

L3(ξ) = ξ(ξ+1)2

L1(ξ) = ξ(ξ−1)2

1 3

-1 1ξ

2

0

C0

f(ξ) =4∑i=1

Li(ξ) fi

f(ξ) =3∑i=1

Li(ξ) fi

f(ξ) =2∑i=1

Li(ξ) fi

(linear: p = 1)

(cubic: p = 3)

(quadratic: p = 2)

Figure 8.5: 1D Lagrange basis functions Ωξ = [−1, 1]

in the Lagrange functions for the quadratic or the cubic local approx-imations. Yet, we know that a quadratic local approximation is quitecapable of linear local approximations over Ωξ. In other words, the linearbehavior is implicitly embedded in the quadratic behavior but the linearbasis functions do not explicitly appear in the basis functions for thequadratic case. This property of quadratic local approximation to con-tain linear approximation is known as an ‘embedding property’. Thus,in general we can say that when 1D local approximations are constructedusing Lagrange interpolation functions, the lower degree basis functionsare implicitly embedded in the higher degree basis functions.

Due to the fact that the embedding property of these basis functions isnot explicit, when computations are performed for progressively increas-ing p-levels (degree of the polynomial), we cannot take advantage of thecomputations performed at the lower p-level when performing computa-tions for higher p-levels. That is, when computing at p-level of (p+ 1),computations performed at p-level of p are of no use i.e. cannot be used.

(4) In order to remedy the shortcomings described in (3), we need to ac-complish the following.

(a) We should be able to increase the p-level, i.e. degree of local ap-proximation, without adding additional nodes. With this featurethe discretizations and the nodes will remain fixed for all p-levels,which is advantageous.


(b) The embedding property of the higher degree local basis functionsneeds to be explicit rather than implicit. That is, when we examinethe basis functions for the quadratic local approximation, the linearbasis functions and the corresponding nodal degrees of freedom mustappear explicitly in the quadratic case. In other words, a linearlocal approximation must be an explicit subset of a quadratic localapproximation and likewise a quadratic local approximation mustbe a subset of a cubic approximation and so on. In general, a localapproximation of degree p must be a complete and explicit subsetof the local approximation of degree (p + 1). This property will bereferred to as the hierarchical property of the local approximations.

Derivation of C0 p-version hierarchical basis functions

Recall that the local approximations derived using Lagrange interpola-tion functions do posses the embedding property, but the embedding prop-erty is implicit. The purpose of this derivation is to make the embeddingproperty explicit and at the same time eliminate the need for additionalnodes for increasing p-levels. We present details of the development in thefollowing. Let (ξi, f(ξi)) = (ξ, fi) (i = 1, . . . , n) be the node locations andfunction values at n nodes of an n-node Lagrange element. Then

f(ξ) =n∑i=1

Li(ξ)fi (8.22)

in which Li(ξ) are Lagrange interpolation functions or polynomials. Considera quadratic case, that is n = 3 and substitute Li(ξ) (i = 1, 2, 3) in (8.22)

f(ξ) = −1

2ξ(1− ξ)f1 + (1− ξ2)f2 +

1

2ξ(1 + ξ)f3 (8.23)

By rearranging terms in (8.23) we can write

f(ξ) =1

2(1− ξ)f1 +

1

2(1 + ξ)f3 +

(ξ2 − 1

2

)(f1 − 2f2 + f3) (8.24)

Differentiating (8.23) or (8.24)twice with respect to ξ we obtain

d2f

dξ2= f1 − 2f2 + f3 (8.25)

Note that for the three node quadratic element, node 2 is located at ξ = 0,so that (8.25) can be written as(d2f

dξ2

)ξ=0

= f1 − 2f2 + f3 (8.26)


Substituting back into (8.24) we obtain

f(ξ) =1

2(1− ξ)f1 +

1

2(1 + ξ)f3 +

(ξ2 − 1

2

)(d2f

dξ2

)ξ=0

(8.27)

where f1 = f |ξ=−1 and f3 = f |ξ=+1. Recall that for the linear approximationof f (two-node element) we have

f(ξ) =(1− ξ

2

)f1 +

(1 + ξ

2

)f2 (8.28)

where f1 = f |ξ=−1 and f2 = f |ξ=+1. Comparing (8.28) with (8.24) wenote that the linear approximation is an explicit subset of the quadratic ap-proximation in the sense that the linear approximation functions and thenodal dofs in (8.28) appear explicitly in the quadratic approximation (8.27).Thus, when using (8.27) and (8.28) for linear and quadratic approximationsof f , when we increase the p-level form 1 to 2 we only need to add the

term(ξ2−1

2

)(d2fdξ2

)ξ=0

to the linear approximation to obtain a quadratic ap-

proximation. Hence, the linear approximation is explicitly embedded in thequadratic approximation. In other words, the linear and quadratic approxi-mations described by (8.27) and (8.28) are hierarchical.

Next, we consider a cubic Lagrange interpolation over Ωξ = [−1, 1] for afour-node element in Ωξ, and we can write

f(ξ) =4∑i=1

Li(ξ)fi (8.29)

or

f(ξ) = − 9

16

(1− ξ

)(1

3+ ξ)(1

3− ξ)f1 +

27

16

(1 + ξ

)(1− ξ

)(1

3− ξ)f2

+27

16

(1 + ξ

)(1− ξ

)(1

3+ ξ)f3 −

9

16

(1

3+ ξ)(1

3− ξ)(

1 + ξ)f4 (8.30)

If we differentiate f(ξ) with respect to ξ twice and thrice and evaluate theseat ξ = 0 and then substitute these into (8.30), we obtain

f(ξ) =1

2(1− ξ)f1 +

1

2(1 + ξ)f3 +

(ξ2 − 1

2!

)(d2f

dξ2

)ξ=0

+(ξ3 − ξ

3!

)(d3f

dξ3

)ξ=0

(8.31)Comparing the cubic approximation (8.31) with the quadratic approximation(8.27), we note that the cubic approximation can be obtained by adding the

term(ξ3−ξ

3!

)(d3fdξ3

)ξ=0

to the quadratic approximation. Another point to

note is that the addition of this term does not require a new node, i.e.for the cubic approximation the same three-node configuration used for the


quadratic case suffices. Clearly, the quadratic approximation is a completesubset of the cubic approximation, i.e. has an explicit embedding propertyand hence is hierarchical.

The linear, quadratic and cubic approximations described by (8.28),(8.24) and (8.31) are hierarchical C0 p-version approximations which re-quire only a three-node element configuration in which the nodes are lo-cated at ξ = −1, ξ = 0 and ξ = 1. Thus, we note that using the proceduredescribed here the two-node, three-node and four-node Lagrange elementshave been reduced to an equivalent three-node configuration for the linear,quadratic and cubic approximations described by (8.28), (8.27) and (8.31)(see Fig. 8.6). In general, if we consider pξ, the degree of Lagrange interpo-lating polynomials over (pξ + 1), nodes in Ωξ and follow a similar procedureas described above, then we find that this can also be reduced to a three-node configuration (with nodes at ξ = −1, ξ = 0 and ξ = 1) for which wecan write the following hierarchical p-version approximation for f(ξ).

1 2

-1 1ξ

1 4

-1 1ξ

2

13

3

−13

1 3

-1 1ξ

2

0

1 3

-1 1ξ

2

0

...

3-node p-version element

can be reduced to

Lagrange elements

Figure 8.6: Lagrange to Hierarchical interpolation functions

f(ξ) =1

2(1− ξ)f1 +

1

2(1 + ξ)f3 +

pξ∑i=2

ξi − ai!

(difdξi

)ξ=0

(8.32)

or

f(ξ) = N1ξ1 (ξ)f |ξ=−1 +N1ξ

3 (ξ)f |ξ=+1 +

pξ∑i=2

N iξ2 (f,ξi)ξ=0 (8.33)

where

N1ξ1 =

(1− ξ2

), N1ξ

3 =(1 + ξ

2

), N iξ

2 =(ξi − a

i!

)(f,ξi)ξ=0 =

dif

dξi

∣∣∣∣ξ=0

=(difdξi

)node 2

(8.34)


and

a =

1, i is even

ξ, i is odd

Equations (8.32) and (8.33) represents hierarchical p-version C0 approxima-tion of f over Ωξ = [−1, 1] in which the element nodal configuration is athree-node configuration with nodes at ξ = −1, 0 and +1 and remains fixedregardless of p-level. The p-version hierarchical approximation of C0 typedescribed by (8.32) or (8.33) can be more conveniently expressed in tabularform in terms of the basis functions and nodal variable operators as opposedto nodal variables, as shown in Table 8.1. Nodal variable operators assignedto specific nodes act on the dependent variables to produce nodal degrees offreedom.

Table 8.1: C0 basis functions and nodal variable operators

1D hierarchical p-version p-level 1D hierarchical p-version nodal

basis functions (C0 type) variable operators (C0 type)

−1 10

2 31ξ

−1 10

2 31ξ

12(1− ξ) 1

2(1 + ξ) 1 1 1

ξ2−12!

2 d2

dξ2

ξ3−ξ3!

3 d3

dξ3

ξ4−14!

4 d4

dξ4

......

...ξpξ−apξ!

pξdpξ

dξpξ

8.2.4 Higher order global differentiability approximations inone dimension: p-version

Let x = x(ξ) define the map of a three-node (equally spaced) elementfrom x-space (Ωe) to ξ-space (Ωξ). Let xi (i = 1, 2, 3) be the coordinates ofthe nodes of the element in x-space. Then,

x(ξ) =(1− ξ

2

)x1 +

(1 + ξ

2

)x3 (8.35)

defines the mapping of the points from ξ-space to x-space. As usual, we have

J =dx

dξ=he2

(8.36)


where he is the element length in x-space. Let φ be the quantity of interest tobe interpolated over Ωe. Ωe is the domain of an element e of the discretizationΩT =

⋃e Ωe. In finite element processes we approximate φ over Ωe by φeh.

Thereby they approximation of φh over ΩT is given by φh =⋃e φ

eh. The

global differentiability of φh is naturally dependent on the differentiabilityachievable by

⋃e φ

eh and is controlled by the inter-element boundaries. Thus,

in this section we consider development of φeh, that is, approximations of φover Ωe that are capable of yielding global differentiability of orders 1, . . . , iof φh. For the sake of simplicity we drop the subscript h and superscript ein the following. We consider approximations of type C1(Ωe), C2(Ωe) andalso of Ci(Ωe).

8.2.4.1 Local approximation of class C1(Ωe)

In the following we switch from f to φ, as in finite element processes wehave commonly used φ for the dependent variable.

The one dimensional C0 p-version hierarchical interpolation of φ over Ωξ

(a three-node element) can be written as (for the 3-node configuration shownin Fig. 8.7)

φ(ξ) =(1− ξ

2

)φ1 +

(1 + ξ

2

)φ2 +

p∑i=2

(ξi − ai!

) diφdξi

∣∣∣∣ξ=0

(8.37)

where

a =

1, i is even

ξ, i is odd

ξ

η

1 3 2

2

ξ = −1 ξ = 0 ξ = 1

Figure 8.7: A C0, p-version hierarchical interpolation for the 3-node configuration

Here, φ1 and φ2 are the nodal values of φ at the end nodes 1 and 2. Differ-entiating φ(ξ) with respect to ξ, we obtain

dφ

dξ= −1

2φ1 +

1

2φ2 +

p∑i=2

( iξi−1 − ai!

) diφdξi

∣∣∣∣ξ=0

(8.38)


where

a =

0, i is even

1, i is odd

Evaluating dφdξ at ξ = −1 (node 1) and at ξ = +1 (node 2) and denoting

these by dφ1dξ and dφ2

dξ , we obtain

dφ

dξ

∣∣∣∣ξ=−1

=dφ1

dξ= −1

2φ1 +

1

2φ2 −

d2φ

dξ2

∣∣∣∣ξ=0

+1

3

d3φ

dξ3

∣∣∣∣ξ=0

+

p∑i=4

( i(−1)i−1 − ai!

) diφdξi

∣∣∣∣ξ=0

(8.39)

dφ

dξ

∣∣∣∣ξ=1

=dφ2

dξ= −1

2φ1 +

1

2φ2 +

d2φ

dξ2

∣∣∣∣ξ=0

+1

3

d3φ

dξ3

∣∣∣∣ξ=0

+

p∑i=4

( i(1)i−1 − ai!

) diφdξi

∣∣∣∣ξ=0

(8.40)

Solving for d2φdξ2

∣∣∣ξ=0

and d3φdξ3

∣∣∣ξ=0

from (8.39) and (8.40) and substituting

these into (8.37), we obtain

φ(ξ) =((1− ξ)

2+

(ξ3 − ξ)4

)φ1 +

((ξ3 − ξ)4

− (ξ2 − 1)

4

)dφ1

dξ

+((1 + ξ)

2− (ξ3 − ξ)

4

)φ2 +

((ξ3 − ξ)4

+(ξ2 − 1)

4

)dφ2

dξ

+

p∑i=4,6,...

((ξi − 1)− 12(ξ2 − 1)

i!

) diφdξi

∣∣∣∣ξ=0

+

p∑i=5,7,...

((ξi − ξ)−(i−1

2

)(ξ3 − ξ)

i!

) diφdξi

∣∣∣∣ξ=0

(8.41)

The interpolation defined in (8.41) for φ(ξ) ensures inter-element continuityof both φ and dφ

dξ in the natural coordinate space ξ. However, in the solutionof differential equations using the finite element method, we require inter-element continuity of φ and dφ

dx . This can be accomplished easily by notingthat

dφ

dx=

1

J

dφ

dξ(8.42)

Hence,dφ1

dξ= J

dφ1

dxand

dφ2

dξ= J

dφ2

dx(8.43)


If the element is a straight line with equally spaced nodes, then J = he2 where

he is the length of element e. Thus, the interpolation (8.41) can be writtenas

φ(ξ) =((1− ξ)

2+

(ξ3 − ξ)4

)φ1 +

((ξ3 − ξ)4

− (ξ2 − 1)

4

)Jdφ1

dx

+((1 + ξ)

2− (ξ3 − ξ)

4

)φ2 +

((ξ3 − ξ)4

+(ξ2 − 1)

4

)Jdφ2

dx

+

p∑i=4,6,...

((ξi − 1)− 12(ξ2 − 1)

i!

) diφdξi

∣∣∣∣ξ=0

+

p∑i=5,7,...

((ξi − ξ)−(i−1

2

)(ξ3 − ξ)

i!

) diφdξi

∣∣∣∣ξ=0

(8.44)

or

φ(ξ) = N01 (ξ)φ1 +N1

1 (ξ)dφ1

dx+N0

2 (ξ)φ2 +N12 (ξ)

dφ2

dx+

p∑i=4

N i3(ξ)

diφ

dξi

∣∣∣∣ξ=0

(8.45)where

N01 (ξ) =

((1− ξ)2

+(ξ3 − ξ)

4

)N1

1 (ξ) =((ξ3 − ξ)

4− (ξ2 − 1)

4

)J

N02 (ξ) =

((1 + ξ)

2− (ξ3 − ξ)

4

)N1

2 (ξ) =((ξ3 − ξ)

4+

(ξ2 − 1)

4

)J

N i3(ξ) =

(

(ξi−1)− 12

(ξ2−1)

i!

)i is even(

(ξi−ξ)− i−12

(ξ3−ξ)i!

)i is odd

(i = 4, 5, . . . , p)

(8.46)

We can illustrate these graphically in terms of nodal approximation functionsand the nodal variable operators as follows (see Fig. 8.8).

The interpolation functions defined in (8.44) are unique and satisfy inter-element continuity of both φ and dφ

dx . To establish uniqueness, it suffices toconsider a special case of (8.44) in which the terms corresponding to i =4, 5, . . . , p are absent. In particular, we need to show that (1) the dimensionof the space of φ(ξ) is the same as the total degrees of freedom for theelement and (2) that if all the degrees of freedom are identically zero thenφ(ξ) = 0 ∀ξ ∈ Ωξ. With i = 4, 5, . . . , p absent in (8.44) we have a completecubic polynomial in ξ with basis functions or monomials 1, ξ, ξ2 and ξ3 which


ξ

η

1 3

N02 (ξ)

N11 (ξ) N1

2 (ξ)

N01 (ξ)

N i3(ξ)

2

C1(Ωe) basis functions

ξ

η

1 3

ddx

di

dξi

∣∣∣ξ=0

2

1 1

i = 4, 5, . . . , pi = 4, 5, . . . , p

ddx

Nodal variable operators forC1(Ωe) interpolations

Figure 8.8: C1 basis functions and nodal variable operators

is a complete set of linearly independent monomials. Hence, the dimensionof the space of φ(ξ) is four (equal to the number of basis functions). Thedegrees of freedom for this element are φ1, dφ1

dx , φ2 and dφ2dx . If φ1 = 0,

dφ1dx = 0, φ2 = 0 and dφ2

dx = 0 and if φ(ξ) is a cubic polynomial in ξ, then

φ(ξ) = 0∀ξ ∈ Ωξ. Alternatively, since φ(ξ) = N10φ1 +N1

1dφ1dx +N0

2φ2 +N12dφ1dx

then φ(ξ) = 0 ∀ξ ∈ Ωξ if all of the degrees of freedom at the nodes are zero.The inter-element continuity follows from the fact that both φ and dφ

dx arenodal degrees of freedom at the end nodes of the element.

8.2.4.2 Interpolations or local approximations of class C2(Ωe):

Following the procedure similar to that used for deriving C1(Ωe) basisfunctions, we can derive C2(Ωe) type p-version interpolation functions (orbasis functions) in one dimension. Let φ(ξ) of class C0(Ωe) be given by (C0

p-version hierarchical for the three node configuration).

φ(ξ) =(1− ξ

2

)φ1 +

(1 + ξ

2

)φ2 +

(ξ2 − 1

2!

) d2φ

dξ2

∣∣∣∣ξ=0

+(ξ3 − ξ

3!

) d3φ

dξ3

∣∣∣∣ξ=0

+(ξ4 − 1

4!

) d4φ

dξ4

∣∣∣∣ξ=0

+(ξ5 − ξ

5!

) d5φ

dξ5

∣∣∣∣ξ=0

+

p∑i=6

N i3(ξ)

diφ

dξi

∣∣∣∣ξ=0

(8.47)

Hence

dφ

dξ= −1

2φ1 +

1

2φ2 + ξ

d2φ

dξ2

∣∣∣∣ξ=0

+

(3ξ2 − 1

3!

)d3φ

dξ3

∣∣∣∣ξ=0

+

(ξ3

6

)d4φ

dξ4

∣∣∣∣ξ=0

+

(5ξ4 − 1

5!

)d5φ

dξ5

∣∣∣∣ξ=0

+

p∑i=1

dN i3

dξ

diφ

dξi

∣∣∣∣ξ=0

(8.48)


d2φ

dξ2=d2φ

dξ2

∣∣∣∣ξ=0

+ ξd3φ

dξ3

∣∣∣∣ξ=0

+

(ξ2

2

)d4φ

dξ4

∣∣∣∣ξ=0

+

(ξ3

6

)d5φ

dξ5

∣∣∣∣ξ=0

+

p∑i=6

d2N i3(ξ)

dξ2

diφ

dξi

∣∣∣∣ξ=0

(8.49)

We use (8.48) and (8.49) to evaluate dφdξ and d2φ

dξ2at ξ = −1 and ξ = 1 and

then define

dφ1dξdφ2dξd2φ1dξ2

d2φ2dξ2

= [B]

φ1

φ2

+ [A]

d2φdξ2

d3φdξ3

d4φdξ4

d5φdξ5

ξ=0

+

p∑i=6

(dN i

3dξ

∣∣∣ξ=−1

diφdξi

∣∣∣ξ=0

)p∑i=6

(dN i

3dξ

∣∣∣ξ=1

diφdξi

∣∣∣ξ=0

)p∑i=6

(d2N i

3dξ2

∣∣∣ξ=−1

diφdξi

∣∣∣ξ=0

)p∑i=6

(d2N i

3dξ2

∣∣∣ξ=1

diφdξi

∣∣∣ξ=0

)

(8.50)

where

dφ1

dξ=dφ

dξ

∣∣∣∣ξ=−1

,dφ2

dξ=dφ

dξ

∣∣∣∣ξ=1

d2φ1

dξ2=d2φ

dξ2

∣∣∣∣ξ=−1

,d2φ2

dξ2=d2φ

dξ2

∣∣∣∣ξ=1

(8.51)

and

[A] =

−1 1

3 −16

130

1 13

16

130

1 −1 12 −

16

1 1 12

16

, [B] =

−1

212

−12

12

0 00 0

(8.52)

From (8.50) we obtain

d2φdξ2

d3φdξ3

d4φdξ4

d5φdξ5

ξ=0

= [A]−1


d2φ2dξ2

−[A]−1[B]

φ1

φ2

−[A]−1

p∑i=6

(dN i

3dξ

∣∣∣ξ=−1

diφdξi

∣∣∣ξ=0

)p∑i=6

(dN i

3dξ

∣∣∣ξ=1

diφdξi

∣∣∣ξ=0

)p∑i=6

(d2N i

3dξ2

∣∣∣ξ=−1

diφdξi

∣∣∣ξ=0

)p∑i=6

(d2N i

3dξ2

∣∣∣ξ=1

diφdξi

∣∣∣ξ=0

)

(8.53)


Next, we rewrite (8.47) in the form

φ(ξ) = [a(ξ)]

φ1

φ2

+ [b(ξ)]

d2φdξ2

d3φdξ3

d4φdξ4

d5φdξ5

ξ=0

+

p∑i=6

N i3

diφ

dξi

∣∣∣∣ξ=0

(8.54)

where

[a(ξ)] =[

1−ξ2

1+ξ2

][b(ξ)] =

[ξ2−1

2!ξ3−ξ

3!ξ4−1

4!ξ5−ξ

5!

] (8.55)

Substituting from (8.50) into (8.54), we obtain

φ(ξ) = [a(ξ)]

φ1

φ2

+ [b(ξ)][A]−1


d2φ2dξ2

− [B]

φ1

φ2

−

p∑i=6

(dN i

3dξ

∣∣∣ξ=−1

diφdξi

∣∣∣ξ=0

)p∑i=6

(dN i

3dξ

∣∣∣ξ=1

diφdξi

∣∣∣ξ=0

)p∑i=6

(d2N i

3dξ2

∣∣∣ξ=−1

diφdξi

∣∣∣ξ=0

)p∑i=6

(d2N i

3dξ2

∣∣∣ξ=1

diφdξi

∣∣∣ξ=0

)

+

p∑i=6

N i3

diφ

dξi

∣∣∣∣ξ=0

(8.56)

Using

dφ

dξ= J

dφ

dxand

d2φ

dξ2= J2d

2φ

dx2(8.57)

we have

[dφ1dξ

dφ2dξ

d2φ1dξ2

d2φ2dξ2

]T=[J dφ1dx J dφ2dx J2 d2φ1

dx2J2 d2φ2

dx2

]T(8.58)


Substituting from (8.58) into (8.56), we obtain

φ(ξ) = [a(ξ)]

φ1

φ2

+ [b(ξ)][A]−1

J dφ1dxJ dφ2dxJ2 d2φ1

dx2

J2 d2φ2dx2

− [B]

φ1

φ2

−

p∑i=6

(dN i

3dξ

∣∣∣ξ=−1

diφdξi

∣∣∣ξ=0

)p∑i=6

(dN i

3dξ

∣∣∣ξ=1

diφdξi

∣∣∣ξ=0

)p∑i=6

(d2N i

3dξ2

∣∣∣ξ=−1

diφdξi

∣∣∣ξ=0

)p∑i=6

(d2N i

3dξ2

∣∣∣ξ=1

diφdξi

∣∣∣ξ=0

)

+

p∑i=6

N i3

diφ

dξi

∣∣∣∣ξ=0

(8.59)

From (8.59) we can write

φ(ξ) =[[a(ξ)]− [b(ξ)][A]−1[B]

]φ1

φ2

+ [b(ξ)][A]−1

J dφ1dxJ dφ2dxJ2 d2φ1

dx2

J2 d2φ2dx2

− [b(ξ)][A]−1

p∑i=6

(dN i

3dξ

∣∣∣ξ=−1

diφdξi

∣∣∣ξ=0

)p∑i=6

(dN i

3dξ

∣∣∣ξ=1

diφdξi

∣∣∣ξ=0

)p∑i=6

(d2N i

3dξ2

∣∣∣ξ=−1

diφdξi

∣∣∣ξ=0

)p∑i=6

(d2N i

3dξ2

∣∣∣ξ=1

diφdξi

∣∣∣ξ=0

)

+

p∑i=6

N i3

diφ

dξi

∣∣∣∣ξ=0

(8.60)

Finally, from (8.60) we can write

φ(ξ) = N01 (ξ)φ1 + N0

2 (ξ)φ2 + N11 (ξ)

dφ1

dx+ N1

2 (ξ)dφ2

dx

+ N21 (ξ)

d2φ1

dx2+ N2

2 (ξ)d2φ2

dx2+

p∑i=6

N i3(ξ)

diφ

dξ

∣∣∣∣ξ=0

(8.61)

This is the desired C2(Ωe) interpolation or approximation of φ over Ωe.These are illustrated for the three-node element in Fig. 8.9.

We can show that the C2(Ωe) type interpolation functions in (8.61) are

unique and satisfy the inter-element continuity of φ, dφdx and d2φdx2

. To show theuniqueness of φ ∈ C2(Ωe), it suffices to consider a special case of (8.61) in


ξ

η

1 3

N02 (ξ)

N11 (ξ) N1

2 (ξ)

N01 (ξ)

2ξ

η

1 3

ddx

2

1 1

ddx

N21 (ξ) N2

2 (ξ) d2

dx2d2

dx2

N i3(ξ)

C2(Ωe) basis functions

di

dξi

∣∣∣ξ=0

i = 6, 7, . . . , pi = 6, 7, . . . , p

Nodal variable operators forC2(Ωe) interpolations

Figure 8.9: C2 basis functions and nodal variable operators in Ωe

which the terms corresponding to i = 6, . . . , p are absent. We need to showthat (1) the dimension of the space of φ(ξ) is the same as the total numberof degrees of freedom for the element, and (2) if all degrees of freedom forthe element are zero then φ(ξ) = 0∀ξ ∈ Ωξ. With i = 6, . . . , p absent,we have a complete fifth degree polynomial in ξ for φ(ξ) with monomials[1 ξ ξ2 ξ3 ξ4 ξ5]; hence, the dimension of the space of φ(ξ) is six (equalto the dofs or the number of monomials). The dofs for this element are

φ1,dφ1dx ,

d2φ1dx2

and φ2,dφ2dx ,

d2φ2dx2

at the two end nodes of the element. Thus,the dimension of the space is equal to the total number of degrees of freedomfor the element. If all dofs are zero, then obviously φ(ξ) = 0 ∀ξ ∈ Ωξ. Thisis obvious from (8.61). Thus, C2(Ωe) interpolation functions in (8.61) areunique. The C2 inter-element continuity is inherent in φ(ξ) due to the fact

that φ, dφdx and d2φ

dx2are degrees of freedom at the end nodes of the element.

8.2.4.3 Local approximations of class Ci(Ωe)

Following the procedure presented for interpolation of types C1(Ωe) andC2(Ωe), we can derive the following for Ci(Ωe) type interpolation functionsin one dimension that ensure continuity of the interpolation of φ of order iover the discretization ΩT of Ω. For φ(ξ) ∈ Ci(Ωe) ∀ξ ∈ Ωξ, we can write

φ(ξ) = N˜ 01(ξ)φ1 +N˜ 1

1(ξ)dφ1

dx+N˜ 2

1(ξ)d2φ1

dx2+ · · ·+N˜ i1(ξ)

diφ1

dxi

+N˜ 02(ξ)φ1 +N˜ 1

2(ξ)dφ2

dx+N˜ 2

2(ξ)d2φ2

dx2+ · · ·+N˜ i2(ξ)

diφ2

dxi

+

pξ∑j=2(i+1)

N˜ j3(ξ)djφ

dξj

∣∣∣∣ξ=0

(8.62)


Proofs of uniqueness and the inter-element continuity follow directly by in-duction using the proofs given for C1(Ωe) and C2(Ωe) type interpolation.Details of (8.62) can be illustrated for the three node configuration shownin Fig. 8.10.

dj

dxjdj

dxj

ξ

η

1 3

N˜ j2(ξ)N˜ j1(ξ)

2ξ

η

1 3 2

Ci(Ωe) basis functions Nodal variable operators forCi(Ωe) interpolations

j = 0, 1, 2, . . . , i

k = 2(i+ 1), . . . , p

j = 0, 1, . . . , i

k = 2(i+ 1), . . . , p

N˜ k3(ξ) dk

dξk

∣∣∣ξ=0

Figure 8.10: Ci basis functions and nodal variable operators in Ωe

8.3 Mapping in two dimensions: quadrilateralelements

In two dimensions, the element geometry may be of triangular shape orof quadrilateral shape with straight or curved sides. Such distorted irreg-ular shapes present difficulty when performing integration over the area ofthe element for calculating coefficients of the element matrices and vectors.Another difficulty that is much more significant is the construction of localapproximation functions for quadrilateral and irregular shapes. Both of thesedifficulties can be overcome by mapping the element physical domains from(x, y)-space into a natural coordinate space (ξ, η)-space. In the following, wefirst consider quadrilateral shapes with curved boundaries. To illustrate themapping of points, lines, and areas between the (x, y) and (ξ, η) systems,consider a four-sided element Ωe in the (x, y) coordinate (physical) space,as shown in Fig. 8.11. Let (xi, yi) (i = 1, . . . , n) be the coordinates of thenodes located on the boundary (and possibly in the interior) of the element.Consider a map of Ωe into a square of two units, Ωξη, with the origin of thecoordinate system (ξ, η) located at the center of Ωξη.

Mapping of points:

Regardless of the precise nature of the mapping, in the abstract senseone could describe this mapping by

x = x(ξ, η), y = y(ξ, η) (8.63)

8.3. MAPPING IN TWO DIMENSIONS: QUADRILATERAL ELEMENTS 517

2 3

4

57

1 1 2 3

567

8 994

2

2

6

8

η

ξ

y

x

Ωe in xy space Element map in the naturalcoordinate space ξη

Figure 8.11: Mapping of points from a physical element to points in a square of twounits in natural coordinates

and if the mapping is one-to-one and onto, then the inverse of the mappingexists and is unique and we could also write

ξ = ξ(x, y), η = η(x, y) (8.64)

In the present case the form (8.63) is preferable over (8.64). In (8.63), wenote that

(a) The mapping (8.63) is explicit in ξ and η but (8.64) is implicit in x andy; that is, given ξ∗ and η∗ in (ξ, η)-space, the mapping allows explicitdetermination of their map (x∗, y∗) in the (x, y)-space.

(b) However, given (x∗, y∗) in (x, y)-space, determination of their map (ξ∗, η∗)in (ξ, η)-space is not explicit but requires solution of a system of simul-taneous equations in ξ and η.

(c) The form of the mapping (8.63) only allows mapping of points from(ξ, η)-space to (x, y)-space. This mapping has no concept of lengths orareas.

Explicit form of the mapping in (8.63) is not too difficult to establish.Let Ni(ξ, η) be the basis functions in the natural coordinate space (ξ, η) suchthat

Ni(ξj , ηj) =

1, j = i

0, j 6= i, i = 1, . . . ,m (8.65)

andm∑i=1

Ni(ξ, η) = 1 (8.66)


Then

x =m∑i=1

Ni(ξ, η)xi

y =m∑i=1

Ni(ξ, η)yi

(8.67)

accomplishes the desired mapping of Ωe in the natural coordinate space ξ, η.The element map in the natural coordinate space is also generally referredto as the master element denoted by Ωm or Ωξη.

Mapping of lengths

In this section we establish a relationship between line lengths dx and dyalong the x and y axes and lengths dξ and dη along the ξ and η axes (seeFig. 8.12). Since, x = x(ξ, η) and y = y(ξ, η) we can write

dx =∂x

∂ξdξ +

∂x

∂ηdη

dy =∂y

∂ξdξ +

∂y

∂ηdη

(8.68)

or in matrix formdxdy

=

[∂x∂ξ

∂x∂η

∂y∂ξ

∂y∂η

]dξdη

=

[xξ xηyξ yη

]dξdη

= [J ]

dξdη

(8.69)

in which

[J ] =

[xξ xηyξ yη

](8.70)

is called the Jacobian of transformation. For the transformation (8.63) to

y

x

η

ξ

dy

dx

dη

dξ

Domain Ωe Domain Ωξ or Ωm

Figure 8.12: Mapping of lengths from a physical element to lengths in a square of twounits in natural coordinates

8.3. MAPPING IN TWO DIMENSIONS: QUADRILATERAL ELEMENTS 519

be one-to-one and onto

det [J ] > 0 ∀(ξ, η) ∈ Ωm or Ωξη

Clearly, (8.70) describes the relationship between the elemental lengths inthe two coordinate systems. Derivatives of x and y with respect to ξ and ηneeded in [J ] can be easily obtained using (8.67).

Mapping of areas

Consider elemental length dx, dy along x and y axes forming an areadΩ = dx dy in the x, y-space. Likewise consider length dξ, dη along ξ andη forming an area dΩm = dξ dη. In this section we establish a relationshipbetween dΩ and dΩm. Let ~i and ~j be the unit vectors along x and y axesand ~eξ and ~eη be the unit vectors along ξ and η axes (see Fig. 8.13).

y

η

dξ

dη

ξ~eη

~j~i

x

~eξ

Domain Ωe Domain Ωξ or Ωm

dx

dy

Figure 8.13: Mapping of areas from a physical element to areas in a square of two unitsin natural coordinates

Then the cross product of the vectors dx~i and dy~j would yield a vectorperpendicular to the plane containing vectors dx~i and dy~j and the mag-nitude of this vector represents the area formed by the two vectors, dΩ.Thus,

dx~i× dy~j = dx dy~i×~j = dx dy~k (8.71)

but

dx~i =∂x

∂ξdξ ~eξ +

∂x

∂ηdη ~eη

dy~j =∂y

∂ξdξ ~eξ +

∂y

∂ηdη ~eη

(8.72)

Substituting from (8.72) into the left side of (8.71) we obtain

dx dy~k =(∂x∂ξdξ ~eξ +

∂x

∂ηdη ~eη

)×(∂y∂ξdξ ~eξ +

∂y

∂ηdη ~eη

)(8.73)


expanding the right side of (8.73) gives

dx dy~k =∂x

∂ξ

∂y

∂ξdξ dξ ~eξ × ~eξ +

∂x

∂η

∂y

∂ξdξ dη ~eη × ~eξ

+∂x

∂ξ

∂y

∂ηdξ dη ~eξ × ~eη +

∂x

∂η

∂y

∂ηdη dη ~eη × ~eη (8.74)

Noting that

~eξ × ~eξ = 0 = ~eη × ~eη~eξ × ~eη = ~eζ = ~k

~eη × ~eξ = −~eζ = −~k

(8.75)

we obtain the following from (8.75)

dx dy~k =

(∂x

∂ξ

∂y

∂η− ∂x

∂η

∂y

∂ξ

)dξ dη ~k (8.76)

Hence

dx dy =

(∂x

∂ξ

∂y

∂η− ∂x

∂η

∂y

∂ξ

)dξ dη (8.77)

but

det [J ] = |J | = ∂x

∂ξ

∂y

∂η− ∂x

∂η

∂y

∂ξ(8.78)

Hencedx dy = |J | dξ dη (8.79)

ordΩe = |J | dΩm = |J | dΩξη (8.80)

Equation (8.80) gives the desired relationship between the elemental areasin the two coordinate spaces.

8.4 Local approximation over Ωm: quadrilateral el-ements

Let φ(ξ, η) is the local approximation of φ over Ωξη, then we can write

φ(ξ, η) =

n∑i=1

Ni(ξ, η) δei (8.81)

orφ(ξ, η) = [N(ξ, η)]δe (8.82)

where Ni(ξ, η) are the local approximation functions corresponding to thenodes of the element and δei are nodal degrees of freedom. Ni(ξ, η) are also

8.4. LOCAL APPROXIMATION OVER QUADRILATERAL ELEMENTS 521

referred to as basis functions for the element e. Explicit details of Ni(ξ, η)(i = 1, . . . , n), n being the total degrees of freedom and the correspondingδei depend upon many considerations.

(1) The first important issue is the choice of the nodal configuration for theelement. That is, how many nodes and their locations.

(2) Means of constructing Ni(ξ, η) for:

(a) C0 local approximation of higher degree

(b) C0 p-version hierarchical local approximations

(c) Ci,j (i, j ≥ 1) for rectangular family elements

(d) Ci,j (i, j ≥ 1) for elements with distorted shapes in the x, y-space

(3) In the development of (2), the choices of δei , nodal dofs are of coursecrucial and require careful considerations.

(4) The developments in (2) and (3) could be based on: (a) Lagrange inter-polation functions, (b) Legendre polynomials or (c) Chebyshev polyno-mials. The specific details presented in the following are based on La-grange polynomials. Their extensions to the other two types are straightforward and the details can be found in the cited references.

Obtaining derivatives of φ(ξ, η) with respect to x and y

With the local approximation of φ defined by (8.82) in which φ = φ(ξ, η),obtaining derivatives of φ(ξ, η) with respect to x and y needed in the integralform is not direct and needs to be considered. First, we note that whenφ(ξ, η) is defined by (8.82) we can write

∂φ

∂x=

n∑i=1

∂Ni(ξ, η)

∂xδei

∂φ

∂y=

n∑i=1

∂Ni(ξ, η)

∂yδei

(8.83)

Thus, obtaining ∂φ∂x and ∂φ

∂y implies establishing ∂Ni∂x and ∂Ni

∂y (i = 1, . . . , n).Once again, since Ni (i = 1, . . . , n) are functions of ξ and η and x = x(ξ, η),y = y(ξ, η), we can proceed as follows.

∂Ni

∂ξ=∂Ni

∂x

∂x

∂ξ+∂Ni

∂y

∂y

∂ξ

∂Ni

∂η=∂Ni

∂x

∂x

∂η+∂Ni

∂y

∂y

∂η

(8.84)


arranging (8.84) into matrix form∂Ni∂ξ∂Ni∂η

=

[∂x∂ξ

∂y∂ξ

∂x∂η

∂y∂η

]∂Ni∂x∂Ni∂y

= [J ]T

∂Ni∂x∂Ni∂y

(8.85)

Therefore, ∂Ni∂x∂Ni∂y

= [JT ]−1


; i = 1, . . . , n (8.86)

Using (8.86), derivatives of Ni with respect to ξ and η can be transformedinto the derivatives of Ni with respect to x and y and, hence, the derivativesof φ with respect to x and y in (8.84) are defined.

8.4.1 C00 local approximations over Ωξη: polynomialapproach

First, we consider interpolation φ(ξ, η) of φ over Ωm of type C00, i.e acrossthe inter subdomain boundaries only φ is continuous but the derivatives ofφ normal to the inter-element boundaries in the physical domain may bediscontinuous. For the master element Ωm, a bilinear behavior of φ (in mostcases this would be lowest degree admissible behavior) would require,

φ(ξ, η) = c1 + c2ξ + c3η + c4ξη (8.87)

Evaluation of constants ci; i = 1, . . . , 4 naturally requires four conditions,and for C00 behavior of φ we can choose

φei = φ(ξi, ηi), i = 1, . . . , 4 (8.88)

in which (ξ, η) are the coordinates of the four corner nodes, as shown inFig. 8.14.

21

(ξ2, η2)(ξ1, η1)

ξ

(ξ3, η3)

34

(ξ4, η4)η

Figure 8.14: A four-node square of two units in the natural coordinates


Using (8.87) and (8.88) we obtainc1

c2

c3

c4

=

1 ξ1 η1 ξ1η1

1 ξ2 η2 ξ2η2

1 ξ3 η3 ξ3η3

1 ξ4 η4 ξ4η4

−1

φe1φe2φe3φe4

= [C]φe (8.89)

Substituting from (8.89) into (8.87) we obtain

φ(ξ, η) =[1 ξ η ξη

][C]φe (8.90)

or

φ(ξ, η) = [N ]φe = [N ]δe =

4∑i=1

Ni(ξ, η)φei (8.91)

in which

Ni(ξj , ηj) =

1, j = i

0, j 6= i, i = 1, . . . , 4 (8.92)

and4∑i=1

Ni(ξ, η) = 1 (8.93)

This approximation of φ is referred to as bilinear Lagrange interpolation.

Higher degree of approximation of φ over Ωξη

Let us consider a quadratic (complete second degree in ξ and η) approx-imation of φ in ξ and η over Ωm. This requires an expression similar to(8.87) but with a complete second degree polynomial in ξ and η and hencea decision on which terms to consider. Secondly, in order to evaluate theconstants ci, we need to know the locations of the nodes over the masterelement Ωm. In case of quadratic behavior of φ over Ωm, it may not be verydifficult to decide on these two aspects, but as the degree of approximationor local interpolation is increased the decision on these two aspects is notstraight forward.

The decision on which ξ and η terms to consider for φ and where thenodes are located over Ωm can be facilitated by using what is called Pascal’srectangle. The family of local interpolations generated in this approach areLagrange family of local interpolations. First, we explain how to interpretthe information in Fig. 8.15.

(a) The locations of terms in the rectangular arrangement are the locationsof the nodes in Ωm configuration.


ξη

1

η

ξ

ξη2 ξ2η2 ξ3η2 ξ4η2

ξη ξ2η ξ3η ξ4η

ξ ξ2 ξ3 ξ4



1

η

η2

η4

η3

1

ξ2

η2 ξ3η2

η ξ3η

ξ2η2

1

η2

ξ

ξηη ξ2η

ξη2

1 ξ3

ξ3η3η3

ξη

ξη2

ξη3

ξ2η

ξ2η2

ξ2η3

ξ2ξ

Rectangular array

bi-cubic; p = 3

bi-quadratic; p = 2

bilinear; p = 1

constant; p = 0

Nodal configuration Degree of approximation p of φMonomials to be used

Rectangular elements of various degree local approximations

Figure 8.15: Pascal’s rectangle and Lagrange family of interpolations


(b) The terms shown in the rectangle are the monomials to be used in theexpression for φ(ξ, η).

(c) For example, in the case of bilinear local approximation one would con-sider only up to linear terms in ξ and η, which would yield a four-nodeelement with terms 1, ξ, η and ξη in the expression for φ(ξ, η).

(d) In the case of bi-quadratic (complete) behavior of φ over Ωm, one wouldconsider up to quadratic monomials in ξ and η, which would yield a nine-node element with terms 1, ξ, ξ2, η, ξη, ξ2η, η2, ξη2, ξ2η2 in the expansion forφ(ξ, η).

(e) Likewise, for a bi-cubic element we would have a sixteen-node elementfor which the corresponding monomials in the interpolation for φ(ξ, η) areshown in Fig. 8.15.

(f) Once we know the monomials to be used in the expansion for φ(ξ, η) andthe locations of the nodes, it is a rather simple matter (following the sameprocedure as used for the bilinear case) to derive

φ(ξ, η) =

n∑i=1

Ni(ξ, η)φei

= [N ]φe = [N ]δe (8.94)

where n is the number of nodes.

Remarks.

(1) The local interpolations of various degrees derived above are referred toas C0 Lagrange family of interpolations in which function value at thenodes are the nodal degrees of freedom.

(2) A serious drawback of this approach is that as the degree of local ap-proximation is increased, so does the size of the matrix to be invertedin deriving Ni(ξ, η).

(3) The most fundamental shortcoming of this approximation is that as thedegree of approximation increases, the number of nodes increase dramat-ically, a very undesirable feature from the point of view of constructingdiscretizations for various p-levels.

(4) In the following section 8.4.2 we correct some of these shortcoming.

8.4.2 C00 Lagrange type local approximation usingtensor product

Recall that in the case of 1D local approximations in ξ for −1 ≤ ξ < 1,the basis functions or local approximations can be easily generated usingLagrange interpolation functions which allows us to bypass the inversion of


the matrices. For an element with l equally spaced nodes in −1 ≤ ξ < 1, wecould write φ(ξ) as

φ(ξ) =

l∑i=1

Ni(ξ)φei (8.95)

in which

Nk(ξ) = Lk(ξ) =l∏

m=1m6=k

(ξ − ξmξk − ξm

), k = 1, . . . , l (8.96)

with

Ni(ξj) =

1, j = i

0, j 6= i, i = 1, . . . , l (8.97)

andl∑

i=1

Ni(ξ) = 1 (8.98)

In this, the degree of approximation is (l−1), that is, p = l−1 and the localapproximations are of type C00 due to the fact that only the function valuesare the unknowns or the degrees of freedom at the nodes.

To understand the basic concept of tensor product, consider a four-nodebilinear element in Ωm and a two-node linear element in ξ natural coordinatesystem with N ξ

1 (ξ) and N ξ2 (ξ) as the basis functions and also a two-node

linear element in η natural coordinate system with Nη1 (η) and Nη

2 (η) as thebasis functions. Then the four-node bilinear element can be thought of as amap of the traces of the two-node linear configuration in η traversing alongthe two-node linear configuration in ξ. This is illustrated in Fig. 8.16 andTable 8.2.

21

ξ

34

η

ξ

2η

(1,−1)(−1,−1)

(1, 1)(−1, 1) (+1)N η2

(−1)N η1

(b)(a) Element map Ωm

1η1ξ 2ξ

N ξ2

(+1)N ξ

1

(−1)

η

Figure 8.16: Derivation of four-node bilinear element as a tensor product of two-nodeline elements

The information expressed in Table 8.2 can also be constructed in thefollowing manner. If we declare N ξ

1 and N ξ2 as column vectors and Nη

1 and


Table 8.2: Correspondence between the basis functions of the bilinear element and the1D linear element

2D configurationProduct of 1D basis functions

Node Basis function

1 N1(ξ, η) = Nξ1 N

η1

2 N2(ξ, η) = Nξ2 N

η1

3 N3(ξ, η) = Nξ2 N

η2

4 N4(ξ, η) = Nξ1 N

η2

Nη2 as a row matrix (with corresponding ξ and η coordinates) then a tensor

product of these two would be a simple product of these twoN ξ

1 : ξ = −1

N ξ2 : ξ = +1

[Nn

1 : η = −1 Nη2 : η = +1

]Nξ

1Nη1

(−1,−1)Nξ

1Nη2

(−1,1)

Nξ2N

η1

(1,−1)Nξ

1Nη1

(1,1)

=

[N1(ξ, η) N4(ξ, η)N2(ξ, η) N3(ξ, η)

](8.99)

Remarks.

(1) When taking the tensor product, we always keep track of ξ η coordinatesassociated with the 1D functions so that the results of the tensor productcan be appropriately associated with the corresponding nodes of the 2Dconfiguration.

(2) Whether we use this classical definition of the tensor product in (8.99)or tabular form given earlier, the end result is the same.

Example 8.3 (Explicit expressions for the standard Lagrange basis func-tions for the four-node bilinear element in Ωm). In the case of the four-nodebilinear element

N ξ1 =

1− ξ2

, N ξ2 =

1 + ξ

2, Nη

1 =1− η

2, Nη

2 =1 + η

2(8.100)

and hence (8.99) would yield1−ξ

2 : ξ = −11+ξ

2 : ξ = +1

[1−η

2 : η = −1 1+η2 : η = +1

]

=

( 1−ξ2 )( 1−η

2 )(−1,−1)

( 1−ξ2 )( 1+η

2 )(−1,1)

( 1+ξ2 )( 1−η

2 )(1,−1)

( 1+ξ2 )( 1+η

2 )(1,1)

=

[N1(ξ, η) N4(ξ, η)N2(ξ, η) N3(ξ, η)

](8.101)


Therefore, we hve

N1(ξ, η) =

(1− ξ

2

)(1− η

2

)N2(ξ, η) =

(1 + ξ

2

)(1− η

2

)N3(ξ, η) =

(1 + ξ

2

)(1 + η

2

)N4(ξ, η) =

(1− ξ

2

)(1 + η

2

)(8.102)

Obviously the basis functions in (8.102) are the correct local approximationfunctions for the four-node bilinear element as they satisfy the properties

Ni(ξj , ηj) =

1, j = i

0, j 6= i

and4∑i=1

Ni(ξ, η) = 1

Example 8.4 (Explicit expressions for the standard Lagrange basis functionsfor the bi-quadratic element in Ωξη). Consider a nine-node (determined usingPascal’s rectangle) bi-quadratic element in Ωm. The basis functions (or localapproximation functions) for this element can be generated by taking thetensor product of 1D quadratic basis functions for three-node configurationsin ξ and η directions (see Fig. 8.17):

N ξ1 : ξ = −1 N ξ

2 : ξ = 0 N ξ3 : ξ = +1

[Nη

1(η=−1)

Nη2

(η=0)Nη

3(η=+1)

]=

Nξ1N

η1

(−1,−1)Nξ

1Nη2

(−1,0)Nξ

1Nη3

(−1,1)

Nξ2N

η1

(0,−1)Nξ

2Nη2

(0,0)Nξ

2Nη3

(0,1)

Nξ3N

η1

(1,−1)Nξ

3Nη2

(1,0)Nξ

3Nη3

(1,1)

=

N1(ξ, η) N8(ξ, η) N7(ξ, η)

N2(ξ, η) N9(ξ, η) N6(ξ, η)

N3(ξ, η) N4(ξ, η) N5(ξ, η)

(8.103)

Explicit expressions for N ξi and Nη

i (i = 1, 2, 3) are

N ξ1 =

ξ(ξ − 1)

2, Nη

1 =η(η − 1)

2

N ξ2 = 1− ξ2, Nη

2 = 1− η2 (8.104)

N ξ3 =

ξ(ξ + 1)

2, Nη

3 =η(η + 1)

2


Hence Ni(ξ, η) (i = 1, . . . , 9) can be determined explicitly using (8.103) and(8.104).

η3η

3ξ

N ξ3

(1)N ξ

1

(−1)

1

8

5

4

32

9

67(1, 0)

(1, 1)

(−1)N η1

2η

2ξ

N ξ2

(0)

(0)N η2

(1)N η3(0, 1)

(0, 0)

(0,−1) (1,−1)

(−1, 1)

(−1, 0)

(−1,−1)

ξ

(b)(a) Element map Ωm

1η1ξ

ξ

η

Figure 8.17: Derivation of nine-node bi-quadratic element as a tensor product of three-node line elements

Higher degree local basis functions of complete degrees in ξ and η canbe easily derived using this procedure. It is rather clear that the tensorproduct procedure avoids inverting matrices to determine constants in thepolynomial expansions of φ(ξ, η). In this approach of constructing the localapproximations one could increase the degree of the polynomial (p-level) inξ and η directions as desired. These local approximations could be referredto as p-version. The two main drawbacks of this approach are: (i) that anincrease in p-level requires a new nodal configuration for the element, i.e. es-sentially new geometric description of additional nodes and (ii) and secondlythe local approximations lack hierarchical property, i.e. the approximationsfor lower p-levels are not an explicit subset of those at higher p-levels. Thesetwo drawbacks can be corrected by deriving the C00 p-version hierarchicallocal approximation functions presented in the next section for quadrilateralelements based on Lagrange polynomials.

8.4.3 C00 p-version hierarchical local approximations basedon Lagrange polynomials

Consider quadrilateral elements with four corner nodes, four mid sidenodes and a node at the center of the element. The element shape andsides can be distorted in the (x, y)-space, as shown in Fig. 8.18(a). Considera map of the element in the (ξ, η)-space into a square of two units withthe origin of the coordinate system located at the center of the element, asshown in Fig. 8.18(b). The C00 p-version hierarchical local approximationsfor the element of Fig. 8.18(b) can be derived by considering 1D p-version


hierarchical local approximation in ξ and η for the three-node element andthen taking their tensor product.

Let φ be the field variable to be interpolation over Ωm or Ωξη. Considera three node element in ξ-coordinate space as shown in Fig. 8.18(c). Thenfor this element we can write

φ(ξ) = N1ξ1 (ξ)(φ)ξ=−1 +N1ξ

3 (ξ)(φ)ξ=1 +

pξ∑i=2

N iξ2 (φ,ξi)ξ=0 (8.105)

where φ,ξi = ∂iφ∂ξi

and pξ is the highest degree of the polynomial in ξ. Sim-

ilarly, for the three-node configuration of Fig. 8.18(c) in η direction we canwrite

φ(η) = N1η1 (η)(φ)η=−1 +N1η

3 (η)(φ)η=1 +

pη∑i=2

N iη2 (φ,ηi)η=0 (8.106)

where φ,ηj = ∂jφ∂ηj

and pη is the highest degree of the polynomial in η.

2 3

4

567

8

1 1 2 3

567

8 994

2

2

(a) (b)

η

ξ

y

x

A nine-node quadrilateralelement map in ξη-space

A quadrilateral element inxy-space

ξ

3

2

1

1 32

η

(c) 1D three-node elements in ξ and η spaces

Figure 8.18: Mapping of nine-node bi-quadratic element in (x, y)-space into (ξ, η)-space


The 1D local approximations given by (8.105) and (8.106) can be ex-pressed more conveniently in terms of 1D functions in ξ and η and the cor-responding nodal variable operators. Keeping in mind that nodal variableoperators act on the dependent variables(s) to produce dofs at the corre-sponding nodes. Figure 8.19 shows the arrangement of the 1D functions inξ and η and Fig. 8.20 show the corresponding nodal variable operators. Theapproximation functions and the nodal variable operators for the nine-node2D element of Fig. 8.18(b) can be constructed by simply taking the tensorproducts of the 1D C0 p-version hierarchical approximation functions andthe corresponding nodal variable operators. The resulting C00 2D p-versionhierarchical approximation functions and the nodal variable operators areshown in Fig. 8.20. Since the 1D C0 p version functions and the nodal vari-able operators used in deriving the 2D C00 approximation are hierarchical,the hierarchical nature of the 2D approximations is preserved. Thus, for thenine-node C00 p-version hierarchical element the dependent variable φ canbe interpolated over Ωm or Ωξη using

φ(ξ, η) = [N(ξ, η)]δe (8.107)

where

[N(ξ, η)] =

[ node 1︷︸︸︷N11

1 ,

node 2︷︸︸︷[N21

2 , N312 , . . . , N

pξ12 ],

node 3︷︸︸︷N11

3 ,

node 4︷︸︸︷[N12

4 , N134 , . . . , N

1pη4 ],

node 5︷︸︸︷N11

5 ,

node 6︷︸︸︷[N21

6 , N316 , . . . , N

pξ16 ],

node 7︷︸︸︷N11

7 ,

node 8︷︸︸︷[N12

8 , N138 , . . . , N

1pη8 ],[ node 9︷︸︸︷

[N229 , N32

9 , . . . , Npξ29 ], [N23

9 , N339 , . . . , N

pξ39 ],

. . . ,

node 9︷︸︸︷[N

2pη9 , N

3pη9 , . . . , N

pξpη9 ]

]](8.108)

and

δeT =

[node 1︷︸︸︷φ1 ,

node 2︷︸︸︷[(d2φ

dξ2

)2,(d3φ

dξ3

)2, . . . ,

(dpξφdξpξ

)2

],

node 3︷︸︸︷φ3 ,


dη2

)4,(d3φ

dη3

)4, . . . ,

(dpηφdηpη

)4

],

node 5︷︸︸︷φ5 ,



dξ2

)6,(d3φ

dξ3

)6, . . . ,

(dpξφdξpξ

)6

],

node 7︷︸︸︷φ7 ,


dη2

)8,(d3φ

dη3

)8, . . . ,

(dpηφdηpη

)8

],

[ node 9︷︸︸︷[( d4φ

dη2dξ2

)9,( d5φ

dη2dξ3

)9, . . . ,

( dpξ+2φ

dη2dξpξ

)9

],

node 9︷︸︸︷[( d5φ

dη3dξ2

)9,( d6φ

dη3dξ3

)9, . . . ,

( dpξ+3φ

dη3dξpξ

)9

],

. . . ,

node 9︷︸︸︷[( dpη+2φ

dηpηdξ2

)9,( dpη+3φ

dηpηdξ3

)9, . . . ,

( dpη+pξφ

dηpηdξpξ

)9

]]](8.109)

For a chosen p-level in ξ and η directions (i.e. pξ and pη), explicit expres-sions for the approximation functions [N ] and the nodal dofs δe are easilyobtained. We remark that the above development is based on Lagrange poly-nomials due to the fact that 1D C0 p-version hierarchical functions in ξ andη directions were based on Lagrange polynomials. Following the approachpresented here, one could easily derive 2D C00 p-version hierarchical localapproximations based on Legendre and Chebyshev polynomials.

8.5 2D C ij(Ωe) p-version local approximations:Rectangular family of elements

In this section we derive 2D Cij(Ωe) type local approximation funtionsin xy coordinate space which possess interelement continuity of orders iand j in x and y directions with complete polynomials of orders pξ andpη and present proofs of necessary and sufficient conditions. Consider anine-node p-version rectangular element in x, y space with sides parallelto x and y axes [Fig. 8.22(a)] and its map in natural coordinate space[Fig. 8.22(b)]. We derive Cij(Ωe) type interpolations for the element shownin Fig. 8.22(a) by using tensor products of one dimensional Ci, Cj interpo-lations in x and y directions [Fig. 8.22(c)] for the three-node configurationsshown in Fig. 8.22(a).

Let pξ be the degree of interpolation in ξ directions (parallel to x-axis andpointing in the same direction) and i be order of continuity in the x direction,then for the three-node 1D configuration of Fig. 8.22(c) in ξ direction we

8.5. 2D CIJ (ΩE) P -VERSION LOCAL APPROXIMATIONS 533

can write interpolations of type Ci in x direction as follows for a dependentvariable φ:

φ(ξ) = N˜ 0

1(ξ)φ1 +N˜ 1

1(ξ)

∂φ1

∂x+N˜ 2

1(ξ)

∂2φ1

∂x2+ · · ·+N˜ i1(ξ)

∂iφ1

∂xi

+N˜ 0

2(ξ)φ2 +N˜ 1

2(ξ)

∂φ2

∂x+N˜ 2

2(ξ)

∂2φ2

∂x2+ · · ·+N˜ i2(ξ)

∂iφ2

∂xi

+

pξ∑i1=2(i+1)

N˜ i13 (ξ)∂i1φ3

∂ξi1∀ξ ∈ Ωξ and ∀x = x(ξ) ∈ Ωe

x (8.110)

(ξ = 1)(ξ = 0)(ξ = −1)

ξ

(η = −1)

(η = 0)

(η = 1)3

2

1

η

N1η3 = 1+η

2

N1η1 = 1−η

2

N2η2 = η2−1

2N3η

2 = η3−η3!

pη = 2pη = 3

b =

1; pη is even

η; pη is odd

N(pη)η2 = ηpη−b

pη !

p = pη pη = 1

1 2 3

N1ξ1 = 1−ξ

2N1ξ

3 = 1+ξ2

N2ξ2 = ξ2−1

2

N3ξ2 = ξ3−ξ

3!

N(pξ)ξ2 = ξ

pξ−apξ!

pξ = 1

pξ = 2

pξ = 3

p = pξ

a =

1; pξ is even

ξ; pξ is odd

Figure 8.19: 1D, C0 hierarchical functions in ξ and η directions.


(ξ = 1)(ξ = 0)(ξ = −1)

ξ

1 2 3

pξ = 1

pξ = 2

pξ = 3

11

p = pξ

∂2

∂ξ2

∂3

∂ξ3

∂pξ

∂ξpξ

(η = −1)

(η = 0)

(η = 1)3

2

1

ηpη = 1

1

1

pη = 2pη = 3p = pη

∂pη

∂ηpη∂3

∂η3∂2

∂η2

Figure 8.20: 1D, C0 hierarchical nodal variable operators in ξ and η directions.

in which (φ1, φ2) are function values at nodes 1 and 2 and (∂φ1∂x ,∂2φ1∂x2

) arefirst and second derivatives of φ with respect to x at node one. Likewise,

(∂φ2∂x ,∂2φ2∂x2

) are the first and second derivatives of φ with respect to x at node2, and so on.

Let pη be the degree of interpolation in η directions and j be the orderof continuity in the y-direction then for the three-node 1D configuration ofFig. 8.22(c) in η direction, we can write the following for the dependentvariable φ:


3

7 6 5

4

1 2

8

7 6 5

4

321

89

9

non-hierarchical nodes

hierarchical nodes

η

ξ

i = 2, . . . , pξj = 2, . . . , pη

N i12 (ξ, η) = (N iξ

2 N1η1 )

i = 2, . . . , pξ

N ij9 (ξ, η) = (N iξ

2 N jη2 )

N i16 (ξ, η) = (N iξ

2 N1η3 )

N113 (ξ, η) = (N1ξ

3 N1η1 )N11

1 (ξ, η) = (N1ξ1 N1η

1 )

N1j8 (ξ, η) = (N1ξ

1 N jη2 )

i = 2, . . . , pξN11

7 (ξ, η) = (N1ξ1 N1η

3 ) N115 (ξ, η) = (N1ξ

3 N1η3 )

j = 2, . . . , pη N1j4 (ξ, η) = (N1ξ

3 N jη2 )

j = 2, . . . , pη

(a) Approximation functions

η

ξ

i = 2, . . . , pξj = 2, . . . , pη

(∂j

∂ηj

)

i = 2, . . . , pξ

(∂i

∂ξi

)

(∂i

∂ξi

)

i = 2, . . . , pξ

j = 2, . . . , pη

(1)

(1) (1)

(1)

(∂i+j

∂ηj∂ξi

)j = 2, . . . , pη

(∂j

∂ηj

)

(b) Nodal variable operators

Figure 8.21: p-version approximation functions and nodal variable operators for a nine-node 2D element


231

ξ = −1 ξ = 0 ξ = 1

ξ

η = −11

3 η = 0

η = 12

η

x

y1 2 3

498

7 6 5 7

8

1 2

9

6

η

5

4

3

ξ

(b) Map of element of (a) in natural co-ordinate space ξη (Ωm)

1D three-node configurations in ξand η directions

(c)

A nine-node rectangular element inxy space (Ωe)

(a)

Figure 8.22: 2D nine-node element and 1D three-node configurations in ξ and η directions

φ(ξ) = N˜ 0

1(η)φ1 +N˜ 1

1(η)

∂φ1

∂y+N˜ 2

1(η)

∂2φ1

∂y2+ · · ·+N˜ j1(η)

∂jφ1

∂yj

+N˜ 0

2(η)φ2 +N˜ 1

2(η)

∂φ2

∂y+N˜ 2

2(η)

∂2φ2

∂y2+ · · ·+N˜ j2(η)

∂jφ2

∂yj

+

pη∑j1=2(j+1)

N˜ j13 (η)∂j1φ3

∂ηj1∀η ∈ Ωη and ∀y = y(η) ∈ Ωe

y (8.111)


k1 = 0, 1, . . . , i

k2 = 2(i+ 1), . . . , pξ

1 3 2

1

2

ξ

η

1 3 2

1

3

2

ξ

η

∂l1∂yl1

∂l2∂yl2

∂l1∂yl1

3

N˜ l12 (η)

N˜ l23 (η)

N˜ l11 (η)

N˜ k11(ξ) N˜ k23

(ξ) N˜ k12(ξ)

l2 = 2(j + 1), . . . , pη

l1 = 0, 1, . . . , j

∂k1

∂xk1∂k2

∂xk2

∂k1

∂xk2

k1 = 0, 1, . . . , i

k2 = 2(i+ 1), . . . , pξ

l2 = 2(j + 1), . . . , pη

l1 = 0, 1, . . . , j

(a) C i, Cj 1D basis functions (b) C i, Cj 1D nodal variable operators

Figure 8.23: 1D higher order continuity basis functions and nodal variable operators

By taking tensor product of 1D functions and the nodal variable opera-tors in (8.110) and (8.111) (shown in Fig. 8.23) we can generate 2D Cij(Ωe)type interpolations for φ = φ(ξ, η) (i.e. interpolation functions and the nodalvariable operators). The resulting Cij(Ωe) 2D approximation functions andthe nodal variable operators are shown in Fig. 8.24 and we can write

φ(ξ, η) = [N(ξ, η)]δe ∀ξ η ∈ Ωm and (x, y) ∈ Ωe (8.112)

In the following, we present a theorem and its proof that ensures necessaryand sufficient conditions for interpolation (8.112) to be of type Cij(Ωe).

N˜ k1

1(ξ)N˜ l23 (η)

N˜ k1

1(ξ)N˜ l12 (η)

N˜ k1

1(ξ)N˜ l11 (η)

∂k1+l2

∂yl2∂xk1

∂k1+l1

∂yl1∂xk1

∂k1+l1

∂yl1∂xk1

k2 = 2(i + 1), . . . , pξ

k1 = 0, 1, . . . , i

l2 = 2(j + 1), . . . , pη

l1 = 0, 1, . . . , j

89

6 57

1 2 3

4

7

8

1

9

6 5

4

32

N˜ k1

2(ξ)N˜ l12 (η)N˜ k2

3(ξ)N˜ l12 (η)

N˜ k1

2(ξ)N˜ l23 (η)

N˜ k1

2(ξ)N˜ l11 (η)N˜ k2

3(ξ)N˜ l11 (η)

N˜ k2

3(ξ)N˜ l23 (η)

(a) Basis functions (b) Nodal variable operators

∂k2+l1

∂yl1∂xk2

∂k2+l1

∂yl1∂xk2

∂k2+l2

∂yl2∂xk2

∂k1+l2

∂yl2∂xk1

∂k1+l1

∂yl1∂xk1

∂k1+l1

∂yl1∂xk1

Figure 8.24: 2D Cij(Ωe) basis functions and nodal variable operators


Theorem 8.2. If φ(ξ) ∈ Ci(Ωex)∀ξ ∈ Ωξ or ∀x = x(ξ) ∈ Ωe

x of degree pξ andφ(η) ∈ Cj(Ωe

y)∀η ∈ Ωη or ∀y = y(η) ∈ Ωey of degree pη are one dimensional

interpolations for the three-node 1D configurations of Fig. 8.22 (c), then thetensor product of generating 2D interpolations for φ(ξ, η) by (8.112) ensuresnecessary and sufficient conditions for φ(ξ, η) ∈ Cij(Ωe)∀x(ξ), y(η) ∈ Ωe forthe nine node element of Fig. 8.22(a).

The proof of this theorem is constructed by induction and is presentedin the following sections.

8.5.1 2D interpolations of type C11(Ωe) with p-levelsof pξ and pη

(a) Proof of the uniqueness of C11(Ωe) interpolations.

It suffices to consider pξ = 3 and pη = 3 in which case there are nodegree of freedom at the midside nodes and the center node. We needto show

(i) That the dimension of the space of φ(ξ, η) is same as the totaldegrees of freedom for the element.

(ii) If all degrees of freedom are identically zero, then φ(ξ, η) = 0 ∀ξ, η ∈Ωm.

For a nine-node element with complete bicubic interpolations in ξ and η,the total number of basis functions required is sixteen (based on Pascalrectangle). Hence, the dimension of the approximation space of φ(ξ, η)is sixteen. For pξ = pη = 3 (bicubic) the element has no degrees of

freedom at the midside nodes and at the center node and φ, ∂φ∂x , ∂φ

∂y and∂2φ∂y∂x (defined as set S) as degrees of freedom at each of the four cornernodes giving rise to a total of sixteen degrees of freedom for the element.Hence the dimension of the space of φ(ξ, η) is equal to the total degrees offreedom. If all degrees of freedom are zero, then it follows directly fromthe expression for φ(ξ, η) that φ(ξ, η) = 0 ∀ξ, η ∈ Ωm. Thus C11(Ωe)interpolations generated using tensor product in (8.112) are unique.

(b) Proof of C11(Ωe) inter-element continuity.

In this case also we can consider pξ = pη = 3 without loss of general-ity. Consider a three-element discretization in which the element onehas elements two and three as neighbors with mating boundaries withcommon sides L14 and L14 (L14 being closed set, i.e. it includes nodes1 and 4; L14 being open set) parallel to y axis and sides L12 and L12

(L12 being closed set, that is, it includes nodes 1 and 2; L12 being openset) parallel to x axis. Then, we need to show that the elements of the


Table 8.3: Degrees of freedom, order of polynomials for φ, ∂φ∂x

and uniqueness of φ and

its derivatives ∀y ∈ L14 and ∀y ∈ L14

Degrees of Third degree Uniqueness Uniquenessfreedom at polynomial ∀y ∈ L14 ∀y ∈ L14

nodes 1 and 4 ∀y ∈ L14

φ, ∂φ∂y

φ φ, ∂iφ∂yi

; i = 1, 2, 3(φ, ∂φ

∂y

): setS1

∂φ∂x, ∂∂y

(∂φ∂x

)∂φ∂x

∂φ∂x, ∂

i

∂yi

(∂φ∂x

); i = 1, 2, 3

(∂φ∂x, ∂∂y

(∂φ∂x

)): setS2

Table 8.4: Degrees of freedom, order of polynomials for φ, ∂φ∂y

and uniqueness of φ and

its derivatives ∀y ∈ L12 and ∀y ∈ L12

Degrees of Third degree Uniqueness Uniquenessfreedom at polynomial ∀x ∈ L12 ∀x ∈ L12

nodes 1 and 2 ∀y ∈ L12

φ, ∂φ∂x

φ φ, ∂iφ∂xi

; i = 1, 2, 3(φ, ∂φ

∂x

): setT1

∂φ∂y, ∂∂x

(∂φ∂y

)∂φ∂y

∂φ∂y, ∂

i

∂xi

(∂φ∂y

); i = 1, 2, 3

(∂φ∂y, ∂∂x

(∂φ∂y

)): setT2

set S, that is, φ, ∂φ∂x , ∂φ∂y and ∂2φ∂y∂x , are unique ∀y ∈ L14 and ∀x ∈ L12. The

proof is best presented by referring to Tables 8.3 and 8.4. First, considerL14 (Table 8.3). With φ and ∂φ

∂y as degrees of freedom at nodes 1 and

4, φ = φ(y) is a cubic polynomial in y ∀y ∈ L14 ensuring uniqueness

of φ, ∂iφ∂yi, i = 1, 2, 3 ∀y ∈ L14 (row 1, column 3 in Table 8.3). However

only φ and ∂φ∂y (set S1) are unique ∀y ∈ L14 due to the fact φ and ∂φ

∂y arethe only dofs at nodes 1 and 4. From row 2 of Table 8.3, we note thatwith ∂φ

∂x and ∂∂y

(∂φ∂x

)(set S2) as degrees of freedom at nodes 1 and 4,

∂φ∂x is also a polynomial of degree three in y ∀y ∈ L14 ensuring that ∂φ

∂x

and ∂i

∂yi

(∂φ∂x

); i = 1, 2, 3 ∀y ∈ L14 (row 2, column 3) are unique. Thus,

set S = S1 ∪ S2 is unique ∀y ∈ L14. Hence, this interpolation is of typeC11(Ωe) ∀y ∈ L14. Next we consider L12 (Table 8.4). With φ and ∂φ

∂x

(set T1) and ∂φ∂y ,

∂∂x

(∂φ∂y

)(set T2) as degrees of freedom at nodes 1 and

2, φ and ∂φ∂y are polynomials of degree three in x ∀x ∈ L12 ensuring

uniqueness of φ, ∂iφ∂xi

; i = 1, 2, 3 and ∂φ∂y ,

∂i

∂xi

(∂φ∂y

); i = 1, 2, 3 ∀x ∈ L12,

however only φ, ∂φ∂x and ∂φ∂y ,

∂∂x

(∂φ∂y

)are unique ∀x ∈ L12 due to the

fact that these are nodal degrees of freedom at nodes 1 and 2. Hence,the elements of the set S = T1 ∪ T2 are unique ∀x ∈ L12. Thus theinterpolation is of type C11(Ωe) ∀x ∈ L12. Since the interpolation isof type C11(Ωe) ∀x ∈ L12 and also of type C11(Ωe) ∀y ∈ L14, theinterpolation is of type C11(Ωe) ∀x, y ∈ Ωe. This completes the proof.


8.5.2 2D interpolations of type C22(Ωe) with p-levelsof pξ and pη

(a) Proof of the uniqueness of C22(Ωe) interpolations.

It suffices to consider pξ = 5 and pη = 5 in which case there are no degreeof freedom at the midside nodes and the center node of the nine-nodeelement. We need to show that:

(i) the dimension of the space is same as the total degrees of freedomfor the element, and

(ii) if all degrees of freedom are identically zero, then φ(ξ, η) = 0 ∀ξ, η ∈Ωm.

For a nine-node p-version element with complete fifth degree polynomialin ξ and η for local approximation, the total number of basis functionsrequired is 36 (based on Pascal rectangle). Hence, the dimension of theapproximation space of φ(ξ, η) is 36. This local approximation has φ,∂φ∂x , ∂2φ

∂x2, ∂φ∂y , ∂

∂y

(∂φ∂x

), ∂∂y

(∂2φ∂x2

), ∂2φ∂y2

, ∂2

∂y2

(∂φ∂x

)and ∂2

∂y2

(∂2φ∂x2

)as degrees of

freedom at each of the four corner nodes (defined as set S) and no degreesof freedom at the mid side and center nodes, giving a total of 36 degreesof freedom for the element. Hence, the dimension of the approximationspace for φ(ξ, η) is equal to the total number of degrees of freedom forthe element. If all degrees of freedom are zero, then it follows directlyfrom the expression for φ(ξ, η) that φ(ξ, η) = 0 ∀ξ, η ∈ Ωm. Thus, theC22(Ωe) interpolations of p-levels of pξ and pη in ξ and η directions areunique.

(b) Proof of C22(Ωe) interelement continuity.

Without loss of generality we can consider pξ = pη = 5. Consider athree-element discretization in which the element one has elements twoand three as neighbors that is with common boundaries (see Fig. 8.26)with common sides L14 and L14 parallel to y axis and sides L12 andL12 parallel to x axis. Then we need to show that the elements of theset S are unique ∀y ∈ L14 and ∀x ∈ L12. For the proof we refer toTables 8.5 and 8.6 corresponding to the interelement boundaries L14

and L12. With φ, ∂φ∂y and ∂2

∂y2as degrees of freedom at nodes 1 and 4,

φ = φ(y) is a fifth degree polynomial in y ∀y ∈ L14 ensuring uniqueness

of φ, ∂iφ∂yi

(i = 1, 2, . . . , 5) ∀y ∈ L14 (row 1, column 3 of Table 8.5), how-

ever only φ, ∂φ∂y and ∂2φ∂y2

(set S1) are unique ∀y ∈ L14 due to the fact thatonly these are the degrees of freedom at nodes 1 and 4. Similarly, fromTable 8.6, we can conclude that elements of sets S2 and S3 are unique∀y ∈ L14. Thus, the set S is unique ∀y ∈ L14 and the interpolation is of


Table 8.5: Degrees of freedom, order of polynomials for φ, ∂iφ∂xi

; i = 1, 2 and uniqueness

of φ and its derivatives ∀y ∈ L14 and ∀y ∈ L14

Degrees of freedom Fifth degree Uniqueness Uniquenessat nodes 1 and 4 polynomial ∀y ∈ L14 ∀y ∈ L14

∀y ∈ L14

φ, ∂φ∂y, ∂

2φ∂y2

φφ, ∂

iφ∂yi

(φ, ∂φ

∂y, ∂

2φ∂y2

): setS1

i = 1, . . . , 5

∂φ∂x, ∂∂y

(∂φ∂x

), ∂

2

∂y2

(∂φ∂x

)∂φ∂x

∂φ∂x, ∂

i

∂yi

(∂φ∂x

) (∂φ∂x, ∂∂y

(∂φ∂x

), ∂

2

∂y2

(∂φ∂x

))i = 1, . . . , 5 : setS2

∂2φ∂x2

, ∂∂y

(∂2φ∂x2

), ∂

2

∂y2

(∂2φ∂x2

)∂2φ∂x2

∂2φ∂x2

, ∂i

∂yi

(∂2φ∂x2

) (∂2φ∂x2

, ∂∂y

(∂2φ∂x2

), ∂

2

∂y2

(∂2φ∂x2

))i = 1, . . . , 5 : setS3

Table 8.6: Degrees of freedom, order of polynomials for φ, ∂iφ∂yi

; i = 1, 2 and uniqueness

of φ and its derivatives ∀x ∈ L12 and ∀x ∈ L12

Degrees of freedom Fifth degree Uniqueness Uniquenessat nodes 1 and 2 polynomial ∀y ∈ L12 ∀y ∈ L12

∀y ∈ L12

φ, ∂φ∂x, ∂

2φ∂x2

φφ, ∂

iφ∂xi

(φ, ∂φ

∂x, ∂

2φ∂x2

); setS1i = 1, . . . , 5

∂φ∂y, ∂∂x

(∂φ∂y

), ∂

2

∂x2

(∂φ∂y

) ∂φ∂y

∂φ∂y, ∂

i

∂xi

(∂φ∂y

) (∂φ∂y, ∂∂x

(∂φ∂y

), ∂

2

∂x2

(∂φ∂y

))i = 1, . . . , 5 ; setS2

∂2φ∂y2

, ∂∂x

(∂2φ∂y2

), ∂

2

∂x2

(∂2φ∂y2

)∂2φ∂y2

∂2φ∂y2

, ∂i

∂xi

(∂2φ∂y2

) (∂2φ∂y2

, ∂∂x

(∂2φ∂y2

), ∂

2

∂x2

(∂2φ∂y2

))i = 1, . . . , 5 ; setS3

type C22(Ωe) ∀y ∈ L14. Next we consider L12 (Table 8.6). With φ, ∂φ∂x

and ∂2φ∂x2

as nodal dofs at nodes 1 and 3, φ(x) is a polynomial of degree

five in x ∀x ∈ L12 ensuring uniqueness of φ, ∂iφ∂xi

; i = 1, 2, . . . , 5 ∀x ∈ L12

(row 1, column 3 of Table 8.6), however only φ, ∂φ∂x and ∂2φ

∂x2(set T1) are

degrees of freedom at nodes 1 and 2 and hence unique ∀x ∈ L12 (row1, column 2 of Table 8.6). Similarly for rows 2 and 3 of Table 8.6, weconclude that elements of sets T2 and T3 are unique ∀x ∈ L12. thus,the elements of the set S = T1 ∪ T2 ∪ T3 are unique ∀x ∈ L12. Thusthe interpolation is of type C22(Ωe) ∀x ∈ L12. Since the interpolationis of type C22(Ωe) ∀x ∈ L12 and also of type C22(Ωe) ∀x ∈ L14, theinterpolation is of type C22(Ωe) ∀(x, y) ∈ Ωe. This completes the proof.


8.5.3 2D Cij(Ωe) interpolations of p-levels pξ and pη

A proof of the necessary and sufficient conditions for the general case inwhich the interpolations are of orders i and j in x and y directions and areof polynomial degrees pξ and pη in ξ and η directions follows by induction.This completes the proof of the theorem.

Remarks.

(1) Since 1D approximation functions and the nodal variable operators usedin the tensor product are only valid when ξ and η are parallel to x and yaxes. The family of 2D Cij(Ωe) local approximations generated by usingthem are only valid when the 2D elements in xy space are rectangularwith their sides parallel to x and y axes with the additional restrictionthat ξ and η axes must be pointing in the same directions as x andy axes. This obviously limits their usefulness in practical applicationsrequiring distorted element shapes in the xy space.

(2) The local approximation functions for a 2D C11(Ωe) element with pξ =pη = 3 will only have basis functions and dofs at the corner nodes. Ifp-levels are increased in ξ and η beyond three (could be unequal), thenadditional dofs will appear at the mid side and center nodes of the ninenode element. We note that pξ = pη = 3 is minimum for C11(Ωe) localapproximation.

(3) If we consider a 2D C22(Ωe) local approximation, then pξ = pη = 5 isthe minimum p-levels needed. For this local approximation, the basisfunctions and the dofs will only appear at the corner nodes. If p-levelsin ξ and η increased beyond five, then additional dofs will appear at themid side and the center node of the nine-node element.

(4) The need for higher order global differentiability local approximationsfor distorted quadrilateral 2D elements is rather obvious. We considerthis in the next section.

8.6 2D C ij(Ωe) approximations for quadrilateralelements: higher order global differentiabilityapproximations (HGDA)

In this section we present development of higher order global differen-tiability local approximations for two dimensional quadrilateral elements ofdistorted geometries in xy space. The distorted quadrilateral elements inphysical coordinate space xy are mapped into a master element in ξη nat-ural coordinate space in a two unit square with the origin at the center ofthe element. For the master element, 2D C00 p-version hierarchical local

8.6. 2D CIJ (ΩE) APPROXIMATIONS FOR QUADRILATERAL ELEMENTS 543

2 1

3

1

4

φ, ∂φ∂x ,∂φ∂y ,

∂2φ∂x∂y

3

2

x

y

φ, ∂φ∂x ,∂φ∂y ,

∂2φ∂x∂y

φ, ∂φ∂x ,∂φ∂y ,

∂2φ∂x∂y

L14 or L14

L12 or L12

Figure 8.25: A C11 element one with neighboring C11 elements two and three

2 1

3

φ, ∂φ∂x ,∂φ∂y ,

∂2φ∂x∂y ,

∂2φ∂x2 ,

∂2φ∂y2 ,

∂∂y

(∂2φ∂x2

), ∂∂x(∂2φ∂y2

), ∂

2

∂x2

(∂2φ∂y2

)φ, ∂φ∂x ,

∂φ∂y ,

∂2φ∂x∂y ,

∂2φ∂x2 ,

∂2φ∂y2 ,

∂∂y

(∂2φ∂x2

), ∂∂x(∂2φ∂y2

), ∂

2

∂x2

(∂2φ∂y2

)1 2

4 3

y

x

φ, ∂φ∂x ,∂φ∂y ,

∂2φ∂x∂y ,

∂2φ∂x2 ,

∂2φ∂y2 ,

∂∂y

(∂2φ∂x2

), ∂∂x(∂2φ∂y2

), ∂

2

∂x2

(∂2φ∂y2

)L14 or L14

L12 or L12

Figure 8.26: A C22 element one with neighboring C22 elements two and three

approximations are considered. The degrees of freedom and the approxi-mation functions from the mid-side nodes and/or center node are borrowedto derive desired derivative degrees of freedom at the corner nodes in theξη space for various higher order global differentiability approximations in


(b)

(c)

2 3

4

5

2

2

(a)

8

1 2 3

57

8 9 4

7 5

4

2 31

9

6

66

1

7

89

Map of the element of (a) innatural coordinate space ξη

Dofs for C00 p-version hierarchical elementwith p-levels pξ and pη directions

y

x

Nine-node 2D distortedquadrilateral in xy-space

η

η

ξ

φ φ

φφ∂iφ∂ξi

∂jφ∂ηj ξ

i = 2, 3, . . . , pξj = 2, 3, . . . , pη

∂iφ∂ξi

∂jφ∂ηj

∂i+jφ∂ηj∂ξi

Figure 8.27: 2D distorted p-version hierarchical element with p-levels pξ and pη in ξ andη directions

the ξη space. These derivative degrees of freedom at the corner nodes inξη space are then transformed from the natural coordinate space (ξ, η) tothe physical coordinate space (x, y) using Jacobians of transformations toobtain the desired higher order global differentiability local approximationsin the xy coordinate space. Pascal rectangle is used to establish a systematicprocedure for the selection of degrees of freedom and the corresponding ap-proximation functions from C00 p-version hierarchical element for the globaldifferentiability of any desired order in xy space.

The higher order global differentiability local approximations for dis-torted geometries cannot be derived using the tensor product approach pre-sented earlier in this chapter and also utilized in reference [1]. Here wepresent derivations of the higher order global differentiability local approx-


imations for distorted quadrilateral elements using a completely differentapproach. Ahmadi, Surana and Reddy [2] developed basic strategy for dis-torted quadrilateral elements using Lagrange monomials.

If the elements in xy space are distorted, we could possibly consider analternative. The distorted element from xy physical coordinate space is firstmapped in a two unit square in ξη natural coordinate space. We can consider1D higher order global differentiability approximations in ξ, η directions. Atensor product of these 1D approximations would yield higher order differ-entiability approximations in ξ, η space. The requirement of higher orderglobal differentiability in xy space necessitates that the derivative degreesof freedom at the corner nodes be transformed from ξη space to xy space.For example, in case of C11 HGDA, ∂

∂ξ , ∂∂η , ∂2

∂ξ∂η need to be transformed to∂∂x , ∂

∂y , ∂2

∂x∂y and for C22 HGDA, ∂∂ξ , ∂

∂η , ∂2

∂ξ2, ∂2

∂η2∂2

∂ξ∂η , ∂3

∂ξ2∂η, ∂3

∂ξ∂η2, ∂4

∂ξ2∂η2

need to be transformed into ∂∂x , ∂

∂y , ∂2

∂x2, ∂2

∂y2∂2

∂x∂y , ∂3

∂x2∂y, ∂3

∂x∂y2, ∂4

∂x2∂y2. Due

to the fact that degrees of freedom (dofs) in ξη space for Cij higher orderapproximations are not a complete set, this transformation is not possible.In case of C11, ∂

∂ξ , ∂∂η can be transformed into ∂

∂x , ∂∂y but there is no feasible

resolution for transforming ∂2

∂ξ∂η into ∂2

∂x∂y . In case of C22, we can transform∂∂ξ , ∂

∂η to ∂∂x , ∂

∂y and ∂2

∂ξ2, ∂2

∂η2, ∂2

∂ξ∂η to ∂2

∂x2, ∂2

∂y2, ∂2

∂x∂y . However, we cannot

transform ∂3

∂ξ2∂η, ∂3

∂ξ∂η2, ∂4

∂ξ2∂η2into their counterparts in xy space. Similar

situation exists for orders higher than two as well. Thus, the derivationof HGDA for 2D distorted elements in xy space requires a fundamentallydifferent approach.

Guidelines

In deriving the desired HGDA for 2D distorted elements of quadrilateralfamily in xy space, we use following guidelines.

(a) The distorted element geometry is mapped from xy physical coordi-nate space into ξη natural coordinate space, a two-unit square (seeFig. 8.27(a) and (b)). The origin of the ξ, η coordinate system is lo-cated at the center of the map in ξη space and we have the following forthe mapping of points,

xy

=

n∑i=1

Ni(ξ, η)

xiyi

(8.113)

In which, (xi, yi) are the Cartesian coordinates of the nodes and Ni(ξ, η)are the shape functions. We could use eight node configuration (i.e.n=8) with serendipity functions (see later section for details) for this


purpose, or standard Lagrange family tensor product basis or shapefunctions.

(b) If possible, we would like to consider C00 p-version hierarchical localapproximation as a starting point in the derivation of 2D HGDA due tothe fact that this would permit increase in p-levels without rediscretiza-tion. Figure 8.27 (c) shows nodal degrees of freedom for a standardC00 p-version hierarchic element in which ϕ is the field variable beinginterpolated. The degrees of freedom at the corner nodes of this elementconsist of only function values.

(c) Different degrees of freedom are needed at the corner nodes than thosefor the 2D HGDA generated using tensor product. This is due to thefact that dofs in tensor product 2D HGDA do not transform from xy toξη or vice versa. Obviously the choices of the dofs at the corner nodesare dictated by the transformations of the derivatives between xy andξη spaces.

(d) The degrees of freedom for 2D HGDA element of distorted shape arechosen such that they can be transformed using standard Jacobiansof transformation from natural coordinate space to physical coordinatespace. The choices of nodal operators (or dofs) at the corner nodes listedin Table 8.7 for C11, C22 and C33 HGDA satisfy this requirement. Wenote that for C11, the derivative operators are a complete set of firstorder operators. For C22 HGDA, the set for C11 is augmented by acomplete second order set and so on.

Table 8.7: Choices of nodal operators at the corner nodes for Cij 2D distorted quadri-lateral elements in xy space

Type of HGDA Nodal operators at the corner nodes

C11 1, ∂∂x ,∂∂y

C22 1, ∂∂x ,∂∂y ,

∂2

∂x2, ∂2

∂y∂x ,∂2

∂y2

C33 1, ∂∂x ,∂∂y ,

∂2

∂x2, ∂2

∂y∂x ,∂2

∂y2, ∂3

∂x3, ∂3

∂y∂x2, ∂3

∂y2∂x, ∂

3

∂y3

(e) Since C00 p-version hierarchical approximations are used as a startingpoint and since in these local approximations only the function values arethe degrees of freedom at the corner nodes, we must establish some rulesand a systematic procedure that allow us to borrow the desired numberof degrees of freedom from C00 p-version hierarchical approximations togenerate the desired dofs at the corner nodes of the 2D HGDA elementof distorted geometry in xy space.


Transformation matrices

In this section we present details of the transformation matrices essentialto derive 2D HGDA for distorted quadrilateral geometries in xy space. Thesetransformations permit transformations of the derivative degrees of freedomfrom ξη space to xy space. Figure 8.27 (c) shows nodal degrees of freedom forC00 p-version hierarchical element in which ϕ is dependent variable. From(8.113), we obtain the following for mapping of lengths in (ξ, η) and (x, y)spaces,

dxdy

= [J ]

dξdη

(8.114)

where

[J ] =

[∂x∂ξ

∂x∂η

∂y∂ξ

∂y∂η

]=[x yξ η

]=

[xξ xηyξ yη

](8.115)

Using the C00 p-version hierarchical approximations for a nine-node p-versionhierarchical element [see Fig. 8.116(c)], the field variable φ can be approxi-mated using

φ(ξ, η) = [N(ξ, η)]δe (8.116)

in which [N(ξ, η)] is a row matrix of C00 p-version hierarchical local ap-proximations and δe are the corresponding nodal dofs (arranged in somesuitable fashion).

We define

φξη1 =[∂φ∂ξ

∂φ∂η

]T(8.117)

φxy1 =[∂φ∂x

∂φ∂y

]T(8.118)

φξη2 =[∂2φ∂ξ2

∂2φ∂η∂ξ

∂2φ∂η2

]T(8.119)

φxy2 =[∂2φ∂x2

∂2φ∂y∂x

∂2φ∂y2

]T(8.120)

φξη3 =[∂3φ∂ξ3

∂3φ∂η∂ξ2

∂3φ∂η2∂ξ

∂3φ∂η3

]T(8.121)

φxy3 =[∂3φ∂x3

∂3φ∂y∂x2

∂3φ∂y2∂x

∂3φ∂y3

]T(8.122)

where T denotes transpose.

We note that (8.117) and (8.118) are a complete set of first order deriva-tives. Equations (8.119) and (8.120) are a complete set of second orderderivatives, and (8.121) and (8.122) are a complete set of third order deriva-

tives in ξη and xy coordinate spaces. In this manner, we can define φξηi andφxyi as complete sets of the derivatives of order i in (ξ, η) and (x, y) spaces.


Next we define the rules of transformation between the sets of different orderderivatives in (ξ, η) and (x, y) spaces.

φξηi = [Ji]φxyi (8.123)

Obviously,

[J1] =

[xξ yξxη yη

]= [J ]T (8.124)

Using chain rule of differentiation, we can determine the transformationmatrices for higher order derivatives of the dependent variable. We will usethe following notations for higher order derivatives of physical coordinate xwith respect to natural coordinates ξ and η

xξ =∂x

∂ξ, xiξ =

(∂x∂ξ

)i, xξi =

∂ix

∂ξi, xξiηj =

∂i+jx

∂ξi∂ηj(8.125)

and

xη =∂x

∂η, xiη =

(∂x∂η

)i, xηi =

∂ix

∂ηi(8.126)

Similarly, we have the following notations for derivatives of physical co-ordinate y with respect to natural coordinates ξ and η.

yξ =∂y

∂ξ, yiξ =

(∂y∂ξ

)i, yξi =

∂iy

∂ξi, yξiηj =

∂i+jy

∂ξi∂ηj(8.127)

and

yη =∂y

∂η, yiη =

(∂y

∂η

)i, yηi =

∂iy

∂ηi(8.128)

Following these notations, the transformation matrices for the secondorder derivatives are as follows:

φxy2 = [J2]−1[φξη2 − [J1

2 ]φxy1

](8.129)

[J2] =

x2ξ 2xξyξ y2

ξ

xξxη xξyη + xηyξ yηyξ

x2η 2xηyη y2

η

[J12 ] =

xξ2 yξ2

xξη yξη

xη2 yη2

(8.130)

Similarly, transformation matrices for the third order derivatives are asfollows:

φxy3 = [J3]−1[φξη3 − [J1

3 ]φxy2 − [J23 ]φxy1

](8.131)

[J3] =

x3ξ 3x2

ξyξ 3xξy2ξ y3

ξ

x2ξxη 2xηxξyξ + x2

ξyη 2xξyηyξ + y2ξxη yηy

2ξ

x2ηxξ 2xηxξyη + x2

ηyξ 2xηyηyξ + y2ηxξ yξy

2η

x3η 3x2

ηyη 3xηy2η y3

η

(8.132)


[J13 ] =

3xξxξ2 3xξ2yξ + 3yξ2xξ 3yξyξ2

xηxξ2 + 2xξxξη xξ2yη + 2xξηyξ + xηyξ2 + 2xξyξη yηyξ2 + 2yξyξη

xξxη2 + 2xηxξη xξyη2 + 2xξηyη + xη2yξ + 2xηyξη yη2yξ + 2yηyξη

3xηxη2 3xη2yη + 3yη2xη 3yηyη2

(8.133)

[J23 ] =

xξ3 yξ3

xξ2η yξ2η

xξη2 yξη2

xη3 yη3

(8.134)

From (8.123), (8.129) and (8.131) we can write the following general expres-sion,

φxyi = [Ji]−1[φξηi − [J1

i ]φxyi−1− [J2i ]φxyi−2− . . .− [J i−1

i ]φxy1

](8.135)

8.6.1 C11 HGDA for 2D distorted quadrilateral elements in xyspace

In order to show specific details of the development, we consider C11

HGDA. Figure 8.28 (a) shows the dofs at the corner nodes of C11 HGDAelement (subscript indicates differentiation). Comparing Figure 8.28 (a) withC00 p-version element of Fig. 8.27 (c), we note that the element of Fig. 8.28(a) requires φx and φy as additional dofs at each of the four corner nodes,

(a) (b)

7 6 5

8

21 3

9 48

7 5

4

2 31

9

6

Nodal dofs at the corner nodes of a2D C11 HGDA element

Dofs at the hierarchical nodes of 2DC11 HGDA element

j = 4, 5, . . . , pηm = 2, 3, . . . , pη

i = 4, 5, . . . , pξl = 2, 3, . . . , pξ

η

ξ

φ, φx, φy φ, φx, φy

φ, φx, φyφ, φx, φy

η

∂iφ∂ξi

∂jφ∂ηj ξ

∂iφ∂ξi

∂jφ∂ηj

∂l+mφ∂ηl∂ξm

Figure 8.28: Nodal dofs for a 2D C11 HGDA distorted quadrilateral element


a total of eight dofs for the four corner nodes. We need to borrow eight dofsand the corresponding C00 p-version approximation functions to generatethese dofs and the corresponding approximation functions for the 2D C11

HGDA element. This would obviously result in reduction of dofs at thehierarchical nodes of the 2D HGDA element. In doing so we must followa systematic procedure. For this case, the choice of dofs from C00 elementis rather straightforward. We borrow two dofs which are associated withthe lowest p-levels (i.e. p = 2 and 3) and the corresponding approximationfunctions from each of the four mid-side nodes. This implies that the degrees

of freedom ∂2φ∂ξ2

, ∂3φ∂ξ3

from nodes 2, 6 and ∂2φ∂η2

, ∂3φ∂η3

from nodes 4 and 8 andthe corresponding approximation functions are borrowed. These dofs mustbe eliminated to generate the derivative dofs at the corner nodes of 2DC11 HGDA element [see Fig. 8.28(a)]. Figure 8.28(b) shows the dofs at thehierarchical nodes of the 2D HGDA element. The remaining details for thederivation of the 2D C11 HGDA element can be obtained from the generalderivation given in a later section.

We note that the dofs removed from the mid side nodes of the C00 p-version element correspond to p-levels 2 and 3 and hence consistent withthe tensor product C11 element. The first degree of freedom at the mid sidenodes of the C11 HGDA element corresponds to p-level of 4. For C11 HGDAelement, we do not need to borrow any dofs from the center node of C00

p-version element.


We consider 2D C22 HGDA for distorted quadrilateral elements in xyspace. Figure 8.29 (a) shows dofs at the corner nodes of the element. Com-paring this with C00 HGDA of Fig. 8.27(c), we note that φx, φy, φx2 , φxyand φy2 are additional dofs at each of the four corner nodes, i.e. a total oftwenty. We need to borrow twenty dofs and the corresponding C00 p-versionapproximation functions from 2D C00 p-version approximation to generatethese dofs and the corresponding approximation functions for the 2D C22

HGDA element. This would obviously result in reduction of dofs at the hi-erarchical nodes of the 2D HGDA element. For this case, the choice of dofsfrom C00 element is not as straight forward as those for C11 HGDA element.We borrow four dofs which are associated with the p-levels 2, 3, 4 and 5 andthe corresponding approximation functions from each of the four mid-sidenodes to maintain conformity with C22 tensor product element. This implies

that the degrees of freedom ∂2φ∂ξ2

, ∂3φ∂ξ3

, ∂4φ∂ξ4

, ∂5φ∂ξ5

from nodes 2, 6 and ∂2φ∂η2

, ∂3φ∂η3

,∂4φ∂η4

, ∂5φ∂η5

from nodes 4 and 8 and the corresponding approximation functionsare borrowed. This would result in a total of 16 degrees of freedom.


(a)(b)

7 6 5

8

21 3

9 4

8

7 5

4

2 31

6

9

Nodal dofs at the corner nodes of a2D C22 HGDA element

Dofs at the hierarchical nodes of 2DC22 HGDA element

l = 2, 3, . . . , pξ

m = 2, 3, . . . , pηl +m ≥ 7

φ, φx, φy

φx2 , φxy, φy2

φ, φx, φy

φx2 , φxy, φy2

φ, φx, φy

φx2 , φxy, φy2

φ, φx, φy

φx2 , φxy, φy2

η

ξ

η

∂iφ∂ξi

∂jφ∂ηj ξ

∂iφ∂ξi

∂jφ∂ηj

i = 6, 7, . . . , pξ j = 6, 7, . . . , pη

∂6φ∂η2∂ξ4

∂l+mφ∂ηl∂ξm

∂6φ∂η4∂ξ2

Figure 8.29: Nodal dofs for a 2D C22 HGDA distorted quadrilateral element

The remaining four degrees of freedom are borrowed from the centernode (node 9). The degrees of freedom that are borrowed from center nodeof the C00 tensor product element are borrowed in such a way that the dofscorresponding to lower p-levels are selected before those corresponding tohigher p-levels. Figure 8.30 shows the dofs generated at the center nodeof a C00 element corresponding to p-levels pξ (in ξ direction) and pη (inη direction). The dofs illustrated with a circle are selected in deriving aC22 HGDA element. They correspond to (pξ, pη) pairs of (2, 2), (3, 2), (2, 3)and (3, 3). Degree of freedom corresponding to p-level pair of (3, 3) is chosenover (4, 2) or (2, 4) to ensure symmetry with respect to pξ and pη. Symmetryin the degrees of freedom pairs is maintained to preserve the symmetry offinite element solutions for symmetric discretizations. These dofs must beeliminated from C00 p-version approximations to generate the derivativedofs at the corner nodes required for 2D C11 HGDA element [Fig. 8.29(a)].Figure 8.29(b) shows the dofs at the hierarchical nodes of the 2D HGDAelement.


We consider 2D C33 HGDA for distorted quadrilateral elements in xyspace. Compared to a C00 HGDA of Fig. 8.27(c), we note that φx, φy, φx2 ,φxy, φy2 , φx3 , φx2y, φxy2 and φy3 are additional dofs at each of the fourcorner nodes, a total of thirty six dofs. Hence, we need to borrow thirty sixdofs from the hierarchical nodes of C00 p-version element, keeping in mindthat the remaining dofs at the mid side nodes must begin with p-level ofeight. This is to ensure that C33 HGDA is in conformity with C33 tensor


2 3 4 5

2

3

4

5

6

6

ξ

ξ∂

∂ 2

η∂

∂η

3

ξ

ξ∂

∂ 4

ξ

ξ∂

∂ 5

ξ

ξ∂

∂ 6

η∂

∂η

2

ξ

ξ∂

∂ 3

η∂

∂η

4

η∂

∂η

5

η∂

∂η

6

1

ξp

ηp

22ηξ 23ηξ 24ηξ 25ηξ 26ηξ



52ηξ 53ηξ54ηξ 55ηξ

56ηξ

62ηξ 63ηξ64ηξ 65ηξ 66ηξ

Additional dofs from the center node of C00 p-version element for C22

HGDA element

Additional dofs from the center node of C00 p-version element for C33

HGDA element

Figure 8.30: Dofs at the center node of a C00 p-version hierarchical element

product element. This allows us to borrow twenty four dofs (correspondingto p-levels of 2, 3, 4, 5, 6 and 7) from each of the mid side nodes (nodes 2,4, 6, 8) of Fig. 8.27(c), making a total of twenty four. The remaining twelvedofs needed to generate the dofs at the corner nodes of C33 HGDA elementmust come from the center node (node 9).

From Fig. 8.30, the dofs illustrated with circle and square are the dofsselected from the center node of C00 tensor product element in derivinga C33 HGDA element. The additional dofs correspond to (pξ, pη) pairs of(2, 2), (3, 2), (2, 3), (3, 3), (2, 4), (2, 5), (3, 4), (3, 5), (4, 2), (4, 3), (5, 2) and(5, 3). These degrees of freedom are chosen in such a way that symmetry isensured with respect to pξ and pη.

With the discussion of the concepts relating to the transformation ma-trices and the selection of the dofs, the general derivation of the Cij ap-proximations for distorted quadrilateral elements is now presented in thefollowing.


8.6.4 Derivation of Cij approximations for distortedquadrilateral elements

In this section, we describe a general methodology which utilizes C00

p-version hierarchical interpolation functions as a starting point (describedin earlier sections) to generate desired higher order global differentiabilityapproximations for distorted quadrilateral elements. Since the approxima-tion functions are functions of natural coordinates ξ, η, Ni = Ni(ξ, η), thedesired derivative degrees of freedom need to be generated first in ξη spaceand then transformed into xy space as described above for C11, C22, andC33 HGDA. The transformations described earlier assist us in transformingthe desired derivative degrees of freedom from ξη coordinate space to xycoordinate space.

The dofs δe of a nine node C00 p-version hierarchical element are sep-arated into those corresponding to corner nodes (denoted by co), mid-sidenodes (m) and center node (c) as follows:

ϕ(ξ, η) = [a]δecor1 + [b]δemcel + [c]δemr2 + [d]δecr3 (8.136)

where the subscript r1 denotes the degrees of freedom retained at the cornernodes. Subscript el corresponds to the degrees of freedom borrowed from themid-side nodes and the center node that are to be eliminated to incorporatethe new derivative degrees of freedom at the corner nodes of a Cij HGDAelement. The subscripts r2 and r3 denote the remaining degrees of freedom(to be retained) from mid-side nodes and center node (after borrowing therequired degrees of freedom). [a], [b], [c] and [d] are row matrices containingC00 p-version local approximations corresponding to the dofs in the r1, el,r2, r3 sets respectively.

For a C11 HGDA element, δemcel consists of dofs from mid-side nodesonly since we do not need any dofs from the center node, i.e. δemcel =δemel,which consists of the following:

δeme =

∂2φ

∂ξ2

∣∣∣∣2

,∂3φ

∂ξ3

∣∣∣∣2

,∂2φ

∂η2

∣∣∣∣4

,∂3φ

∂η3

∣∣∣∣4

,∂2φ

∂ξ2

∣∣∣∣6

,∂3φ

∂ξ3

∣∣∣∣6

,∂2φ

∂η2

∣∣∣∣8

,∂3φ

∂η3

∣∣∣∣8

T(8.137)

The subscript in (8.137) indicates node number. For classes higher thanC11, δemcel will consist of dofs from mid-side nodes as well as center nodeas described in sections 8.6.2 and 8.6.3.

Let the desired new derivative dofs at the corner nodes of a Cij HGDAelement be denoted by δexyn . In case of a C11 HGDA element, these dofswill consist of complete set of first order derivatives of the dependent variableevaluated at the four corner nodes, i.e.

δeξηn =φξ|1 , φη|1 , φξ|3 , φη|3 , φξ|5 , φη|5 , φξ|7 , φη|7

T(8.138)


where subscripts ξ and η on φ denote differentiations, φξ|1 = ∂φ∂ξ |node 1 =

∂φ∂ξ |(ξ=−1,η=−1).

For classes higher than C11, the new derivative dofs at the corner nodeswill be augmented with the complete sets of derivatives up to the class beingderived. Differentiating (8.136) with respect to ξ and η and evaluating theresulting expression at each of the four corner nodes, we get

δeξηn = [A]δecor1 + [B]δemcel + [C]δemr2 + [D]δecr3 (8.139)

Solving for the degrees of freedom to be eliminated i.e. δemcel in (8.139),we get

δemcel = [B]−1δeξηn − [B]−1[A]δecor1 − [B]−1[C]δemr2 − [B]−1[D]δecr3(8.140)

Substituting Jacobian of transformation from (8.135) into the aboveequation, we can transform the new derivative dofs from ξη space to xyspace. Equation (8.140) can thus be written as

δemcel = [B]−1[Ji]δexyn −[B]−1[A]δecor1−[B]−1[C]δemr2−[B]−1[D]δecr3(8.141)

Now, substituting δemcel from (8.141) into (8.136),

φ(ξ, η) = [a]δecor1 + [b]

([B]−1[Ji]δexyn − [B]−1[A]δecor1

− [B]−1[C]δemr2 − [B]−1[D]δecr3)

+ [c]δemr2 + [d]δecr3 (8.142)

Collecting terms in the (8.142), we get the final form of the Cij HGDAlocal approximations as follows:

φ(ξ, η) =([a]− [b][B]−1[A]

)δecor1 + [b][B]−1[Ji]δexyn

+([c]− [b][B]−1[C]

)δemr2 +

([d]− [b][B]−1[D]

)δecr3 (8.143)

8.6.5 Limitations of 2D C11 global differentiability localapproximations for distorted quadrilateral elements

In the proposed framework, 2D Cij global differentiability local approx-imations are derived by borrowing appropriate degrees of freedom and thecorresponding approximation functions from the hierarchical nodes of C00

element. In (8.143), [a], [c] and [d] contain C00 local approximations whichare retained at corner, mid-side and center nodes whereas [b] contains C00

local approximation functions which are borrowed from mid-side and cen-ter nodes. [A], [B], [C] and [D] are matrices containing derivatives of C00


approximations collected in [a], [b], [c] and [d] with respect to ξ and η evalu-ated at the corner nodes. The approximation functions for the C11 distortedelement at the corner, mid-side and center nodes (which are retained) areobtained by modifying the corresponding functions for the C00 element by[b][B]−1[A], [b][B]−1[C] and [b][B]−1[D] respectively.

In case of 2D C11 HGDA element, the new derivative degrees of freedomintroduced at the corner nodes are first order derivatives with respect tox and y. The nature of the C00 local approximation functions and thecoordinates of the corner nodes (ξ and η coordinates are either +1 or −1)always result in all of the coefficients of [D] matrix to be zero regardlessof the p-level. The coefficients of matrices [A], [B], [C] however are not allzero. This results in the approximation functions at the center node of C11

distorted element to be exactly same as those corresponding to C00 element(since [b][B]−1[D] is a row matrix containing all zeros). When we deriveapproximation functions for C22 and higher order elements, the derivativedegree of freedoms introduced at the corner nodes include mixed derivativeswith respect to x and y. The mixed derivatives of the C00 approximationfunctions in [d] (center node) evaluated at the corner nodes are not all zeroand hence coefficients in [D] matrix are not all zero. This possibly resultsin loss of complete basis for C11 HGDA elements. This is currently underinvestigation. An alternative way to generate C11 HGDA element is beingconsidered.

Remarks.

1. The derivation presented above is general and is independent of the natureof the C00 interpolation functions. Hence, the row matrices [a], [b] , [c]and [d] can contain approximation functions of any kind (for example,Lagrange, Legendre or Chebyshev functions). The different choices ofC00 approximation functions would yield the corresponding Cij HGDAelements. This could be important from the point of view of conditioningnumber of the resulting matrices in applications.

2. The matrices [A], [B], [C], [D] contain derivatives of C00 approximationfunctions with respect to ξ and η evaluated at the corner nodes. Theycan be precomputed once and used to generate approximation functionsof any order Cij ; i, j ≥ 2 element. This is important from the point ofview of efficiency of computations.

3. The approximation functions that are borrowed from the mid-side nodesand center node of C00 elements should be such that: (i) lowest degree ad-missible functions (corresponding to a lower p-level) are selected first (ii)and a symmetric pattern maintained in their selection so that symmetricdiscretizations for symmetric behaviors would yield numerical solutionsthat are also symmetric.


8.7 Interpolation theory for 2D triangular elements:basis functions of class C00 based on Lagrangeinterpolation

8.7.1 Langrange family C00 basis functions based on Pascaltriangle

Following a procedure similar to that for quadrilateral elements usingPascal’s rectangle, local approximations of class C00 can be established fortriangular elements as well. However, in this case the mapping of the ele-ment from the physical space x, y or x, y, z to a natural coordinate space issomewhat different than for the quadrilateral shapes. First, let us consider atriangular subdomain or element with straight sides in the physical coordi-nate space. A local approximation φ(x, y) over Ωe would require a decisionon (just like quadrilateral elements):

(a) The choice of the monomial terms in x, y and xy etc. to be used in thelinear combination to describe φ(x, y) over Ωe based on complete linear,quadratic, cubic, etc. polynomials in x and y.

(b) The choice of the locations of the nodes on the boundary of the elementas well as its interior.

This decision can be greatly facilitated by considering what is known asPascal’s triangle. Pascal’s triangle is a systematic arrangement of monomialsof various degrees in x and y and their interaction such that the resultingpolynomials of any fixed degree in x and y can be readily constructed. Alongone side of the triangle we have increasing powers of x starting with zerowhile on the other side of the triangle we have increasing powers of y alsostarting with zero. The horizontal lines connecting the corresponding likepower terms in x and y provides the interaction terms between x and y.Using the Pascal’s triangle local approximation functions of any degree canbe generated for the triangular elements. First, we note the following:

(a) The location of the terms in the triangular arrangement are the locationsof the nodes for the triangular element Ωe.

(b) Also, the terms themselves are the functions to be used in the linearcombination for φ(x, y).

(c) Based on these two criteria, elements with constant, bilinear, bi-quadraticand bi-cubic behavior of φ over Ωe with 1, 3, 6 and 10 nodes (see Fig. 8.31)can be easily generated.

Example 8.5. Consider the six-node bi-quadratic element in Fig. 8.32,which shows the configuration of nodes as well as the monomials to be con-

8.7. INTERPOLATION THEORY FOR 2D TRIANGULAR ELEMENTS 557

6

5

4

3

2

0

Pascal’s triangle

1

1

3

6

10

15

21

28

Element with nodes

yx

x3

x2 xy

1

y2

y3

y4

x5 x3y2 x2y3 xy4 y5x4y

xy2x2y

y3xy3x3yx4

x3y3 x2y4 xy5 y6x5yx6 x4y2

Degree of thepolynomial

Number of termsin the polynomial

Figure 8.31: Pascal’s triangle for the C00 Lagrange family of triangular elements.

sidered in the polynomial representation of φ over Ωe. For the local approx-imation of φ over Ωe, we can immediately write

φ(x, y) = c1 + c2x+ c3y + c4xy + c5x2 + c6y

2 (8.144)

Let (xi, yi); i = 1, . . . , 6 be the coordinates of the nodes of the element andlet φei (i = 1, . . . , 6) be the function values of φ at these nodes. Then

φ(x, y) =[1 x y xy x2 y2

][C]−1

φe1...φe6

= [N(x, y)]φe =6∑i=1

Ni(x, y)φei

(8.145)in which

[C] =

1 x1 y1 x1y1 x21 y

21

......

......

......

1 x6 y6 x6y6 x26 y

26

(8.146)

6

4

1 32

5

MonomialsNodes

yx

x2 xy

1

y2

Figure 8.32: Six-node triangular element

Once again, we could verify that

Ni(xj , yj) =

1, j = i

0, j 6= i, i = 1, . . . , 6 (8.147)


and6∑i=1

Ni(x, y) = 1 (8.148)

Remarks.

(1) The shortcomings of this approach are similar to those for the quadri-lateral elements based on Pascal’s rectangle. With increasing p-levels,number of nodes per element increase requiring a new discretization. Es-tablishing the local approximation functions requires inverse of progres-sively increasing size coefficient matrices with progressively increasingp-levels.

(2) The basis functions derived using this approach are clearly of C0 type(as function values are the only nodal dofs) of Lagrange family.

(3) It is significant to note that regardless of the shortcomings of this ap-proach, Pascal’s triangle provides a systematic means of the nodal con-figurations and the selection of monomials based on p-levels.

8.7.2 Lagrange family C00 basis functions based onarea coordinates

The concept of area coordinates is the most natural way to derive basisfunctions for triangular elements. Consider a three node triangular elementwith straight sides shown in Fig. 8.33. Let the sides opposite to node i belabelled side i for i = 1, 2, 3. Consider a point inside the element and connectit with straight lines to the nodes of the element. This divides the area A ofthe triangle into three triangular areas A1, A2 and A3, noting that area Aiis along the side i, i = 1, 2, 3. Let Li be the ratio of Ai

A (i = 1, 2, 3).

Li =AiA, i = 1, 2, 3 (8.149)

usingA = A1 +A2 +A3 (8.150)

and dividing both sides by A and then using (8.149)

1 = L1 + L2 + L3 (8.151)

Li (i = 1, 2, 3) are called area coordinates or natural coordinates for thetriangular region of area A. The properties of L1, L2 and L3 are importantto note.

(i) Li has a value of 1 at node i and has a value of zero on side i and thushas a value of zero at the other nodes. That is, L1 is one at node 1,zero on side 1 and, hence zero at nodes 2 and 3.


y

x

Side 1

1

Side 2

Side 3

A1

A3

A2

3

2

Figure 8.33: Triangular coordinates

(ii) Their sum is obviously one at any point within the element ((8.151)).

The area coordinates Li can be used to define the Cartesian coordinates ofany point in Ωe. If (xi, yi) are the Cartesian coordinates of the nodes of theelement, then we can write

x = L1x1 + L2x2 + L3x3

y = L1y1 + L2y2 + L3y3(8.152)

using (8.151) and (8.152)1xy

=

1 1 1x1 x2 x3

y1 y2 y3

L1

L2

L3

= [C]

L1

L2

L3

(8.153)

Therefore L1

L2

L3

= [C]−1

1xy

=

N1(x, y)N2(x, y)N3(x, y)

(8.154)

We could easily confirm that Ni(x, y) (i = 1, 2, 3) are same as those usingPascal triangle. We note that det[C] = A

2 , hence [C]−1 in (8.154) is unique.Thus, for the triangular element with three nodes (see Fig. 8.33) we canuse L1, L2, L3 as basis functions instead of Ni(x, y) (i = 1, 2, 3). In the nextsection, we extend this concept of area coordinate to basis functions of higherp-level.

8.7.3 Higher degree C00 basis functions using areacoordinates

Consider the triangular elements with straight sides shown in Fig. 8.34with vertex nodes 1,2 and 3. First, we explain how the figures in Fig. 8.34 areconstructed. Consider Fig. 8.34(a). Draw m equally spaced lines parallel toside 1 labelled 0, 1, . . . ,m, ‘0’ being side 1. Similarly we also draw m equally


Equally spaced m lines parallelto side 1

(a)


(c) Identification of a point basedon lines parallel to sides 1 to 3

(d)


(b)

3

Side 1

1

1 0

2

0

m− 1

1

m m− 1 p

m

r r

p

2

Side 2

0

3

11 mm− 1

q

p q r

21 Side 3

3 3

21

p

Figure 8.34: Identification of element nodes based on area coordinates

spaced lines parallel to sides 2 and 3 [see Fig. 8.34(b) and (c)], sides 2 and3 being marked ‘0’. If we consider typical lines p, q and r parallel to sides1, 2 and 3, then their intersection marked pqr [Fig. 8.34(d)] defines a pointwithin the element. The interaction of these parallel lines with the sides 1, 2and 3 and amongst themselves define the locations of the nodes. These arein agreement with the Pascal triangle. Consider triangle in Fig. 8.34(a). Wenote that L1 = 0 on side 1 and has a value 1 at node 1. We can setup La-grange type interpolations to define functions N0(L1), N1(L1), . . . , Nm(L1)corresponding to the (m+ 1) parallel lines. We define Ni(L1) by

Ni(L1) =

i∏j=1

(mL1 − j + 1

j

), for j ≥ 1; i = 1, 2, . . . ,m

= 0, for i = 0

(8.155)

Similarly in the other two directions we setup Ni(L2) and Ni(L3)

Ni(L2) =i∏

j=1

(mL2 − j + 1

j

), for j ≥ 1; i = 1, 2, . . . ,m

= 0, for i = 0

(8.156)


Ni(L3) =i∏

j=1

(mL3 − j + 1

j

), for j ≥ 1; i = 1, 2, . . . ,m

= 0, for i = 0

(8.157)

The two dimensional basis function for a node located at p, q, r i.e. Npqr(L1, L2, L3)can now be defined

Npqr(L1, L2, L3) = Np(L1)Nq(L2)Nr(L3) (8.158)

Np(L1), Nq(L2) and Nr(L3) are defined by (8.155) to (8.157). We considersome examples in the following.

Three-node triangular element (p-level of one)

In this case m = 1 and we have the pqr values at the nodes shown inFig. 8.35, and

N0(L1) = 1

N1(L1) = L1, m = 1, j = 1, i = 1 in (8.155)(8.159)

N0(L2) = 1

N1(L2) = L2, m = 1, j = 1, i = 1 in (8.156)(8.160)

N0(L3) = 1

N1(L3) = L3, m = 1, j = 1, i = 1 in (8.157)(8.161)

Side 3 (pqr) = (010)(pqr) = (100)

Side 2 Side 1

3

1 2

(pqr) = (001)

Figure 8.35: pqr values for three node element

The basis functions Ni(L1, L2, L3) (i = 1, 2, 3) for the three-node trian-gular element are now obtained using the following:

N1(L1, L2, L3) = N pqr(100)

= N1(L1)N0(L2)N0(L3) = L1 (8.162)

N2(L1, L2, L3) = N pqr(010)

= N0(L1)N1(L2)N0(L3) = L2 (8.163)

N3(L1, L2, L3) = N pqr(001)

= N0(L1)N0(L2)N1(L3) = L3 (8.164)


Six-node triangular element (p-level of two)

Figure 8.36 shows a six-node triangular element for p = 2 in x and y(based on Pascal’s triangle) and pqr values at the nodes of the element. Wegenerate Ni(L1), Ni(L2) and Ni(L3) functions using (8.155) to (8.157). Inthis case m = 2.

N0(L1) = 1

N1(L1) =

1∏j=1

(2L1 − j + 1

j

)= 2L1

N2(L1) =2∏j=1

(2L1 − j + 1

j

)= L1(2L1 − 1)

(8.165)

3

5

6 4

2 31

(002)

(011)

(020)(110)(200)

(101)

21

vertex nodes for L1, L2, and L3 linesand locations of 1, 2, . . . , 6 nodes ofthe six-node triangular element

Figure 8.36: pqr values at the nodes of a six-node quadratic triangular element

Similarly,

N0(L2) = 1

N1(L2) =1∏j=1

(2L2 − j + 1

j

)= 2L2

N2(L2) =

2∏j=1

(2L2 − j + 1

j

)= L2(2L1 − 1)

(8.166)

N0(L3) = 1

N1(L3) =1∏j=1

(2L3 − j + 1

j

)= 2L3

N2(L3) =

2∏j=1

(2L3 − j + 1

j

)= L3(2L3 − 1)

(8.167)


The basis functions Ni(L1, L2, L3) (i = 1, 2, . . . , 6) for the six-node quadraticelement can be generated using the following:

N1(L1, L2, L3) = N pqr(200)

(L1, L2, L3) = N2(L1)N0(L2)N0(L3) = L1(2L1 − 1)

N2(L1, L2, L3) = N pqr(111)

(L1, L2, L3) = N1(L1)N1(L2)N0(L3) = 4L1 L2)

N3(L1, L2, L3) = N pqr(020)

(L1, L2, L3) = N0(L1)N2(L2)N3(0) = L2(2L1 − 1)

N4(L1, L2, L3) = N pqr(011)

(L1, L2, L3) = N0(L1)N1(L2)N1(L3) = 4L2 L3

N5(L1, L2, L3) = N pqr(002)

(L1, L2, L3) = N0(L1)N0(L2)N2(L3) = L3(2L3 − 1)

N6(L1, L2, L3) = N pqr(101)

(L1, L2, L3) = N1(L1)N2(0)N3(1) = 4L1 L3

(8.168)

Ten-node triangular element (p-level of three)

Figure 8.37 shows a ten-node cubic triangular element (p-level of three)in x and y (see Pascal triangle also) and pqr values at the nodes of theelement. We generate Ni(L1), Ni(L2) and Ni(L3) functions using (8.155) to(8.157). In this case m = 3.

3

21

9

8

2 3 4

(210) (030)

(012)

(021)(201)

(102)

(003)7

6

5

1

10

(300) (120)

(111)

1, 2, . . . , 10 nodes for p-level ofthree

vertex nodes for L1, L2, and L3 lines

Figure 8.37: pqr values at the nodes of a ten-node cubic triangular element

We have

N0(L1) = 1

N1(L1) =1∏j=1

(3L1 − j + 1

j

)= 3L1


N2(L1) =

2∏j=1

(3L1 − j + 1

j

)=

3

2L1(3L1 − 1)

N3(L1) =3∏j=1

(3L1 − j + 1

j

)=L1

2(3L1 − 1)(3L1 − 2) (8.169)

Similarly,

N0(L2) = 1

N1(L2) = 3L2

N2(L2) =3

2L2(3L2 − 1)

N3(L2) =L2

2(3L2 − 1)(3L2 − 2)

(8.170)

N0(L3) = 1

N1(L3) = 3L3

N2(L3) =3

2L3(3L3 − 1)

N3(L3) =L3

2(3L3 − 1)(3L3 − 2)

(8.171)

The basis functions Ni(L1, L2, L3) (i = 1, 2, . . . , 10) for the ten-node cubictriangular element (see Fig. 8.37) can be generated using the following:

N1(L1, L2, L3) = N pqr(300)

(L1, L2, L3) = N3(L1)N0(L2)N0(L3)

=L1

2(3L1 − 1)(3L1 − 2)

N2(L1, L2, L3) = N pqr(210)

(L1, L2, L3) = N2(L1)N1(L2)N0(L3)

=9

2L1 L2(3L1 − 1)

N3(L1, L2, L3) = N pqr(120)

(L1, L2, L3) = N1(L1)N2(L2)N0(L3)

=9

2L1 L2(3L2 − 1)

N4(L1, L2, L3) = N pqr(030)

(L1, L2, L3) = N0(L1)N3(L2)N0(L3)

=L2

2(3L2 − 1)(3L2 − 2)


N5(L1, L2, L3) = N pqr(021)

(L1, L2, L3) = N0(L1)N2(L2)N1(L3)

=9

2L2 L3(3L2 − 1)

N6(L1, L2, L3) = N pqr(012)

(L1, L2, L3) = N1(L1)N2(L2)N0(L3)

=9

2L2 L3(3L3 − 1)

N7(L1, L2, L3) = N pqr(003)

(L1, L2, L3) = N0(L1)N0(L2)N3(L3)

=L3

2(3L3 − 1)(3L3 − 2)

N8(L1, L2, L3) = N pqr(102)

(L1, L2, L3) = N1(L1)N0(L2)N2(L3)

=9

2L1 L3(3L3 − 1)

N9(L1, L2, L3) = N pqr(201)

(L1, L2, L3) = N2(L1)N0(L2)N1(L3)

=9

2L1 L3(3L1 − 1)

N10(L1, L2, L3) = N pqr(111)

(L1, L2, L3) = N0(L1)N0(L2)N3(L3)

= 27L1 L2 L3 (8.172)

Using the procedure used for linear, quadratic and cubic triangular elements,higher degree C00 local approximation functions can be derived easily.

Remarks.

(1) As in case of quadrilateral family of 2D elements based on Lagrangefunctions derived using tensor product, these triangular element havesimilar short comings with increasing p-level the number of nodes perelement increase requiring a new discretization. Secondly the approxi-mation functions do not have hierarchical property.

(2) In applications distorted triangular elements are essential, hence we needto establish a mechanism to map the distorted triangular element intoa regular master element that has straight sides.

(3) Many other details of the derivatives of Ni(L1, L2, L3) with respect tox, y as needed in the integral forms and the details of integration overthe element area also need to be considered.

(4) The developments needed in (1)–(3) are facilitated if we consider Leg-endre polynomials instead of Lagrange polynomials in the derivations ofthe basis functions.


8.8 1D and 2D local approximations based onLegendre polynomials

8.8.1 Legendre polynomials

Legendre polynomials1 are given by Rodriguez formula [3]

Pn(ξ) =1

n!2ndn

dξn[(ξ2 − 1)n

]; ξ ∈ [−1, 1]; n = 0, 1, 2, . . . (8.173)

For example, we have

P0(ξ) = 1

P1(ξ) = ξ

P2(ξ) =3ξ2 − 1

2

P3(ξ) =5ξ3 − 3

2

P4(ξ) =35ξ4 − 30ξ2 + 3

8

(8.174)

These polynomials satisfy Legendre differential equation.((1− ξ2)P ′n(ξ)

)′+ n(n+ 1)Pn(ξ) = 0, ξ ∈ (−1, 1) (8.175)

The Legendre polynomials have the following orthogonal property

1∫−1

Pn(ξ)Pm(ξ) dξ =

2

2n+ 1, for m = n

0, for m 6= n(8.176)

The Legendre polynomials can also be represented by a recursive relation.

P0(ξ) = 1

P1(ξ) = ξ

Pi+1(ξ) =1

i+ 1

((2i+ 1)ξ Pi(ξ)− iPi−1(ξ)

), i = 1, 2, . . .

(8.177)

The Legendre polynomials can be used to define 1D p-version hierarchicalapproximation functions.

1Adrien-Marie Legendre (1752–1833) was a French mathematician, who made numer-ous contributions to mathematics, such as the Legendre polynomials and Legendretransformation. Legendre polynomials are solutions to Legendre’s differential equation:ddξ

[(1− ξ2) d

dξPn(ξ)

]+ n(n+ 1)Pn(ξ) = 0.

8.8. 1D AND 2D APPROXIMATIONS BASED ON LEGENDRE POLYNOMIALS 567

8.8.2 1D p-version C0 hierarchical approximation functions(Legendre polynomials)

Babuska and Szabo [4] have shown that for the three-node 1D configura-tion of Fig. 8.38 the C0 p-version hierarchical interpolation for a dependentvariable φ can be written as (for an element e)

φeh(ξ) = N1(ξ)φe1 +N2(ξ)φe2 +

pξ∑i=2

N i3(ξ) δei (8.178)

in which

N1(ξ) =1− ξ

2

N2(ξ) =1 + ξ

2

Ni(ξ) =1√

2(2i− 1)

(Pi(ξ)− Pi−2(ξ)

), i = 2, 3, . . .

(8.179)

φe1 and φe2 are function values at nodes 1 and 2 and δei ; i = 2, . . . are thenodal degrees of freedom at the hierarchical node 3 corresponding to p-levels of 2, 3, . . .. We can show that these approximation functions satisfythe desired properties. Based on the C00 p-version hierarchical derivation

presented earlier using Lagrange polynomials, we can designate δei = ∂iφ∂ξi

(i = 2, 3, . . .).

1 2

-1 1ξ

3

0

Figure 8.38: 1D three-node configuration

8.8.3 2D p-version C00 hierarchical interpolation functionsfor quadrilateral elements (Legendre polynomials)

If we consider a none-node element configuration in ξη space (in a twounit square), then using the 1D functions and nodal variable operators de-fined by (8.178) in ξ and η directions and by taking their tensor products,we can construct 2D p-version C00 hierarchical local approximations or in-terpolations for the nine-node element. This procedure is identical to whathas been used for Lagrange family of 2D C00 p-version hierarchical localapproximations based on tensor product. Details are straight forward.


8.8.4 2D Cij p-version interpolations functions for quadrilat-eral elements (Legendre polynomials)

The derivation of these interpolation functions and nodal dofs followsexactly same procedure as used for Cij 2D p-version interpolations based onLagrange family except that the corresponding C00 2D p-version hierarchicalinterpolation functions of Lagrange type are replaced by those based onLegendre type. The details are straight forward.

8.8.5 2D C00 p-version interpolation functions for triangularelements (Legendre polynomials)

We consider a seven-node (three vertex, three mid side and a centernode) distorted triangular element in xy space [see Fig. 8.39(a)]. The dis-torted element of Fig. 8.39 is mapped into ξη space in a two-unit equilateraltriangle [Fig. 8.39(b)]. The midside and the center nodes are hierarchicalnodes whereas the corner nodes are non-hierarchical nodes. The origin ofthe ξη coordinate system is located at node 2 of the equilateral triangle. Themapping of points can be defined using

xy

=

n∑i=1

Ni(L1, L2, L3)

xiyi

(8.180)

in which Ni(L1, L2, L3) are standard shape functions based on area coordi-nates derived using seven-node configuration of Fig. 8.39(b). We could alsouse six-node configuration with parabolic shape functions derived earlier.

(b)

2 3

(a)

1 31

1

46

5

7

2

1

46 7

5

A seven-node master ele-ment in ξη space

y

x

A seven-node distorted tri-angular element in xy space

ξ

η

Figure 8.39: A seven-node distorted triangular element in xy space and its map in themaster element in ξη


The area coordinates L1, L2, L3 can be related to the orthogonal naturalcoordinates ξ, η through the relations introduced by Szabo [4]

L1 =1

2

(1− ξ − η√

3

)L2 =

1

2

(1 + ξ − η√

3

)L3 =

η√3

(8.181)

Relations (8.181) can be used to convert the interpolation functions given in(8.180) from L1, L2, L3 to natural coordinates ξ, η.

xy

=

n∑i=1

Ni(ξ, η)

xiyi

(8.182)

We now present C00 p-version hierarchical interpolations over the elementof Fig. 8.39(b). Following Babuska and Szabo [4], we can consider seven-node triangular element in the ξη space (Fig. 8.40). The nodal degrees offreedom for pξ = pη = p in ξ and η are also shown in Fig. 8.40:

1. Interpolation functions for vertex nodes

N1 = L1

N3 = L2

N5 = L3

L1 + L2 + L3 = 1

(8.183)

2. Interpolation functions for mid-side nodes

(p− 1) mid-side interpolation functions are defined in terms of Legendrepolynomials (Pi) at each of the three mid-side nodes (2, 4 and 6), thusgiving a total of 3(p− 1) interpolation functions

N i2 = L1L2ψi(L2 − L1)

N i4 = L2L3ψi(L3 − L2)

N i6 = L1L3ψi(L1 − L3)

, i = 2, 3, . . . , p (8.184)

where ψi(·) is defined by

ψi(α) =

(Pi(α)− Pi−2(α)√

2(2i− 1)

)4

(1− α2), i = 2, 3, . . . , p (8.185)


1 3

7

5

4

2

6

ξφ

η

φ

φ

δjs1; j = 2, . . . , p

δjs3; j = 2, . . . , p

δin; i = 1, 2, . . . , (p−1)(p−2)2

δjs2; j = 2, . . . , p

Figure 8.40: Degrees of freedom for C00 p-version hierarchical triangular element

in which α assumes a value of (L2−L1), (L3−L2), (L1−L3) for mid-sideinterpolation functions corresponding to nodes 2, 4 and 6 respectively.First few terms of ψi(α) are

ψ2(α) = −√

6

ψ3(α) = −√

10α

ψ4(α) = −√

7

8(5α2 − 1) and so on

(8.186)

3. Internal interpolation function

From the Pascal triangle the total number of interpolation functions cor-responding to p-level of p are 1

2(p+ 1)(p+ 2) for completeness. From thesum of the interpolation functions for the vertex and the mid-side nodes,we require 1

2(p−1)(p−2) additional interpolation functions for complete-ness. These are defined at the internal node (node 7). these are non-zeroonly in the interior of the triangular domain and vanish on all three sidesof the triangular element. The internal approximation functions associ-ated with node 7 (center node) of the master element for p ≥ 3 are asfollows:

N j7 (L1, L2, L3) = L1L2L3Pp−i−2(L2 − L1)Pi−1(2L3 − 1)

where i = 1, 2, . . . , p− 2

j = 1, 2, . . . ,1

2(p− 1)(p− 2)

(8.187)

For example, for a p-level of 3, we only have one internal interpolation


function.N1

7 (L1, L2, L3) = L1L2L3 (8.188)

For p-level of 4, we have the following three internal interpolation func-tions.

N17 (L1, L2, L3) = L1L2L3

N27 (L1, L2, L3) = L1L2L3P1(L2 − L1)

N37 (L1, L2, L3) = L1L2L3P1(2L3 − 1)

(8.189)

For p-level of 9 the triangular C00 p-version element contains 55 interpo-lation functions: 3 at vertex nodes, 24 at mid-side nodes and 28 internal.We transform the C0 p-version interpolation functions in L1, L2, L3 to ξηspace using (8.181).

Remarks.

1. The integration of the coefficients of the element matrix and vectors en-countered in finite element processes needs to be considered (see AppendixA).

2. The degrees of freedom at the mid-side nodes in fact are the tangentialderivatives of various orders.

8.8.6 2D Cij interpolation functions for triangular elements(Legendre polynomials)

In deriving higher order continuity interpolations in xy space for trian-gular elements, we use C0 p-version interpolations based on Legendre familyusing area coordinates as starting point. We follow some guidelines.

Selection of the derivative degrees of freedom at the vertices of a 2DHGDA triangular element is dictated by the transformation rules for thederivatives of various orders between xy and ξη spaces. The following choicesof nodal operators (or dofs) at the vertices listed in Table 2.1 for C11, C22 andC33 HGDA satisfy the requirements. We note that for C11 HGDA element,the derivative operators at the vertex nodes are a complete set of first orderoperators. For C22 HGDA, the set of C11 is augmented by a complete secondorder set of derivatives and so on. This selection of degrees of freedomis consistent with the framework developed for 2D distorted quadrilateralelements presented in a previous section and reference [2].

Since C00 p-version hierarchical approximations are used as a startingpoint, which have only function value as a degree of freedom at the vertexnodes, we must establish some rules that allow us to borrow some dofs fromC00 p-version hierarchical approximations to generate the desired dofs at thevertices of the 2D HGDA for the distorted triangular element.


The details presented in for quadrilateral elements for transforming deriva-tives of various orders between ξη and xy spaces also holds for 2D distortedtriangular elements between ξη and xy spaces and hence not repeated. Theonly difference being that we need to use the geometry mapping equations(8.180) in their construction.

Table 8.8: Choices of dofs at the corner nodes for Cij 2D distorted triangular elementsin xy space

Type of HGDA Nodal Operators at the corner nodes

C11 1, ∂∂x ,∂∂y

C22 1, ∂∂x ,∂∂y ,

∂2

∂x2, ∂2

∂y∂x ,∂2

∂y2

C33 1, ∂∂x ,∂∂y ,

∂2

∂x2, ∂2

∂y∂x ,∂2

∂y2, ∂3

∂x3, ∂3

∂y∂x2, ∂3

∂y2∂x, ∂

3

∂y3

C11 HGDA for 2D distorted triangular elements in xy space

In this section we present specific details for C11 HGDA element. Fig-ure 8.41 (a) shows the dofs at the vertices of C11 HGDA element (subscriptindicates differentiation). Comparing Fig. 8.41 (a) with C00 p-version el-ement of Fig. 8.40, we note that the C11 element requires φx and φy asadditional dofs at each of the three vertices i.e. a total of six dofs for thethree vertices. We borrow six dofs and the corresponding C00 p-versionapproximation functions to generate the desired derivative dofs and the cor-responding approximation functions for the 2D C11 HGDA element. Thiswould obviously result in reduction of dofs at the hierarchical nodes of theC0 p-version element. In doing so we must follow a systematic procedure.

For all HGDA element, the choice of dofs from C00 element is ratherstraightforward. We borrow dofs corresponding to p-levels of 2 and 3 frommid-side nodes 2, 4 and 6. These dofs must be eliminated from C00 p-version approximations to generate the derivative dofs at the vertices of 2DC11 HGDA element as shown in Fig. 8.41 (a). Figure 8.41 (b) shows thedofs at the hierarchical nodes of the 2D HGDA element.

The first degree of freedom at the mid-side nodes of the C11 HGDAelement corresponds to p-level of 4. For C11 HGDA element, we do not needto borrow any dofs from the internal node of C00 p-version element.

C22 HGDA for 2D distorted triangular elements in xy space

Here we consider 2D C22 HGDA for distorted triangular elements inxy space. Figure 8.42 (a) shows dofs at the corner nodes of the element.Comparing this with C00 p-version element of Fig. 8.40, we note that φx,


6

5

1 2 3

47 6

5

1 3

4

2

7

(a) Nodal dofs at the corner nodes of a2D C11 HGDA element

i = 1, 2, . . . , (p−1)(p−2)2

j = 4, . . . , p

(b) Nodal dofs at the hierarchical nodesof a 2D C11 HGDA element

ξ

η

φ, φx, φyφ, φx, φy

φ, φx, φy

ξ

η

δjs1

δjs2 δjs3δji

Figure 8.41: Nodal dofs for C11 2D distorted triangular element

φy, φx2 , φxy and φy2 are additional dofs at each of the three vertex nodes, atotal of fifteen. Hence, we need to borrow fifteen dofs from the hierarchicalnodes of C00 element shown in Fig. 8.40, keeping in mind that the remainingdofs at the mid-side nodes of C00 element must begin with p-level of six. Thisis due to the fact that a quintic polynomial describes a C22 approximationin 1D. This allows us to borrow four dofs (corresponding to p-levels of 2,3, 4 and 5) from each of the mid-side nodes of Fig. 8.40, making a total oftwelve. The remaining three dofs needed to generate the dofs at the cornernodes of C22 HGDA element must come from the internal node.

The degrees of freedom are borrowed from internal node of the C00 p-version element in such a way that the dofs corresponding to a lower p-levelare selected before those corresponding to higher p-levels. Figure 8.42 (b)shows the dofs at the hierarchical nodes of the 2D HGDA element.

With the discussion of the concepts relating to the selection of the dofs forC11 and C22 HGDA, we now present a derivation of the Cij approximationsfor distorted triangular elements.

Cij HGDA for 2D distorted triangular elements in xy space

We propose a new methodology (parallel to that used for 2D quadrilat-eral elements in the previous section and reference [2]) which utilizes C00

p-version hierarchical interpolation functions as a starting point and gener-ates desired order global differentiability approximations for distorted trian-gular elements. Since the approximation functions are functions of naturalcoordinates ξ, η, i.e. Ni = Ni(ξ, η), the desired derivative degrees of freedomcan be easily generated first in ξη space and then transformed into xy space.


6

5

1 2 3

47 6

5

1 3

4

2

7

(a) Nodal dofs at the corner nodes of a2D C22 HGDA element

i = 3, 4, . . . , (p−1)(p−2)2

j = 6, . . . , p

(b) Nodal dofs at the hierarchical nodesof a 2D C22 HGDA element

φ, φx, φyφx2 , φxy, φy2



ξ

η

ξ

η

δjs1

δjs3δji

δjs2

Figure 8.42: Nodal dofs for C22 2D distorted triangular element

The transformation matrices similar to those presented for quadrilateral el-ements assist us in transforming the desired derivative degrees of freedomfrom ξη to xy space.

Using the C00 p-version hierarchical approximations for a seven nodeelement (see Fig. 8.40), the field variable φ can be approximated as,

φeh(ξ, η) = [N(ξ, η)]δe (8.190)

in which [N(ξ, η)] is a row matrix of C00 p-version hierarchical local ap-proximations and δe are the corresponding nodal dofs (arranged in somesuitable fashion). The dofs in δe of (8.190) are grouped into those corre-sponding to vertex or corner nodes (denoted by co), mid-side nodes (denotedby m) and internal node (denoted by i) as follows:

φeh(ξ, η) = [a]δecor1 + [b]δemiel + [c]δemr2 + [d]δei r3 (8.191)

where a subscript r1 denotes the degrees of freedom retained from cornernodes. Subscript el corresponds to the degrees of freedom borrowed fromthe mid-side nodes and the internal node that are to be eliminated to derivethe new derivative degrees of freedom at the corner nodes of a Cij HGDAelement. Subscripts r2 and r3 denote the degrees of freedom remaining atthe mid-side nodes and internal node (after borrowing the required degreesof freedom), and [a], [b], [c] and [d] are vectors containing C00 p-versionlocal approximations corresponding to the dofs in the r1, el, r2 and r3 setsrespectively.

For a C11 HGDA element, δemiel consists of dofs from mid-side nodesonly, since we do not need any from the internal node, δemiel = δemel,


which would contain the following dofs:

δeme =δ2|2, δ3|2, δ2|4, δ3|4, δ2|6, δ3|6

T(8.192)

For classes higher than C11, δemiel will consist of dofs from mid-side nodesas well as internal node.

Let the desired new derivative dofs at the corner nodes of a Cij HGDAelement be denoted by δexyn . In case of a C11 HGDA element, these dofs inξη space will consist of complete set of first order derivatives of the dependentvariable with respect to ξ and η evaluated at the three corner nodes,

δeξηn =φξ|1, φη|1, φξ|3, φη|3, φξ|5, φη|5

T(8.193)

where subscript denotes differentiation i.e. φξ|1 = ∂φ∂ξ |node 1 = ∂φ

∂ξ |ξ=−1,η=0.

For classes higher than C11, the new derivative dofs constituting completesets are added at the corner nodes. Differentiating (8.191) with respect toξ and η and evaluating the resulting expression at each of the three cornernodes, we get

δeξηn = [A]δecor1 + [B]δemiel + [C]δemr2 + [D]δei r3 (8.194)

Solving for degrees of freedom to be eliminated i.e. δemiel in (8.194), weget

δemiel = [B]−1δeξηn − [B]−1[A]δecor1 − [B]−1[C]δemr2 − [B]−1[D]δei r3(8.195)

Substituting Jacobian of transformation [Ji] into the above equation, wecan transform the new derivative dofs from ξη space to xy space. Equation(8.195) can thus be written as

δemiel = [B]−1[Ji]δexyn −[B]−1[A]δecor1−[B]−1[C]δemr2−[B]−1[D]δei r3(8.196)

Now, substituting δemiel from (8.196) into (8.191)

φeh(ξ, η) = [a]δecor1 + [b]([B]−1[Ji]δexyn − [B]−1[A]δecor1

− [B]−1[C]δemr2 − [B]−1[D]δei r3)

+ [c]δemr2 + [d]δei r3 (8.197)

Collecting terms in the (8.197), we get the final form of the Cij HGDA localapproximations as follows:

φeh(ξ, η) =([a]− [b][B]−1[A]

)δecor1 + [b][B]−1[Ji]δexyn

+([c]− [b][B]−1[C]

)δemr2 +

([d]− [b][B]−1[D]

)δei r3 (8.198)


Limitations of 2D C11 global differentiability local approximationsfor distorted triangular elements

In the proposed framework, 2D Cij global differentiability local approx-imations are derived by borrowing appropriate degrees of freedom and thecorresponding approximation functions from the hierarchical nodes of C00

element. In (8.198), [a], [c] and [d] contain C00 local approximations whichare retained at corner, mid-side and internal nodes whereas [b] contains C00

local approximations functions which are borrowed from mid-side and inter-nal nodes. [A], [B], [C] and [D] are matrices containing derivatives of C00

approximations with respect to ξ and η collected in [a], [b], [c] and [d] evalu-ated at the corner nodes. The approximation functions for the C11 distortedelement at the corner, mid-side and internal nodes (which are retained) areobtained by modifying the corresponding functions for the C00 element by[b][B]−1[A], [b][B]−1[C] and [b][B]−1[D] respectively.

In case of 2D C11 HGDA element, the new derivative degrees of free-dom introduced at the corner nodes are first order derivatives with respectto x and y. The nature of the C00 local approximation functions and thecoordinates of the corner nodes (ξ and η coordinates are either +1, 0,

√3

or −1) always result in all the coefficients of [D] matrix to be zero regard-less of the p-level. The coefficients of matrices [A], [B], [C] however are notall zero. This results in the approximation functions at the internal nodeof C11 distorted element to be exactly same as those corresponding to C00

element (since [b][B]−1[D] is a row matrix containing all zeros). As a conse-quence, it appears that we have an incomplete C11 HGDA distorted element,which may result in inaccurate behaviors for coarser discretizations. Whenwe derive approximation functions for C22 and higher order elements, thederivative degree of freedoms introduced at the corner nodes include mixedderivatives with respect to x and y. The mixed derivatives of the C00 ap-proximation functions in [d] (internal node) evaluated at the corner nodes arenot all zero and hence coefficients in [D] matrix are not all zero. Alternateways of deriving C11 local approximations is under investigation.

Remarks.

(1) The derivation presented here is general. Hence, C00 p-version hierar-chical local approximation can be Legendre, Lagrange or Chebyshev.The choice of these functions changes the matrices [a], [b], [c] and [d]accordingly, but the details of the development remains unaffected.

(2) The matrices [A], [B], [C] and [D] contain derivatives of C00 p-versionhierarchical approximations with respect to ξ and η evaluated at thevertex nodes. These can be precomputed once and stored and hence canbe reused. This results in substantial computational efficiency.

8.9. 1D AND 2D INTERPOLATIONS BASED ON CHEBYSHEV POLYNOMIALS 577

(3) The C00 p-version functions borrowed from the mid-side nodes and theinternal node must be so chosen that (i) lowest degree admissible func-tions (corresponding to lower p-levels) are selected first before those cor-responding to the higher p-levels (ii) the symmetric pattern is maintainedin their choice so that symmetry of computed solutions for symmetricdiscretization is preserved.

(4) Mapping of the distorted triangular element in ξη coordinate space andthe transformation of the C00 approximation functions from L1, L2, L3

area coordinates to ξη is essential. The derivative degrees of freedomgenerated at the vertex nodes in ξη space can be easily transformed intothe corresponding derivative degrees of freedom in the xy space neededfor Cij HGDA elements.

8.9 1D and 2D interpolations based on Chebyshevpolynomials

8.9.1 Chebyshev polynomials

Chebyshev polynomials satisfy the Chebyshev differential equation [3]The polynomials Ti; i = 0, 1, 2, . . . are given by

T0 = 1

T1 = ξ

Ti+1 = 2ξ Ti − Ti−1, i = 1, 2, . . .

(8.199)

8.9.2 1D C0 p-version hierarchical interpolations based onChebyshev polynomials

For nodes 1 and 2 (non-hierarchical) we have the standard functions

N1(ξ) =1− ξ

2

N2(ξ) =1 + ξ

2

(8.200)

at node 3, we introduce the interpolation functions N i3(ξ) defined by

N i3(ξ) =

Ti − 1 ; if i is even

Ti − ξ ; if i is oddfor i = 2, 3, . . . , pξ (8.201)

If φ is the dependent variable, then approximation of φ(ξ), φeh(ξ), is givenby

φeh(ξ) = N1φe1 +N2φ

e2 +

pξ∑i=2

N i3 δ

ei (8.202)


We can choose δei = ∂iφ∂ξi

∣∣∣ξ=0

; i = 2, 3, . . . , pξ. As an example if pξ = 2 then

we haveN2

3 = 2ξ2 − 2 (8.203)

If pξ = 3, then we have

N23 = 2ξ2 − 2

N33 = 4ξ3 − 4ξ etc.

(8.204)

We note that the functions in (8.203) and (8.204) are indeed zero at ξ =±1. Thus (8.202) with (8.200) and (8.201) is the desired 1D C0 p-versionhierarchical interpolation based on Chebyshev polynomials.

1

0

3

1

2

−1ξ

Figure 8.43: 1D three-node p-version element in natural coordinate space

8.9.3 2D p-version C00 hierarchical interpolation functions forquadrilateral elements (Chebyshev polynomials)

If we consider nine-node element configuration in ξη space (in a two-unitsquare), then by using the 1D functions and nodal variable operators definedby (8.202) in ξ and η directions and by taking their tensor products we canconstruct 2D p-version C00 hierarchical local approximation (or interpola-tions) for the nine-node element. The procedure is identical to what has beenused for Lagrange family 2D C00 p-version hierarchical local approximationsbased on tensor product. Details are straight forward.

8.9.4 2D Cij p-version interpolation functions for quadrilateralelements (Chebyshev polynomials)

The derivation of these interpolation functions and nodal dofs follows ex-actly the same procedure as used for Cij 2D p-version interpolations basedon Lagrange family except that the corresponding C00 2D p-version hierar-chical interpolation functions of Lagrange type are replaced by those basedon Chebyshev type. The details are straight forward.

8.10 Serendipity family of C00 interpolations oversquare subdomains

“Serendipity” means discovery by chance. Thus, this family of elementshas very little theoretical or mathematical basis other than the fact that in

8.10. SERENDIPITY FAMILY OF C00 INTERPOLATIONS 579

generating approximation functions for these elements we only utilize thetwo fundamental properties of the approximation functions,

Ni(ξj , ηj) =

1, j = i

0, j 6= i(i = 1, . . . ,m) (8.205)

andm∑i=1

Ni(ξ, η) = 1 (8.206)

(a) The main motivation in generating these basis functions is to possiblyeliminate some or many of the internal nodes that appear in generating theinterpolations using tensor product for family of higher degree interpolationfunctions.

(b) For example, in the case of a bi-quadratic local approximation requiringa nine-node element, the corresponding serendipity element will contain eightboundary nodes, as shown in Fig. 8.44.

1 2 3

567

84

1 2 3

567

8 94

η

ξ

η

ξ

Nine-node Lagrangebi-quadratic element

Eight-node serendipityelement

Figure 8.44: Nine-node Lagrange and eight-node serendipity elements

(c) In the case of a bi-cubic element requiring 16-nodes with four internalnodes, the corresponding serendipity element will contain 12 boundary nodes(see Fig. 8.45)

(d) While in the case of bi-quadratic and bi-cubic local approximations itwas possible to eliminate the internal nodes and thus serendipity elementswere possible. This may not always be possible for higher degree local ap-proximations than three.

8.10.1 Method of deriving serendipity interpolation functions

We use the two basic properties that the approximation functions mustsatisfy (stated by (8.205) and (8.206)). Let us consider a four-node bilinear


η

ξ

η

ξ

16-node bi-cubicelement

12-node cubic serendipityelement

Figure 8.45: Sixteen-node Lagrange and twelve-node serendipity elements

element. In this case, obviously, non-serendipity and serendipity approxima-tions are identical. Nonetheless, we derive the approximation functions forthis element using the approach used for serendipity basis functions.

21

ξ

34

η 1− η = 0

1− ξ = 0

1 + ξ = 0

1 + η = 0

Figure 8.46: Derivation of 2D bilinear serendipity element

(a) First, we note that the four sides of the elements are described by theequations of the straight lines as shown in the figure. Consider node 1.N1(ξ, η) is one at node 1 and zero at nodes 2,3 and 4. Hence, equations ofthe straight lines connecting nodes 2 and 3 and nodes 3 and 4 can be usedto derive N1(ξ, η). That is,

N1(ξ, η) = c1(1− ξ)(1− η) (8.207)

in which c1 is a constant. But N1(−1,−1) = 1, hence using (8.207) we get

N1(−1,−1) = 1 = c1(1− (−1))(1− (−1)) ⇒ c1 =1

4(8.208)

Thus, we have

N1(ξ, η) =1

4(1− ξ)(1− η) (8.209)


which is the correct approximation function for node 1 of the bilinear ele-ment. Similarly, for node 2, 3 and 4 we can write

N2(ξ, η) = c2(1 + ξ)(1− η)

N3(ξ, η) = c3(1 + ξ)(1 + η)

N4(ξ, η) = c4(1− ξ)(1 + η)

(8.210)

But

N2(1,−1) = 1 ⇒ c2 =1

4

N3(1, 1) = 1 ⇒ c3 =1

4

N4(−1, 1) = 1 ⇒ c4 =1

4

(8.211)

Thus, from (8.210) and (8.211) we obtain

N2(ξ, η) =1

4(1 + ξ)(1− η)

N3(ξ, η) =1

4(1 + ξ)(1 + η)

N4(ξ, η) =1

4(1− ξ)(1 + η)

(8.212)

(8.209) and (8.212) are the correct approximation functions for the four-nodebilinear element.

(b) In the above derivations we have only utilized the property (8.205), hencewe must show that the interpolation functions in (8.209) and (8.212) satisfy(8.206). In this case, obviously they do. However, this may not always bethe case.

Eight-node serendipity element:

Consider node 1 first. We have N1(ξ, η)|(−1,−1) = 1 and zero at all theremaining nodes. Hence, for node 1 we can write

N1(ξ, η) = c1(1− ξ)(1− η)(1 + ξ + η) (8.213)

Since

N1(ξ, η)|(−1,−1) = 1 ⇒ c1 = −1

4(8.214)

we obtain

N1(ξ, η) = −1

4(1− ξ)(1− η)(1 + ξ + η) (8.215)


ξ

η 1− η = 0

1− ξ = 0

1 + ξ = 0

1 + η = 0

4

567

1 + ξ + η = 08

31 2

Figure 8.47: Derivation of 2D bilinear serendipity element: node 1

1− η = 0

1− ξ + η = 03

5 7

1 + η = 0 1 + η = 0

1 + ξ − η = 01− ξ − η = 0

1 + ξ = 01 + ξ = 0 1− ξ = 0

for node 3 for node 5 for node 7

Figure 8.48: Derivation of 2D bi-quadratic serendipity element: nodes 3, 5, and 7

For nodes 3, 5 and 7 one may use the equations of the lines indicated inFig. 8.48 and the conditions similar to (8.214) for N2, N3 and N4.

For the mid side nodes, the product of the equations of straight lines notcontaining the mid side nodes provide the needed expressions and we have

N1 =1

4(1− ξ)(1− η)(−1− ξ − η)

N2 =1

2(1− ξ2)(1− η)

N3 =1

4(1 + ξ)(1− η)(−1 + ξ − η)

N8 =1

4(1− ξ)(1− η2)

N4 =1

2(1ξ)(1− η2)

N7 =1

4(1− ξ)(1 + η)(−1− ξ + η)

N6 =1

2(1− ξ2)(1 + η)

N5 =1

4(1 + ξ)(1 + η)(−1 + ξ + η)


In this case also we must show that8∑i=1

Ni(ξ, η) = 1, which holds.

Twelve-node serendipity elements:

Using procedures similar to the four-node bilinear and eight-node bi-quadratic element (see Fig. 8.49) we can also derive the interpolation func-tions for the twelve-node serendipity element.

η

ξ

431 2

7

9

8

6

10

5

11 12

Figure 8.49: Derivation of 2D bi-cubic serendipity element

N1 =1

32(1− ξ)(1− η)[−10 + 9(ξ2 + η2)]

N2 =9

32(1− ξ2)(1− η)(1− 3ξ)

N3 =9

32(1− ξ2)(1− η)(1− 3ξ)

N4 =1

32(1 + ξ)(1− η)[−10 + 9(ξ2 + η2)]

N5 =9

32(1− ξ)(1− η2)(1− 3η)

N6 =9

32(1 + ξ)(1− η2)(1− 3η)

N7 =9

32(1− ξ)(1− η2)(1 + 3η)

N8 =1

32(1 + ξ)(1− η2)(1 + 3η)

N9 =1

32(1− ξ)(1 + η)[−10 + 9(ξ2 + η2)]

N10 =9

32(1− ξ2)(1 + η)(1− 3ξ)

N11 =9

32(1− ξ2)(1 + η)(1 + 3ξ)

N12 =1

32(1 + ξ)(1 + η)[−10 + 9(ξ2 + η2)]


Remarks.

(1) Serendipity interpolations are obviously incomplete polynomials in ξ andη, hence have poorer local approximation compared to the local approx-imations based on Pascal rectangle.

(2) There is no particular theoretical basis for deriving them.

(3) In view of p-version hierarchical elements presented in the previous sec-tions, serendipity elements are precluded and are of no practical signifi-cance.

8.11 Interpolation functions for 3D elements

For BVPs in R3 the domain of definition of the BVP is a subspace ofR3, i.e. a volume. Discretizations of such domain naturally leads to 3Dsubdomains, i.e. 3D elements. In this section we consider hexahedron andtetrahedral families of subdomains (i.e. elements).

8.11.1 Hexahedron elements

Figure 8.50 shows hexahedron elements with distorted faces and edgesand their maps in the natural coordinate space ξ, η, ζ in a two unit cube withthe origin of the coordinate system located at the center of the element.

8.11.1.1 Mapping of points

In the abstract sense, the mapping of points between ξ, η, ζ and x, y, z-spaces is defined by

x = x(ξ, η, ζ)

y = y(ξ, η, ζ)

z = z(ξ, η, ζ)

(8.216)

The inverse of the mapping is given by

ξ = ξ(x, y, z)

η = η(x, y, z)

ζ = ζ(x, y, z)

(8.217)

As in the case of 2D elements, (8.216) is preferable over (8.217). The explicitform of the mapping defined by (8.216) can be established. Let Ni(ξ, η, ζ)be basis functions in the natural coordinate space such that ∀ξ, η, ζ ∈ Ωξηζ

or Ωm we have


1, j = i

0, j 6= i(8.218)

8.11. INTERPOLATION FUNCTIONS FOR 3D ELEMENTS 585

27-node element in x, y, z space 27-node element in ξ, η, ζ space

ζ

ξ

η

x

y

z

z

x

y

Eight-node element in x, y, z space Eight-node element in ξ, η, ζ space

ζ

ξ

η

Figure 8.50: Hexahedral 3D elements

n∑i=1

Ni(ξ, η, ζ) = 1 (8.219)

in which Ωm is the map of Ωe in ξ, η, ζ space, i.e. mapping (8.216) is suchthat Ωm → Ωe or by (8.217), Ωe → Ωm. Using Ni(ξ, η, ζ) we can write,

x = x(ξ, η, ζ) =

n∑i=1

Ni(ξ, η, ζ)xi

y = y(ξ, η, ζ) =

n∑i=1

Ni(ξ, η, ζ) yi

z = z(ξ, η, ζ) =

n∑i=1

Ni(ξ, η, ζ) zi

(8.220)

in which (xi, yi, zi) are coordinates of node i in x, y, z space. Equation (8.220)map a point (ξ∗, η∗, ζ∗) into (x∗, y∗, z∗).


8.11.1.2 Mapping of lengths

In this section we establish a relationship between length dx, dy, dz inx, y, z space and dξ, dη, dζ in ξ, η, ζ space. Based on (8.216), we can write

dx =∂x

∂ξdξ +

∂x

∂ηdη +

∂x

∂ζdζ

dy =∂y

∂ξdξ +

∂y

∂ηdη +

∂y

∂ζdζ

dz =∂z

∂ξdξ +

∂z

∂ηdη +

∂z

∂ζdζ

(8.221)

or dxdydz

[J ]

dξdηdζ

(8.222)

where

[J ] =

∂x∂ξ

∂x∂η

∂x∂ζ

∂y∂ξ

∂y∂η

∂y∂ζ

∂z∂ξ

∂z∂η

∂z∂ζ

(8.223)

is called the Jacobian of transformation. For the mapping to be unique, i.e.one-to-one and onto,

det [J ] > 0 ∀(ξ, η, ζ) ∈ Ωm (8.224)

must hold. Equations (8.222) is the desired relationship for mapping oflengths. The elements of [J ] can easily be obtained using (8.220) in whichNi(ξ, η, ζ) and (xi, yi, zi) are known.

8.11.1.3 Mapping of volumes

In this section we derive a relationship that relates elemental volumedx dy dz in x, y, z space to the elemental volume dξ dη dζ in ξ η ζ space. Let~i,~j,~k be the unit vectors in x, y, z space and let ~eξ, ~eη, ~eζ be unit vectors inξ, η, ζ space, then we can write

dx~i =∂x

∂ξdξ ~eξ +

∂x

∂ηdη ~eη +

∂x

∂ζdζ ~eζ

dy~j =∂y

∂ξdξ ~eξ +

∂y

∂ηdη ~eη +

∂y

∂ζdζ ~eζ

dz ~k =∂z

∂ξdξ ~eξ +

∂z

∂ηdη ~eη +

∂z

∂ζdζ ~eζ

(8.225)

We note that

dx~i · (dy~j × dz ~k) = dx~i · dy dz~i = dx dy dz (8.226)


Substituting for dx~i, dy~j, dz ~k from (8.225) into (8.226) and using propertiesof the dot product and cross products of the unit vectors in x, y, z and ξ, η, ζspaces, we obtain

dx dy dz = det[J ] dξ dη dζ (8.227)

Based on (8.227), det[J ] > 0 must hold otherwise the volume in xyz spacecould be negative or zero if det[J ] = 0.

8.11.1.4 Obtaining derivatives of φeh(ξ, η, ζ) withrespect to x, y, z

With the local approximation φeh(ξ, η, ζ) of φ defined by (8.232), and as-suming thatNi(ξ, η, ζ) are known functions, obtaining derivatives of φeh(ξ, η, ζ)with respect to x, y, z needed in the integral form is not directly possible dueto the fact that Ni(·) are functions of ξ, η and ζ. First, we note that whenφeh(ξ, η, ζ) is defined by (8.232) we have

∂φeh∂x

=n∑i=1

∂Ni

∂xδei

∂φeh∂y

=n∑i=1

∂Ni

∂yδei

∂φeh∂z

=n∑i=1

∂Ni

∂zδei

(8.228)

Thus, obtaining∂φeh∂x ,

∂φeh∂y and

∂φeh∂z implies establishing ∂Ni

∂x , ∂Ni∂y and ∂Ni∂z ; i =

1, . . . , n. Since Ni = Ni(ξ, η, ζ) and x = x(ξ, η, ζ), y = y(ξ, η, ζ) and z =z(ξ, η, ζ), we proceed as follows.

∂Ni

∂ξ=∂Ni

∂x

∂x

∂ξ+∂Ni

∂y

∂y

∂ξ+∂Ni

∂z

∂z

∂ξ

∂Ni

∂η=∂Ni

∂x

∂x

∂η+∂Ni

∂y

∂y

∂η+∂Ni

∂z

∂z

∂η

∂Ni

∂ζ=∂Ni

∂x

∂x

∂ζ+∂Ni

∂y

∂y

∂ζ+∂Ni

∂z

∂z

∂ζ

, i = 1, . . . , n (8.229)

or ∂Ni∂ξ∂Ni∂η∂Ni∂ζ

= [J ]T

∂Ni∂x∂Ni∂y∂Ni∂z

, i = 1, . . . , n (8.230)

Therefore ∂Ni∂x∂Ni∂y∂Ni∂z

=[[J ]T

]−1

∂Ni∂ξ∂Ni∂η∂Ni∂ζ

, i = 1, . . . , n (8.231)


Using (8.231) derivatives of Ni(ξ, η, ζ) with respect to ξ, η and ζ can betransformed into the derivatives of Ni(ξ, η, ζ) with respect to x, y, z and,hence, the derivatives of φeh(ξ, η, ζ) with respect to x, y, z in (8.228) aredefined.

8.11.2 Local approximation for a dependent variableφ over Ωm

8.11.2.1 Hexahedron elements

Let φeh(ξ, η, ζ) be the local approximation of φ over Ωm. Then symboli-cally we can write

φeh(ξ, η, ζ) =n∑i=1

Ni(ξ, η, ζ) δei = [N ]δe (8.232)

where Ni(ξ, η, ζ) are the local approximation functions or interpolation func-tions corresponding to the nodes of the element and δei are the nodal degreesof freedom. Functions Ni(ξ, η, ζ) are also referred to as basis functions. Ex-plicit forms of Ni(ξ, η, ζ) depend upon many considerations (similar to thesefor 2D elements).

(1) The first important issue is the choice of nodal configuration for theelement. That is, how many nodes and their locations.

(2) Means of constructing Ni(ξ, η, ζ) for

(a) C000 local approximations of higher degree

(b) C000 p-version hierarchical local approximations

(c) Cijk, i, j, k ≥ 1 local approximations for prism family

(d) Cijk, i, j, k ≥ 1 local approximations for distorted geometries in R3

(3) In the developments in (2), the choices of δei , nodal dofs are crucial andrequire careful considerations.

(4) The developments in (2) and (3) could be based on (i) Lagrange interpo-lation functions (ii) Legendre polynomials or (iii) Chebyshev polynomi-als. The specific details presented in the following are based on Lagrangepolynomials. The extensions to the other two types are straight forward.We consider details in the following

C000(Ωe) polynomial approximations: linear approximation

We consider local approximations of type C000(Ωe), i.e. across the inter-element boundaries φ is continuous but the derivative of φ normal to theinter-element boundaries may be discontinuous in the physical domain. For


C000(Ωe), local approximation function values at the nodes suffice as degreesof freedom. Consider a two-unit cube in ξ, η, ζ coordinate space. For thisgeometry the least possible nodal configuration is eight nodes located at thecorner points. This element in the physical space x, y, z could have distortedshape with flat faces and straight edges (see Fig. 8.51).

x

y

z

Eight-node element in R3 Map of an eight-node elementin ξ, η, ζ space

ζ

φe3

ξ

η

φe1

φe5

φe8 φe7

φe2

1 2

5 6

8

φe4

7

3

φe6

Figure 8.51: Eight-node trilinear element in 3D

In the polynomial approach, we consider φ(ξ, η, ζ) as a linear combinationof monomials in ξ, η ζ. Noting that we have eight function values at theelement nodes as dofs, φ(ξ, η, ζ) can be a polynomial in ξ, η and ζ containingup to eight constants,

φ(ξ, η, ζ) = c1 + c2 ξ + c3 η + c4 ζ + c5 ξη + c6 ηζ + c7 ξζ + c8 ξηζ (8.233)

Equation (8.233) is a complete linear polynomial in ξ, η and ζ. The constantsc1, . . . , c8 in (8.233) are evaluated using

φei = φ(ξi, ηi, ζi), i = 1, . . . , 8 (8.234)

in which (ξi, ηi, ζi) are coordinates of the eight nodes in ξ, η, ζ space. first,we rewrite (8.233) as

φ(ξ, η, ζ) =[1 ξ η ζ ξη ηζ ξζ ξηζ

]c1

c2...c8

(8.235)

Substituting from (8.234) into (8.233) and solving for ci; i = 1, . . . , 8 in terms


of φei (i = 1, . . . , 8)c1

c2...c8

=

1 ξ1 η1 ζ1 . . . ξ1η1ζ1

1 ξ2 η2 ζ2 . . . ξ2η2ζ2...

......

.... . .

...1 ξ8 η8 ζ8 . . . ξ8η8ζ8

−1

φe1φe2...φe8

= [C]φe (8.236)

Substituting from (8.236) into (8.235),

φ(ξ, η, ζ) =[1 ξ η ζ . . . ξηζ

][C]φe (8.237)

or

φ(ξ, η, ζ) =8∑i=1

Ni(ξ, η, ζ)φei (8.238)

in which


1, j = i

0, j 6= i(8.239)

and8∑i=1

Ni(ξ, η, ζ) = 1 (8.240)

The approximation φeh(ξ, η, ζ) in (8.238) is linear in ξ, η and ζ and isof class C000(Ωe) and is based on linear Lagrange interpolation functions(Ni(ξ, η, ζ)), p-level of one in ξ, η and ζ.

8.11.2.2 Higher degree approximations of φ over Ωm

Based on the derivation of linear approximation presented in the previoussection, it is clear that if we wish φeh(ξ, η, ζ) to be a quadratic approximationin ξ, η and ζ then in (8.233) we need additional monomials in ξ, η and ζand we also need to decide on the locations of the additional nodes (as manyas the number of additional terms included in (8.233)). For progressivelyincreasing degree (p-level) of approximation in ξ, η and ζ, the decision onthe choices of additional monomials and locations of the additional nodescan be facilitated by considering Pascal’s prism. The family of interpolationsgenerated using this approach is Lagrange type. Consider Fig. 8.52. First,we explain how to interpret the information in this figure.

(1) There are three independent coordinates: ξ, η and ζ. For each coordinatewe have monomials of orders 0, 1, . . . , pξ, pη and pζ .

(2) First, we construct Pascal’s rectangle in ξ and η, similar to what wasdone for 2D quadrilateral elements. Locations of the terms are the location


ζξη

ζ

ζη

ζξ

ξη

1

η

ξ

ζξ2

ζξ2η2

ζ1 ζξ

ζη

ζξη2

ξ2

ξ2η2

1

η2

ξ

ξηη ξ2η

ξη2

ζξη ζξ2η

ζη2

ζ2ξ2η2

ζ2ξ2η

ζ2ξ2ζ2ξ

ζ2ξη

ζ2ξη2

ζ2η

ζ2η2

ζ21


ξη ξ2η ξ3η ξ4η

ξ ξ2 ξ3 ξ4



1

η

η2

η4

η3

ζ

ζ2

ζ3

ζ4

Degree of approximation p of φMonomials to be usedNodal configuration

p = 2

p = 1

(a) Pascal prism

(b) nodal configurations and monomial terms

Figure 8.52: Pascal’s prism, nodal configurations and monomial terms

of the nodes for the 2D case and the terms themselves are to be used in thelinear combination like the one in (8.233).

(3) We let this configuration traverse along the ζ direction to the desired


p-level. As it encounters a term in ζ it produces a trace of nodes (same as inξη) and each term in ξη configuration gets multiplied by the correspondingζ term. Thus, for ζ = 1 the configuration in ξη as shown in the figure holds.At location ζ all terms in ξη configuration get multiplied with ζ. Similarly,at location ζ2 we multiply the terms in ξη configuration by ζ2 and so on.

(4) Thus, in this scheme, one chooses the desired p-levels in ξ, η and ζ. Thiscontrols up to what terms we need to consider in ξ, η and ζ.

(5) For example, if pξ = pη = pζ = 2, then in ξ, η and ζ we have 1, ξ, ξ2,1, η, η2 and 1, ζ, ζ2 monomials and the result is a 27-node hexahedron elementshown in Fig. 8.52 (b).

(6) Likewise, for cubic φeh(ξ, η, ζ) in ξ, η and ζ we would need a 64-nodeelement for which the approximation will be generated using the monomials1, ξ, ξ2, ξ3, 1, η, η2, η3 and 1, ζ, ζ2, ζ3.

(7) Once we know the monomials to be used in the linear combination forφ(ξ, η, ζ) and the locations of the nodes, it is straight forward to derive(following the same procedure as shown for pξ = pη = pζ = 1),

φeh(ξ, η, ζ) =n∑i=1

Ni(ξ, η, ζ)φei = [N ]φe = [N ]δe (8.241)

where n is the number of nodes (same as the total number of degrees offreedom due to the fact that each node only has function value as degree offreedom). Here Ni(ξ, η, ζ) are Lagrange interpolation functions.

Remarks.

(1) The local approximations discussed here are of class C000 based on La-grange family of interpolation functions.

(2) A serious drawback of this approach is that as the degree of local ap-proximation increases, so does the size of the matrix to be inverted inderiving Ni(ξ, η, ζ).

(3) Another serious shortcoming of this approach is that as the degree oflocal approximation increases, the number of nodes for an element in-creases dramatically. This necessitates a new discretization for eachp-level change.

In the following sections we correct both of these drawbacks.

8.11.2.3 C000 Lagrange type local approximationsusing tensor product

Recall that in the case of 1D local approximations in ξ for −1 ≤ ξ ≤ 1,the basis functions or local approximation functions can be easily generated


1

η

N ξn

ξ

n

N ξ2

2

N ξ1

1

2

m

1

2

N ζ2

N ζq

q

ζ

N ζ1

N η1

N η2

N ηm

Figure 8.53: 1D Lagrange interpolation functions in ξ, η and ζ

N ξ2 N

η1 =

(1+ξ

2

)(1−η

2

)

N ξ2 N

η2 =

(1+ξ

2

)(1+η

2

)N ξ

1 Nη2 =

(1−ξ

2

)(1+η

2

)

N ξ1 N

η2 =

(1−ξ

2

)(1−η

2

)

N ζ2 = 1+ζ

2

N ζ1 = 1−ζ

2

ξ

1

2

η

Figure 8.54: Tensor product of 1D functions in ξ, η with 1D functions in ζ for pξ = pη =pζ = 1

using Lagrange interpolation functions which allows us to bypass inversionof matrices. For a 1D element with n equally spaced nodes in −1 ≤ ξ ≤ 1


ζ

N4

N7

ξN6

η

5 6

78

2

34

1

N2

N3

N1

N7N8

Figure 8.55: 3D element with pξ = pη = pζ = 1

N ξnN

η2

N ξ1 N

η1

ξ

N ξnN

ηm−1

N ξnN

ηm

N ξn−1N

ηm

N ξn−1N

η1N ξ

2 Nη1

N ξ2 N

ηm

N ξ1 N

ηm

N ξ1 N

η2

N ξ1 N

η1

N ζ2

N ζq

q

2

1

ζ

η

N ζ1

N ξ1 N

ηm−1

Figure 8.56: Tensor product in ξ, η and 1D functions in ζ

we could write φeh(ξ) as

φeh(ξ) =n∑i=1

N ξi (ξ)φei (8.242)


1

1

2

η

ξ

2

1 2

ζ

Nη2 = 1+η

2

Nη1 = 1−η

2

N ζ2 = 1+ζ

2

N ζ1 = 1−ζ

2

N ξ1 = 1−ξ

2 N ξ2 = 1+ξ

2

Figure 8.57: 1D approximation functions in ξ, η and ζ for pξ = pη = pζ = 1

in which

N ξk (ξ) = Lk(ξ) =

n∏m=1m6=k


), k = 1, . . . , n (8.243)

with

N ξi (ξj) =

1, j = i

0, j 6= i(8.244)

and

n∑i=1

N ξi (ξ) = 1 (8.245)

In this, the degree of approximation is pξ = n− 1 and the local approxima-tions are of C0 type due to the fact that only the function values are theunknowns at the end nodes (at ξ = ±1).

The concept of tensor product used for 2D quadrilateral elements de-scribed earlier can be easily extended to the 3D case. consider 1D Lagrangetype interpolation in ξ, η and ζ with p-levels of pξ = n−1, pη = m−1, pζ =q− 1. Here, n,m, q are the number of nodes for ξ, η and ζ 1D local approx-


imations. Then the following holds in ξ, η and ζ.

φeh(ξ) =n∑i=1

N ξi (ξ)φei

φeh(η) =m∑i=1

N ξi (η)φei

φeh(ζ) =

q∑i=1

N ξi (ζ)φei

(8.246)

Schematically, we can show these in Fig. 8.53.Following the procedure described for 2D elements, we first take the

tensor product of 1D functions in ξ and η. This gives us (n×m) configurationof nodes in ξ, η and the corresponding 2D approximation functions and the1D functions in ζ remain as they were in Fig. 8.53. Now we take the tensorproduct of (n×m) configuration in ξ, η with the 1D q-node configuration inζ. This will yield an element with (n×m×q) nodes with explicit expressionsof the approximation functions for each node.

As an example, if pξ = pη = pη = 1 (i.e., n = m = q = 2) then the 1DLagrange interpolation functions in ξ, η and ζ to be considerred in the tensorproduct to generate eight-node linear hexahedron C000 element are shown inFig. 8.55. Tensor product of 1D functions in ξ and η gives 2D functions inξη but leaving 1D functions in ζ as they are in Fig. 8.56. Tensor product of2D ξη interpolation functions of Fig. 8.56 with 1D interpolation functionsin ζ gives the eight-node hexahedron linear 3D element of Fig. 8.57.

The explicit expressions for the nodal interpolation functions are givenin the following with pξ = pη = pζ = 1:

N4(ξ, η, ζ) = N ζ1N

ξ1N

η1 =

(1− ζ2

)(1− ξ2

)(1− η2

)N3(ξ, η, ζ) = N ζ

1Nξ2N

η1 =

(1− ζ2

)(1 + ξ

2

)(1− η2

)N7(ξ, η, ζ) = N ζ

1Nξ2N

η2 =

(1− ζ2

)(1 + ξ

2

)(1 + η

2

)N8(ξ, η, ζ) = N ζ

1Nξ1N

η2 =

(1− ζ2

)(1− ξ2

)(1 + η

2

)N1(ξ, η, ζ) = N ζ

2Nξ1N

η1 =

(1 + ζ

2

)(1− ξ2

)(1− η2

)N2(ξ, η, ζ) = N ζ

2Nξ2N

η1 =

(1 + ζ

2

)(1 + ξ

2

)(1− η2

)N6(ξ, η, ζ) = N ζ

2Nξ2N

η2 =

(1 + ζ

2

)(1 + ξ

2

)(1 + η

2

)N5(ξ, η, ζ) = N ζ

2Nξ1N

η2 =

(1 + ζ

2

)(1− ξ2

)(1 + η

2

)

(8.247)


Using this approach we can generate interpolation or approximation func-tions of degrees (n− 1), (m− 1) and (q− 1) in ξ, η and ζ directions. Tensorproduct avoids inversion of matrices as required in the polynomial approach.Still, there are two drawbacks in this approach: (i) increase in p-level requiresa new nodal configuration, i.e. essentially a new geometric description of ad-ditional nodes and (ii) the local approximations lack hierarchical property.These drawbacks are corrected in the next section by considering C000 p-version hierarchical local approximations.

8.11.2.4 C000 p-version 3D hierarchical localapproximations: using tensor product

Consider a three-node p-version hierarchical local approximations in ξ,η and ζ directions and the corresponding nodal variable operators shownin Figs. 8.58 and 8.59. As in the previous section, we first take the tensorproducts of 1D approximation functions and the nodal variable operators in ξand η. This yields 2D nodal configuration, the corresponding approximationfunctions and the nodal variable operator in ξ, η space as shown in Figs. 8.60and 8.61. The 1D functions and nodal variable operators in ζ remain thesame as in Figs. 8.58 and 8.59.

ξ

η

ζ

2

3

N 1ζ3

Nkζ2

(k = 2, . . . , pζ)

1

N 1η3

N 1η3

N 1ξ3N 1ξ

1 N iξ2

N 1ζ1

N jη2

(j = 2, . . . , pη)

1 2 3

2

1

3

(i = 2, . . . , pξ)

Figure 8.58: C000 1D p-version hierarchical approximation functions

The tensor product of basis functions in ξ, η configuration in Fig. 8.58with 1D basis functions in ζ gives approximation functions for 27-node p-version hierarchical hexahedron element. The corresponding nodal variableoperators are obtained by similar processes for 2D nodal variable operatorsin ξ, η configuration and 1D nodal variable operators in ζ in Fig. 8.58. Thedetails are straight forward and are left as an exercise. We note that 1D


ξ

η

ζ

1

1

(j = 2, . . . , pη)

1 2 3

2

1

3

1 1

2

3

1

1

1

∂j

∂ηj

∂i

∂ξi

(i = 2, . . . , pξ)

∂k

∂ζk

(k = 2, . . . , pζ)

Figure 8.59: C0 1D p-version hierarchical nodal variable operators

N 1ζ3

η

N 1ξ3 N 1η

3

ζ

N 1ξ1 N jη

2

j = 2, . . . , pη

N iξ2 N 1η

3

j = 2, . . . , pη

i = 2, . . . , pξN 1ξ

1 N 1η3

N 1ξ3 N 1η

1

N 1ζ1

N iζ2

N iξ2 N jη

2

N 1ξ1 N 1η

1 N iξ2 N 1η

1

i = 2, . . . , pξ

i = 2, . . . , pξj = 2, . . . , pη

k = 2, . . . , pζ

N 1ξ3 N jη

2

ξ

Figure 8.60: C0 1D p-version hierarchical approximation functions in ξ and C00 2Dp-version hierarchical approximation functions in ξ, η

p-version hierarchical basis functions in ξ, η and ζ are given by

N1ξ1 =

(1− ξ2

), N1ξ

3 =(1 + ξ

2

)(8.248)

N iξ2 =

(ξi − ai!

), a =

1, if i is even

ξ, if i is odd


1

η

ζ

j = 2, . . . , pη

i = 2, . . . , pξ

1

ξ

1

∂k

∂ζk

j = 2, . . . , pη

i = 2, . . . , pξ

∂i+j

∂ξi∂ηj

11

1

∂j

∂ηj

∂j

∂ηj

k = 2, . . . , pζ

∂i

∂ξi

j = 2, . . . , pη

i = 2, . . . , pξ

∂i

∂ξi

Figure 8.61: C0 1D p-version hierarchical nodal variable operators

N1η1 =

(1− η2

), N1η

3 =(1 + η

2

)(8.249)

N jη2 =

(ηj − bj!

), b =

1, if j is even

η, if j is odd

N1ζ1 =

(1− ζ2

), N1ζ

3 =(1 + ζ

2

)(8.250)

Nkζ2 =

(ζk − ck!

), c =

1, if k is even

ζ, if k is odd

Thus, the approximation functions for the 27-node p-version hierarchicalhexahedron element are completely defined. The nodal degrees of freedomare obtained by letting the nodal variable operators act on the dependentvariable(s).

8.11.2.5 3D Cijk(Ωe) p-version local approximations: Hexahedronelements

3D Cijk(Ωe) p-version family of interpolations in xyz space which possessinterelement continuity of orders i, j and k in xyz directions with completepolynomials of orders pξ, pη and pζ are considered in this section. We con-sider a twenty seven node hexahedron element (two-unit cube) in ξηζ space


with sides parallel to xyz axes and pointing in the same direction as xyzaxes. We consider map of this element in xyz space, a prism where sidesare also parallel to xyz axes and pointing in the same direction. By usingtensor products of Ci, Cj and Ck one-dimensional interpolations in xyz di-rections, the local approximation functions and nodal variable operators forthe 27 node prism family of hexahedron can be derived in a straight forwardmanner. These elements have same limitations and restrictions as discussedin section 8.5 for similar local approximations for 2D case in xy space. Weremark that these interpolation functions can be of Lagrange, Legendre orChebyshev type by choosing 1D interpolations based on these.

8.11.2.6 3D Cijk(Ωe) p-version interpolations fordistorted hexahedron elements:27 node element

In deriving the local approximations for this family of hexahedron el-ements we can follow the procedure similar to that used for 2D elements(section 8.6). The decision of the choice of dofs at the corner is straightforward due to the fact that these must constitute a transformable set be-tween ξηζ and xyz space. However, from which nodes to borrow the neededdegrees of freedom is not as straight forward but can be facilitated by exam-ining Pascal prism and ensuring that lower order admissible terms are chosenfirst. These interpolations can also be of Lagrange, Legendre, or Chebyshevtype depending upon the choice of 1D interpolations.

8.11.2.7 Interpolation theory for 3D tetrahedronelements: basis functions of class C000(Ωe)based on Lagrange interpolations

Parallel to Pascal prism, in case of tetrahedron elements we have Pascalpyramid (see Fig. 8.62). The locations of the terms are the locations of thenodes and the terms themselves are to be used in the linear combination forthe local approximations. For example a linear tetrahedron will contain fourvertex nodes with the local approximation (for a dependent variable φ)

φeh(x, y, z) = c1 + c2x+ c3y + c4z (8.251)

Using Cartesian coordinates of the vertex nodes (xi, yi, zi) are the functionvalues φ(xi, yi, zi) = φei (i = 1, 2, . . . , 4). In (8.251), it is straight forward toestablish

φeh(x, y, z) =4∑i=1

Ni(x, y, z)φei (8.252)


z

z2

xz2

x2z

x2

x

x2

x3

z2yzy2

z3

y3

y2

y

xy

zy

xy2x2y

1

Figure 8.62: Pascal pyramid

in which Ni(x, y, z) are the interpolation functions with the properties

Ni(xj , yj , zj) =

1, if j = i

0, if j 6= i

4∑i=1

Ni(x, y, z) = 1

(8.253)

Using this procedure higher degree local approximations of class C000(Ωe)can be constructed for the tetrahedron elements. As in case of hexahe-dron elements based on Pascal prism, here also with increasing degree oflocal approximations the number of nodes for an element increase and werequire inverse of progressively increasing size coefficient matrices to deter-mine Ni(x, y, z).

8.11.2.8 Lagrange family C000 interpolations basedon volume coordinates

Parallel to the area coordinates for 2D triangular elements, in case oftetrahedron elements we introduce the concept of volume coordinates. Con-sider a four-node tetrahedron element shown in Fig. 8.63. Consider a pointP in the interior of the element. Connect point P with the vertices of thetetrahedron by straight lines. By doing so we divide the volume V into fourvolumes V1, V2, V3 and V4. Let the side opposite to node i be side i withvolume Vi.


2

3

4

1

P

Figure 8.63: Tetrahedron element with volume coordinates

Let

L1 =ViV, i = 1, 2, . . . , 4

with

4∑i=1

Li = 1(8.254)

be the volume coordinates. Clearly Li is equal to 1 at node i but zero on sidei. The volume coordinates can be used to define the Cartesian coordinatesof any point on the element. If (xi, yi, zi) are the Cartesian coordinates ofthe nodes of the element, then

x = L1x1 + L2x2 + L3x3 + L4x4

y = L1y1 + L2y2 + L3y3 + L4y4

z = L1z1 + L2z2 + L3z3 + L4z4

(8.255)

Using (8.255) and the second equation of (8.254), we can write1xyz

=

1 1 1 1x1 x2 x3 x4

y1 y2 y3 y4

z1 z2 z3 z4

=

L1

L2

L3

L4

= [C]

L1

L2

L3

L4

(8.256)

Therefore L1

L2

L3

L4

= [C]−1

1xyz

=

N1(x, y, z)N2(x, y, z)N3(x, y, z)N4(x, y, z)

(8.257)

We can show that Ni(x, y, z) here in (8.257) are same as these using Pascalpyramid. We note that det[C] is related to the volume of tetrahedron hence


det[C] > 0 holds implying that Ni(x, y, z) in (8.257) are unique. Thus forthe four-node tetrahedron we can use L1, L2, L3, L4 as interpolation functionsinstead of Ni(x, y, z); i = 1, 2, . . . , 4.

8.11.2.9 Higher degree C000 basis functions usingvolume coordinates

Consider the four-node tetrahedron element shown in Fig. 8.63 with ver-tex nodes 1, 2, 3 and 4. We draw m equally spaced planes parallel to side1 labelled 0, 1, . . . , p . . . ,m, 0 being label for side 1. Similarly, we also drawm equally spaced planes parallel to sides 2, 3 and 4, sides 2, 3 and 4 beingmarked ‘0’ and q, r, s being intermediate locations of the planes parallel tosides 2, 3, and 4. If we consider typical planes p, q, r and s (parallel tosides 1, 2, 3 and 4) then their intersection pqrs defines a point within thetetrahedron element. The intersection of these planes with sides 1, 2, 3 and4 and amongst themselves define the location of the nodes in agreement withthe Pascal pyramid. Consider side 1 and planes parallel to it. We note thatL1 is one at node 1 and zero on side 1, hence we can setup Lagrange type in-terpolations to define functions N0(L1), N1(L1), . . . , Nm(L1) correspondingto (m+ 1) parallel planes. We can define Ni(L1) by

Ni(L1) =i∏

j=1

(mL1 − j + 1

j

), for j ≥ 1; i = 1, 2, . . . ,m

= 0 for i = 0

(8.258)

Similarly in the other three directions, we setup Ni(L2, Ni(L3) and Ni(L4)

Ni(L2) =i∏

j=1

(mL2 − j + 1

j

), for j ≥ 1; i = 1, 2, . . . ,m

= 0 for i = 0

(8.259)

Ni(L3) =i∏

j=1

(mL3 − j + 1

j

), for j ≥ 1; i = 1, 2, . . . ,m

= 0 for i = 0

(8.260)

Ni(L4) =

i∏j=1

(mL4 − j + 1

j

), for j ≥ 1; i = 1, 2, . . . ,m

= 0 for i = 0

(8.261)

The 3D interpolations or basis functions for a node located at p, q, r, s i.e.Npqrs(L1, L2, L3, L4) can now be defined using

Npqrs(L1, L2, L3, L4) = Np(L1)Nq(L2)Nr(L3)Ns(L4) (8.262)


in which Np, Nq, Nr and Ns are defined by (8.258) to (8.261). We considersome examples in the following.

8.11.2.10 Four-node linear tetrahedron element (p-level of one)

In this case m = 1 and we have p, q, r, s have values of 0 and 1.

N0(L1) = 1

N1(L1) = L1, m = 1, j = 1, i = 1 in (8.258)(8.263)

N0(L2) = 1

N1(L2) = L2, m = 1, j = 1, i = 1 in (8.259)(8.264)

N0(L3) = 1

N1(L3) = L3, m = 1, j = 1, i = 1 in (8.260)(8.265)

N0(L4) = 1

N1(L4) = L4, m = 1, j = 1, i = 1 in (8.261)(8.266)

The basis or interpolation functions Ni(L1, L2, L3, L4) (i = 1, 2, . . . , 4) forthe four-node tetrahedron can now be constructed using the following.

N1(L1, L2, L3, L4) = N pqrs(1000)

= N1(L1)N0(L2)N0(L3)N0(L4) = L1

N2(L1, L2, L3, L4) = N pqrs(0100)

= N0(L1)N1(L2)N0(L3)N0(L4) = L2

N3(L1, L2, L3, L4) = N pqrs(0010)

= N0(L1)N0(L2)N1(L3)N0(L4) = L3

N4(L1, L2, L3, L4) = N pqrs(0001)

= N0(L1)N0(L2)N0(L3)N1(L4) = L4

(8.267)

8.11.2.11 A ten-node tetrahedron element (p-level of 2)

In this case m = 2 in each of the three directions L1, L2, L3 and L4.The element will contain 10 nodes. Using (8.263) to (8.266) with m = 2 weestablish Ni(Lj) (i = 0, 1, 2) for j = 1, 2, 3 and 4 and then use (8.267) todetermine Ni(L1, L2, L3, L4) for i = 1, 2, . . . , 10.

8.12 Summary

Basic elements of mapping and the interpolation theory are presentedin this chapter. The finite elements in physical spaces R1, R2, and R3 aremapped into natural coordinate spaces ξ; ξ, η; and ξ, η, ζ in predefined ge-ometric configurations. Mapping of points, lengths, areas, and volumes areestablished. The interpolation theories in R1, R2, and R3 are presented us-ing natural coordinate space with fixed element geometries using Lagrange

PROBLEMS 605

interpolation; tensor product and Pascal’s triangle, pyramid, and rectan-gle. For triangular and tetrahedron families of elements, area and volumecoordinates are utilized. Interpolation theories, hence local approximationsare also presented using Legendre and Chebyshev polynomials. In all casesp-version hierarchical local approximations of class C0 as well as of higherclasses in R1 are established first and then utilized through tensor productor otherwise to derive C0 and higher order continuity local approximationsfor regular and distorted geometries (in physical space) in R2 and R3.

Problems

8.1 Figure 8.64(a) shows a three-node parabolic one-dimensional element in one-dimensionalCartesian coordinate space.

1 32

y

x

x1 = 2.0 1 3x2 = 3.0 x3 = 6.0

η

ξ2

2

(b) Schematic in ξη space(a) Schematic in xy space

Figure 8.64: Element map in physical space xy and natural coordinate space ξη

Figure 8.64(b) shows a map of the element in the natural coordinate space. The elementapproximation functions are defined in the book in the natural coordinate system.

(1) Write equations describing the element mapping between xy and ξη spaces.(2) Derive an expression for the determinant of the Jacobian.(3) Calculate the value of the determinant of the Jacobian at the element nodes.(4) Derive expressions for the derivatives of the approximation functions with respect to

x.(5) Calculate the derivatives of the approximation functions with respect to x at the

element nodes.(6) If the element of Fig. 8.64(a) was used in stress analysis, can you comment on the

nature of the strain and stress at the element nodes.

8.2 Consider two-dimensional finite elements shown in Fig. 8.65 (a), (b) and (c). TheCartesian coordinates of the nodes are given. The elements are mapped into ξ, η naturalcoordinate space into a two-unit square.

(I) Determine the Jacobian matrix of transformation and its determinant for each ele-ment. Calculate and tabulate the value of the determinant of the Jacobian at thenodes of the element.

(II) Calculate the derivatives of the approximation function with respect to x and y for

node 3 (i.e. ∂N3(ξ,η)∂x

and ∂N3(ξ,η)∂y

) for each of the three elements shown above.

8.3 Consider a two-dimensional eight-node finite element shown in Fig. 8.66. The Carte-sian coordinates of the nodes of the element are given in Fig. 8.66. The element is mapped


3

3

12

4

5

3

5

1 2 3

4

8

567

(5,6)

1.5

3 3

(10,6)

(6.5,7)

0.5

3

2

4

1

10

10

y

x

y

x

(b) (c)

y

(a)

x

Figure 8.65: 2D elements in xy space

into natural coordinate space ξ, η into a two-unit square with the origin of the ξη coordinatesystem at the center of the element.

6

(5,6) (10,6)

1.5

5

3 3

4

3

2

8

3

3

1

1

1

7

0.5

y

x

Figure 8.66: A 2D element in xy space

(a) Write a computer program (or calculate otherwise) to determine the Cartesian coordi-nates of the points midway between the nodes. Tabulate the x, y coordinates of thesepoints. Plot the sides at the element in xy space by taking more intermediate points.

(b) Determine the area of the element using Gaussian quadrature. Select and use theminimum number of quadrature points in ξ and η directions to calculate the areaexactly. Show that increasing the order of the quadrature does not affect the area.

(c) Determine the locations of the quadrature points (used in (b)) in the Cartesian space.Provide a table of these points and their locations in xy space. Also mark theirlocations on the plot generated in part (a).

Provide program listing, results, tables and plots along with a write-up on the equationsused as part of the report. Also provide a discussion of your results.

[1–11]

References for additional reading[1] K. S. Surana, S. R. Petti, A. R. Ahmadi, and J. N. Reddy. On p-version hierarchi-

cal interpolation functions for higher-order continuity finite element models. Int. J.Comp. Eng. Sci., 2(4):653–673, 2001.


[2] A. R. Ahmadi, K. S. Surana, Maduri, A. Romkes, and J. N. Reddy. Higher orderglobal differentiability local approximations for 2d distorted quadrilateral elements.Int. J. Comp. Meth. in Eng. Sci. and Mech., 10:1–19, 2009.

[3] I. S. Sokolnikoff and R. M. Redheffer. Mathematics of Physics and Modern Engineer-ing. McGraw-Hill, 2nd edition, 1966.

[4] B. A. Szabo and I. Babuska. Finite Element Analysis. Wiley-Interscience, 1991.



[7] K. S. Surana, S. Allu, and J. N. Reddy. The k-version of finite element method forinitial value problems: Mathematical and computational framework. Int. J. Comp.Eng. Sci., 8(3):123–136, 2007.

[8] G. Strang. Variational crimes in the finite element method. In A. K. Aziz, editor, TheMathematical Foundations of the Finite Element Method with Applications to PartialDifferential Equations, pages 689–710. Academic Press, 1972.

[9] J. Patera and F. T. Pettman. Isoparametric hermite elements. Int. J. Num. Meth.Eng., 37:3489–3519, 1994.

[10] D. W. Wang, I. N. Katz, and B. A. Szabo. Implementation of c1 triangular elementbased on the p-version of the finite element method (structural analysis computercode). Comp. Struct., 19(3):381–392, 2011.

[11] R.H. Gallagher. Finite Element Analysis: Fundamentals. Prentice Hall, New Jersey,1975.

9

Linear Elasticity using thePrinciple of Minimum

Total Potential Energy

9.1 Introduction

In chapters 3 and 5 we had considered finite element processes basedon GM/WF and least squares methods for BVPs described by self-adjointdifferential operators. The approach in GM/WF is based on fundamentalLemma whereas the LSPs are based on residual functional. In this chapter wesummarize GM/WF for BVPs described by self-adjoint differential operatorsand present an alternate approach in which the details of the finite elementprocesses can be derived directly without using the fundamental Lemmaand integration by parts. While the mathematical basis for the approachpresented here remains the same as GM/WF, but the approach presented inthis chapter is perhaps more appealing as it deals directly with the physics.First, we revisit some basic concepts (chapters 3 and 5). Consider a BVP

Aφ− f = 0 in Ω (9.1)

in which A is a self-adjoint differential operator, which contains only even or-der derivatives of the dependent variables (we consider homogeneous bound-ary conditions for the sake of simplicity). More specifically A only needs tobe linear and its adjoint A∗ needs to be same as A. In classical GM weconsider

(Aφn − f, v)Ω = 0 (9.2)

in which φn is an approximation of φ over Ω and v = δφn. In (9.2), for theeven order derivative terms in φn we transfer half of the differentiation fromφn to the test function v. Simplification of the concomitant using BCs andregrouping terms gives the weak form

(Aφn − f, v)Ω = B(φn, v)− l(v) = 0 (9.3)

In (9.3), B(φn, v) is bilinear and symmetric, B(φn, v) = B(v, φn), andl(v) is linear. Hence, there exists a functional I(φn)

I(φn) =1

2B(φn, φn)− l(φn) = 0 (9.4)

609

610 ELASTICITY USING THE PRINCIPLE OF MINIMUM TOTAL POTENTIAL ENERGY

such that the first variation of I(φn) gives the weak form

δI(φn) = B(φn, v)− l(v) = 0 (9.5)

andδ2I = B(v, v) > 0 (9.6)

is a unique extremum principle and hence the weak form is a variationallyconsistent integral form. Inequality (9.6) implies that φn obtained fromδI(φn) = 0 minimizes I(φn) in (9.4).

In case of linear elasticity and linear structural mechanics, I(φn) repre-sents the total potential energy of the deforming body in which 1

2B(φn, φn)is the strain energy or elastic energy stored by the body and l(φn) is thepotential energy of loads. Thus minimization of I(φn) is in fact the wellknown principle of minimization of total potential energy. Stated simply:If a deformed elastic body is in equilibrium then its total potential energyis minimum. Thus, if we could construct the total potential energy of adeformed body (integral over the volume of the body) then, its minimiza-tion will yield the integral form defining stable equilibrium of the deformedbody. This indeed is the weak form, which in this case is variationally con-sistent integral form hence will yield unconditionally stable computationalprocesses.

9.2 New notation

To maintain consistency with the notation used in linear elasticity, struc-tural mechanics, and energy, we introduce the following new notation. Wefirst consider classical methods of approximation. Let the quadratic func-tional I(φ) representing total potential energy of a deformed body in equi-librium be denoted by Π(φ). Then, for approximation φn of φ over Ω wehave I(φn) = Π(φn). Let Π1(φn) and Π2(φn) be the strain energy storedby the deformed body and the potential energy of loads respectively for anapproximation φn of φ over Ω. Then

Π1(φn) =1

2B(φn, φn)

Π2(φn) = l(φn)(9.7)

Hence, based on (9.4) and (9.7) we can write

Π(φn) = Π1(φn)−Π2(φn) (9.8)

First variation of Π(φn) set to zero, δΠ(φn) = 0, gives the necessaryconditions from which we determine φn and δ2Π(φn) > 0 is the extremumprinciple which when unique ensures that φn obtained from δΠ(φn) = 0 infact minimizes Π(φn).

9.3. APPROACH 611

9.3 Approach

Based on the details presented above, we can proceed as follows.

(1) Derive an expression for the total potential energy Π of the deformedbody directly from the physics of deformation. This establishes expres-sions for Π1 and Π2.

(2) Minimization of the total potential energy Π would yield the necessaryconditions, namely, the equations of equilibrium for the deformed bodyfrom which we determine the solution.

(3) First, (1) and (2) need to be converted to the discretization ΩT =⋃e Ωe.

ConsiderΠ = Π1 −Π2 for ΩT (9.9)

andΠ =

∑e

Πe for Πe over Ωe (9.10)

minimization of Π implies

δΠ = δ∑e

Πe =∑e

δΠe = 0 (9.11)

Equation (9.11) describes equilibrium of the whole discretizion of thedomain Ω (i.e. ΩT ). In (9.11), we only need to consider δΠe for anelement e with domain Ωe. This would yield the desired discretizedequations for an element e that can be assembled or summed to obtainthe discretized equations for the whole discretization ΩT . We presentdetails of the discretized equations for an element e with domain Ωe

using Πe and δΠe. The total potential energy is

Πe = Πe1 −Πe

2 (9.12)

9.4 Element equations

9.4.1 Local approximation of the displacement field

Let

φeh = ueh or

uehveh

or

uehvehweh

(9.13)

be the local approximation of the displacement field for ∀x or x, y or x, y, z ∈Ωe → Ωm and hence, ∀ξ or ξ, η or ξ, η, ζ ∈ Ωm.

φeh = [N ]δe (9.14)

in which [N ] is the local approximation function matrix and δe are nodaldegrees of freedom.


9.4.2 Stresses and strains

Assuming small deformation, small strain (linear elasticity), the strainsat a point can be expressed in terms of the gradients of the displacements u,v and w in x, y and z directions. Expressing strain tensor as a vector (Voigt’snotation), we can write strains as functions of the gradients of displacements

ε =ε(u v wx y z

)(9.15)

and by substituting from (9.14) into (9.15), we can express the strains interms of nodal degrees of freedom δe.

ε = [B]δe (9.16)

Let σ0 be initial stress in the body before the application of externalloads and also let ε0 be the initial strain. Then the total stress field in thebody before deformation is given by the expression

−[D]ε0 − σ0 (9.17)

in which [D] is the matrix of material constants relating strains to stresses.The stresses due to deformation, ε, are given by (a suitable constitutivelaw)

σ = [D]ε (9.18)

Hence, the total stress is given by

σtotal = σ − [D]ε0 − σ0 = [D]ε − [D]ε0 − σ0 (9.19)

Equation (9.18) is generalized Hooke’s law. [D] is a symmetric matrixcontaining material coefficients.

9.4.3 Strain energy Πe1 and potential energy of loads Πe

2

If Πe1 is total strain energy or elastic energy stored in an element e with

domain Ωe, then

Πe1 =

1

2

∫Ωe

εT σ dΩ (9.20)

Substituting for σ from (9.18) into (9.20),

Πe1 =

1

2

∫Ωe

εT [D]ε dΩ (9.21)

9.4. ELEMENT EQUATIONS 613

Substituting for ε from (9.16) into (9.21)

Πe1 =

1

2

∫Ωe

δeT [B]T [D][B]δe dΩ (9.22)

Next we consider potential energy of loads Πe2 over Ωe. When the elas-

tic body undergoes deformation, the points of application of external loadsexperience motion. The body may be subjected to body forces such as ac-celeration and gravitational loads and/or pressure loads on some boundariesof it. The work is done by all of these disturbances acting on the volumeΩe. When an element is isolated from the discretization ΩT , it must be inequilibrium (free body diagram) under the action of external loads and theinternal forces on its boundaries exerted by the remaining discretization (cutprinciple of Cauchy).

Let us introduce the following notation:

P e be a vector of secondary variables at the element nodes (i.e. internalnodal forces).

p be a vector of distributed loads (say pressure) in x, y, z direction perunit length or area on a boundary Γe∗ of an element e with domain Ωe.

f b be a vector of distributed body forces (due to gravity or accelerationand/or centrifugal forces) per unit volume.

Then, the potential energy of loads can be written as

Πe2 = δeT P e+

∫Γe∗

φehT p dΓ +

∫Ωe

φehT f b dΩ

+

∫Ωe

εT [D]ε0 dΩ +

∫Ωe

εT σ0 dΩ (9.23)

Substituting for ε from (9.16) and φeh from (9.14) in (9.23)

Πe2 =δeT P e −

∫Γe∗

δeT [N ]T p dΓ +

∫Ωe

δeT [N ]T f b dΩ

+

∫Ωe

δeT [B]T [D]ε0 dΩ +

∫Ωe

δeT [B]T σ0 dΩ

(9.24)

This is the final expression for the potential energy of loads.


9.4.4 Total potential energy Πe for an element e

Using (9.22) and (9.24) and noting that Πe = Πe1 −Πe

2, we can write

Πe =1

2

∫Ωe

δeT [B]T [D][B]δe dΩ− δeT P e −∫Γe∗

δeT [N ]T p dΓ

−∫Ωe

δeT [N ]T f b dΩ−∫Ωe

δeT [B]T [D]ε0 dΩ−∫Ωe

δeT [B]T σ0 dΩ

(9.25)

We note that δe is independent of coordinates and hence can be takenoutside of the integrals.

Πe =1

2δe

[ ∫Ωe

[B]T [D][B] dΩ]δe − δeT P e

− δeT(∫

Γe∗

[N ]T p dΓ)− δeT

(∫Ωe

[N ]T f b dΩ)

− δeT(∫

Ωe

[B]T [D]ε0 dΩ)− δeT

(∫Ωe

[B]T σ0 dΩ)

(9.26)

We introduce the following notation:

[Ke] =

∫Ωe

[B]T [D][B] dΩ (9.27)

F ep =

∫Γe∗

[N ]T p dΓ

F eb =

∫Ωe

[N ]T f b dΩ

F eε0 =

∫[B]T [D]ε0 dΩ

F eσ0 = [B]T σ0 dΩ

(9.28)

in which [Ke] is called the element stiffness matrix and F ep, F eb, F eε0and F eσ0 are the equivalent nodal load vectors due to: pressure loading onthe element faces, body forces, initial strain and initial stress. Substitutingfrom (9.27) and (9.27) into (9.26)

Πe = δeT [Ke]δe − δeT P e − δeT F ep− δeT F eb − δeT F eε0 − δeT F eσ0 (9.29)

9.4. ELEMENT EQUATIONS 615

We note thatΠe = Πe

(δe

)(9.30)

andΠ =

∑e

Πe(δe

)= Π

(δ)

(9.31)

whereδ = ∪eδe (9.32)

Therefore

(Π)min ⇒∂Π

∂δ= 0 (9.33)

or∂Π

∂δ=∑e

( ∂Πe

∂δe

)= 0 (9.34)

From (9.29), we can obtain ∂Πe

∂δe

∂Πe

∂δe= [Ke]δe − P e − F ep − F eb − F eε0 − F eσ0 (9.35)


∂Π

∂δ=∑e

([Ke]δe−P e−F ep−F eb−F eε0−F eσ0

)= 0 (9.36)

or∑e

([Ke]δe

)=∑e

P e+∑e

F ep +∑e

F eb +∑e

F eε0 +∑e

F eσ0

(9.37)or [∑

e

Ke]δ = P+ Fp + Fb + Fε0 + Fσ0 (9.38)

or[K]δ = P+ Fp + Fb + Fε0 + Fσ0 (9.39)

in which∑

e[Ke] = [K]. Sums over e (elements of ΩT ) represent assembly

of element equations symbolically. Forces on the right side of (9.39) are dueto assembly of element equivalent nodal loads and secondary variables.

Remarks.

(1) The derivation presented in this chapter is based on minimization oftotal potential energy of an elastic body in equilibrium under the actionof external disturbances. Since the differential operator is self-adjoint,the total potential energy is in fact quadratic functional I(φ).


(2) This direct approach of constructing the total potential energy func-tional using the physics of deformation is quite general and is perhapsmore appealing in applications such as linear elasticity and structuralmechanics.

(3) Various types of 1D, 2D and 3D elements in linear solid mechanics (i.e.linear elasticity), can be easily derived using the general derivation pre-sented above. We outline the main steps in the following.

(a) Identify displacement components.

(b) Identify stress and strain components.

(c) Establish the nature of local approximations over a subdomain Ωe,i.e [N ], the approximation function matrix and the dofs δe.

(d) Express strains in terms of the derivatives of displacements and sub-stitute local approximations to derive [B] matrix.

(e) Identify constitutive behavior [D] expressing material behavior re-lating strains to stresses.

(f) Express initial strains. If thermal strains, then express them in termsof temperature difference with respect to stress free temperature andthe coefficients of thermal expansion.

(g) All other external loadings such as body forces (acceleration andcentrifugal loads) etc. can also be expressed as a function of basisfunctions and subdomain or element volume or mass.

With (a)–(g), all matrices and vectors in the element equations are de-fined except secondary variables for the elements as well as in assembledequations.

(4) Using the approach outlined above it is straight forwards to derive thedetails of the finite element equations for applications such as

(a) plane stress

(b) plane strain

(c) axi-symmetric deformation

(d) 2D beam bending

(e) axi-symmetric shells

(f) 3D membranes

(g) plate bending

(h) 3D solids

(i) 3D thick shells

(j) 3D shells

(k) 3D beam bending

(l) 1D, 2D and 3D axial rods, spars and others

9.5. FINITE ELEMENT FORMULATION FOR 2D LINEAR ELASTICITY 617

What one needs to recognize is that the generic nature of the basicderivation remains the same in all cases. What changes from one typeof element (i.e. physics) to another are the specific contents of variousmatrices and vectors.

9.5 Finite element formulation for 2D plane stresslinear elasticity

To illustrate details of various matrices and vectors in the direct approachdescribed above we consider plane stress linear elasticity as an example. Inplane stress behavior a material particle in the current (deformed) config-uration experiences displacements u and v in x, y directions of the o − xyfixed frame.

We consider a distorted quadrilateral element in x, y space and its mapΩm or Ωξη in the ξ, η natural coordinate space in the two-unit square (seeFig. 9.1).

23

4

5

7

1 1 2 3

567

8 94

2

2

6

8

9ξ

y

xu

v

Ωe

Element map in the naturalcoordinate space ξη

Element geometry in xy space

Ωm or Ωξ,η

η

Figure 9.1: Distorted 2D quadrilateral element and its map in ξ, η space

Geometry:

The mapping can be described usingxy

=

n∑i=1

Ni(ξ, η)

xiyi

with Ni(ξj , ηj) =

1, if j = i

0, if j 6= i

andn∑i=1

Ni(ξ, η) = 1

(9.40)


where n could be nine if the element shape is distorted or four (in which case,it is a four node quadrilateral element) if the sides are straight. Additionally,

we have [J ] =[xyξη

].

9.5.1 Local approximation of u and v over Ωe or Ωξη

Consider C0 local approximations of displacements u and v based onLagrange family of interpolation functions, hence the nodal dofs are values ofu and v at the element nodes (it could be p-version hierarchical also). Equaldegree and equal order local approximations for u and v is valid because theGDEs for the plane stress deformation contain same order derivatives of uand v. Let ueh, v

eh be local approximations of u and v over an element Ωe,

then uehveh

=

n∑i=1

Ni(ξ, η)

ueivei

= [N ]δe (9.41)

in which for a typical node i, uei , vei are the degrees of freedom. The sub-

matrix [Ni] for node i can be written as

[Ni] =

[Ni(ξ, η) 0

0 Ni(ξ, η)

](9.42)

with

δei =

ueivei

(9.43)

9.5.2 Stresses, strains and constitutive equations

Using Voigt’s notation, the strain and stress, as vectors ε and σcan be written as follows. Also, the constitutive theory for stresses can beexpressed in terms of strains:

ε =

εxεyγxy

=

∂u∂x∂v∂y

∂u∂y + ∂v

∂x

(9.44)

σ =

σxσyτxy

= [D]ε (9.45)

If we consider homogeneous and isotropic material then global x, y direc-tion suffice for defining stresses, strains and the material matrix [D]. It is


straight forward to express strains in terms of stresses, modulus of elasticityE and Poisson’s ratio ν

εxεyγxy

=

1E − ν

E 0− νE

1E 0

0 0 2(1+ν)E

=

σxσyτxy

(9.46)

Inverse of (9.46) givesσxσyτxy

=E

(1− ν2)

1 ν 0ν 1 00 0 1−ν

2

εxεyγxy

= [D]ε (9.47)

9.5.3 [B] matrix relating strains to nodal degrees of freedom

Using local approximation (9.41) for u and v

∂ueh∂x

=∑ ∂Ni

∂xuei ,

∂ueh∂y

=∑ ∂Ni

∂yuei

∂veh∂x

=∑ ∂Ni

∂xvei ,

∂veh∂y

=∑ ∂Ni

∂yvei

(9.48)


ε = [B]δe (9.49)

in which [Bi] for a node i is given by

[Bi] =

∂Ni∂x 0

0 ∂Ni∂y

∂Ni∂y

∂Ni∂x

(9.50)

9.5.4 Element stiffness matrix [Ke]

[Ke] =

∫Ωe

[B]T [D][B] dΩ (9.51)

in which [B] and [D] are defined by (9.51) and (9.47). We note that

dΩ = t(ξ, η) det[J ] dξ dη (9.52)

t(ξ, η), thickness at a point ξ, η can be defined by

t(ξ, η) =n∑i=1

N(ξ, η) ti (9.53)


where ti are nodal thicknesses. A typical [Keij ] of [Ke] for nodes i and j can

be written as

[Keij ] =

1∫−1

1∫−1

[Bi]T [D][Bj ] t(ξ, η) det[J ] dξ dη, i, j = 1, . . . , n (9.54)

or

[Keij ] =

1∫−1

1∫−1

∂Ni∂x 0

0 ∂Ni∂y

∂Ni∂y

∂Ni∂x

T D11 D12 0

D12 D22 0

0 0 D33

∂Nj∂x 0

0∂Nj∂y

∂Nj∂y

∂Nj∂x

t(ξ, η) det[J ] dξ dη

(9.55)in which n is the number of nodes. Numerical values of [Ke] are obtainedusing Gauss quadrature.

9.5.5 Transformations from (ξ, η) to (x, y) space

Using x = x(ξ, η), y = y(ξ, η) defining mapping between ξη and xy spaceswe can write the following for mapping of lengths between the two spaces

dxdy

= [J ]

dξdη

(9.56)

where

[J ] =

[∂x∂ξ

∂x∂η

∂y∂ξ

∂y∂η

](9.57)

in which

∂x

∂ξ=

n∑i=1

∂Ni

∂ξxi,

∂x

∂η=

n∑i=1

∂Ni

∂ηxi

∂y

∂ξ=

n∑i=1

∂Ni

∂ξyi,

∂y

∂η=

n∑i=1

∂Ni

∂ηyi

(9.58)

and ∂Ni∂x∂Ni∂y

= [JT ]−1


, i = 1, . . . , n (9.59)

9.5.6 Body forces

Body forces in solid and structural mechanics consist of acceleration orgravity loads and centrifugal forces in spinning objects. Recall that

F eb =

∫Ω

[N ]Tf bxf by

dΩ (9.60)


in which Gx and Gy are body forces percent volume in x and y directions.In ξ, η space, equation (9.60) can be written as

F eb =

1∫−1

1∫−1

[N ]Tf bxf by

det[J ] t(ξ, η) dξ dη (9.61)

Let gx and gy be accelerations in x and y directions and wz be the angularvelocity about the z-axis, then

f bxf by

=

ρ gx + ρ xw2

z

ρ gy + ρ y w2z

=

ρ(gx + xw2

z)ρ(gy + y w2

z)

(9.62)

in which ρ is mass density of the material. For a typical node i

F ei b =

F exi

F eyi

b

=

1∫−1

1∫−1

[Ni(ξ, η) 0

0 Ni(ξ, η)

]ρ(gx + xw2

z)ρ(gy + y w2

z)

t(ξ, η) det[J ] dξ dη (9.63)

9.5.7 Initial strains (thermal loads)

Let α be the coefficient of thermal expansion and Tref be the stress freetemperature, then

ε0 =

α(T (ξ, η)− Tref )α(T (ξ, η)− Tref )

0

=

εx0εy00

(9.64)

in which εx0 and εy0 are thermal strains in the x and y directions. Thetemperature T (ξ, η) may be defined as

T (ξ, η) =∑

N(ξ, η)T ei (9.65)

in which T ei are nodal temperatures for an element e. Therefore

F eε0 =

∫Ωe

[B]T [D]

εx0εy00

dΩ (9.66)

or

F eε0 =

1∫−1

1∫−1

[B]T [D]

εx0εy00

t(ξ, η) det[J ] dξ dη (9.67)


For a typical node i, we have

F ei ε0 =

F exiF eyi

ε0

=

1∫−1

1∫−1

[Bi]T [D]

εx0εy00

t(ξ, η) det[J ] dξ dη (9.68)

9.5.8 Equivalent nodal loads F ep due to pressure acting nor-mal to the element faces

The distributed pressure loads are assumed to act normal to the facesof the element as shown in Fig. 9.2. A quadrilateral element has four facesF1, F2, F3, F4 corresponding to ξ = −1, ξ = 1, η = −1 and η = 1. Therefore,the details of the computations of F ep must be considered for all fourfaces. If Γe∗ is a face or side of the element, then

F ep =

∫Γe∗

[N ]T p dΓ (9.69)

Let

p =

pxpy

=

lxly

p(ξ, η) (9.70)

in which lx, ly are the direction cosines of the unit exterior normal to theboundary Γe∗ on which pressure p(ξ, η) is acting. We need to consider Γe∗ tobe Γei ; i = 1, . . . , 4, that is faces Fi ; i = i = 1, . . . , 4. As an illustrationconsider face F2 of the element e. On this face, ξ = +1 is constant. A vector~t2 tangent to the face F2 is given by

~t2 =

∂x∂η∂y∂η

0

(9.71)

Exterior normal to the face F2 (i.e. ~n2˜ ) can be obtained by taking cross

product of ~t2 with ~k, a unit vector in the z direction

~n2˜ =

∣∣∣∣∣∣∣i j k∂x∂η

∂y∂η 0

0 0 1

∣∣∣∣∣∣∣ =

∂x∂η

−∂y∂η

0

(9.72)

Therefore

~n2 =

~n2˜||~n2˜ ||

=1√(

∂x∂η

)2+(∂y∂η

)2

∂x∂η

−∂y∂η

0

=

lxlylz

(9.73)


and

dΓ =√

(dx)2 + (dy)2 (9.74)

or

dΓ =

√(∂x∂ξ

dξ +∂x

∂ηdη)2

+(∂y∂ξ

dξ +∂y

∂ηdη)2

(9.75)

For face F2 (ξ = 1), we have

dΓ =

(√(∂x∂η

)2+(∂y∂η

)2)dη (9.76)

Substituting from (9.73) and (9.76) into (9.69) for face F2 and definingF eF2

p as equivalent load vector due to pressure on face F2, we obtain

F eF2p =

1∫−1

[N ]Tξ=1

1√(∂x∂η

)2+(∂y∂η

)2

∂y∂η

−∂x∂η

√(∂x∂η

)2+(∂y∂η

)2p(ξ, η) dη

(9.77)or

F eF2p =

1∫−1

[N ]Tξ=1

∂y∂η

−∂x∂η

p(ξ, η) dη (9.78)

p(ξ, η) is known on face F2 (applied pressure loading). We note thatpositive pressure acts in the direction of the unit exterior normal to the face(i.e. away from the face). Expressions similar to (9.78) can also be derivedfor faces F1, F3 and F4. Final expressions for all four faces are given in theTable 9.1.

23

4

1

9

8 9

7

567

8

1 32

4

56

η

y

x

η = 1

η = −1

~n2

~t2

ξ = 1 face

η = −1 face

ξ = 1ξ = −1ξ = −1 face

η = 1 face

ξ

p(ξ, η)

F2 (Γe2)

F3 (Γe3)

F1 (Γe2)

F4 (Γe4)

Element in xy spaceElement map in ξ, η space(Element faces F1, . . . , F4)

Figure 9.2: Pressure loads acting normal to the element faces in physical coordinatespace and the element map in natural coordinate space


Table 9.1: Expressions for equivalent nodal forces F ep for faces Fi ; i = 1, 2, 3, 4

Face F ep

F1 F eF1p =

1∫−1

[N ]Tξ=−1

−∂y∂η∂x∂η

p(ξ, η) dη

F2 F eF2p =

1∫−1

[N ]Tξ=1

∂y∂η

−∂x∂η

p(ξ, η) dη

F3 F eF3p =

1∫−1

[N ]Tη=−1

∂y∂ξ

−∂x∂ξ

p(ξ, η) dη

F4 F eF4p =

1∫−1

[N ]Tη=1

−∂y∂ξ∂x∂ξ

p(ξ, η) dη

9.6 Summary

In this chapter, derivations of the details of finite element formulationshave been presented for linear structural and linear solid mechanics usingthe principle of minimum potential energy. It is shown that minimization oftotal potential energy is identical to Galerkin method with weak form for self-adjoint differential operators (A is linear and A∗ = A) that are encounteredin linear structural and linear solid mechanics. In this approach, we directlywrite a statement of total potential energy from physical consideration andthus avoid using differential mathematical models as well as integral formsderived using GM/WF. In section 9.4, a general formulation is presented forvarious matrices and vectors resulting from this approach that can be usedfor any application in linear structural and linear solid mechanics (listed atthe end of section 9.4). To illustrate how to utilize the general formulation ofsection 9.4 for specific applications, details of the finite element formulationfor a plane stress problem are presented in section 9.5.

10

Linear and Nonlinear SolidMechanics using thePrinciple of Virtual

Displacements

10.1 Introduction

In the study of continuum mechanics [1,2] and variational methods [3–5],we often use principle of virtual work to derive Hamilton’s principle in La-grangian description and Euler-Lagrange equations in Lagrangian as wellas Eulerian descriptions which in fact are the momentum equations in therespective descriptions. In solid mechanics we refer to these as equations ofequilibrium. The principle of virtual work is applicable to linear as well asnon-linear processes. This is rather obvious from the resulting momentumequations that hold for all processes that are linear as well as non-linear.We have seen in the earlier chapters in this book that once we have a math-ematical description of a physical process in terms of a differential model,i.e. a system of partial differential equations, the formulations of the finiteelement computational process using methods of approximation, discretiza-tions and local approximations is rather straight forward. Many of the detailsand intricacies in this approach for non-linear differential operators describ-ing reversible processes encountered in solid mechanics can be avoided ifwe use principle of virtual work to derive details of the finite element pro-cesses. In doing this we utilize the principle of virtual work in deriving thefinite element formulation at a much earlier stage, long before the differentialmathematical models are extracted from the equations of virtual work.

This approach eliminates the use of PDEs and hence construction ofintegral form from the PDEs in the mathematical models as the statementof virtual work itself is an integral statement. Thus, PDEs are not requiredwhen using principle of virtual work. If we pursue the principle of virtualwork to derive differential models then the result is momentum equations orequations of equilibrium. These equations are a statement of force balanceand hence cannot differentiate between reversible and irreversible processes.

625

626 LINEAR AND NONLINEAR SOLID MECHANICS

As we know, in a reversible process the rate of mechanical work on a volumeof matter does not result in entropy production. Whereas in an irreversibleprocess the rate of mechanical work results in rate of entropy productionwhich in turn influences specific internal energy. To describe this physicswe need energy equation and entropy inequality. This is beyond the scopeof virtual work. Thus, within the thermodynamic framework consisting ofconservation and balance laws, the principle of virtual work only addressesmomentum equations hence can only address reversible processes as theirreversibility requires other balance laws. More specifically, in this approachstrictly speaking we can only consider restricted class of solid matter such aselastic solids. The thermoelastic solids with and without memory cannot beconsidered within the framework of principle of virtual work as such solidshave mechanisms of dissipation, hence entropy production due to mechanicalrate of work must be accounted for. The elastic behavior can be linear ornonlinear. Furthermore, since the momentum equations resulting from theprinciple of virtual work are valid for linear as well as nonlinear processes aslong as additional physics of rate of entropy is not considered, this allows usto consider elastoplastic deformation using principle of virtual work so longas we only consider material nonlinearly due to plasticity without accountingfor rate of entropy production. This is rather popular and straight forwardapproach to incorporate elastoplastic deformation in finite element processesusing principle of virtual work.

10.2 Principle of virtual displacements

Definition 10.1. When a deformed solid in equilibrium is subjected to vir-tual displacements then the virtual work done by these virtual displacementsin moving through the actual forces is zero. This is known as the principleof virtual displacements (see Reddy [1, 6, 7]).

Thus, virtual displacements are admissible displacements such that dueto applications of these the equilibrium of the deformed body is not dis-turbed. This allows us to use principle of virtual work to derive equationsdescribing the stable equilibrium state of the deformed solids. In this chap-ter, we use principle of virtual work to derive details of the finite elementcomputational process for finite deformation and finite strain solid mechanicsapplications in which the deformation process is reversible.

When solid continua are disturbed this results in kinetic energy, strainenergy due to deformation, strains, and stresses, and work done by bodyforces and the surface tractions. In this chapter we only consider stationaryprocesses in which kinetic energy can be neglected, i.e. the virtual work dueto the inertial forces can be neglected. In most of the derivations presentedhere we only consider Lagrangian description.

10.3. VIRTUAL WORK STATEMENTS 627

10.3 Virtual work statements

Let V (x, t) be the volume of the solid matter in the reference con-figuration in which x are the coordinates of the material points at timet = 0. Let V be the deformed volume at time t (current configuration)with deformed material point locations x = x+ u(x, t); u(x, t)being displacements of the material point at location x in the referenceconfigurations.

Consider the work done by all forces on the volume V over certain virtualdisplacements δu defined as a function of the deformed position x. Sincethe volume of matter V with surface A is in equilibrium under the actionof body forces and surface forces, the virtual work due to these forces mustequilibrate with the virtual work due to internal stress field. Let

φ =

uvw

(10.1)

in which u, v and w are displacements of a material point at location xin the reference configuration. Let δφ represent the variation of φ i.e.virtual displacements at material point x.

δφ =

δuδvδw

(10.2)

Consider the virtual work done by the body forces, distributed loads on thesurface of the body and the internal stress field. Then we can write thefollowing in Lagrangian description using principle of virtual work for finitedeformation.∫

VδφTρF b dV +

∫AδφT

(dAdA

)p dA =

∫VδεT σ dV (10.3)

in which σ and ε are conjugate stress and strain tensors. We considerσ to be the second Piola-Kirchhoff stress tensor and ε, the Green’sstrain tensor. The second Piola-Kirchhoff stress measure is valid for finitedeformation [2]. In (10.3), the first term on the left side is the virtual workdue to body forces. The second term is the virtual work due to distributedpressure loads and the right side is the virtual work of the internal stressfield. Traditionally in the principle of virtual work we continue further with(10.3) to derive the associated Euler-Lagrange equations (in this case in theabsence of inertial effects), but here instead we begin derivation of the detailsof finite element processes using (10.3).


Let ΩT be the discretization of V such that

ΩT =⋃e

Ωe (10.4)

in which Ωe is the domain of an element e such that Ωe = Ωe∪Γe, closure ofdomain Ωe with closed boundary (or surface) Γe. Note that henceforth anover bar on Ω or Ωe does not refer to the current or deformed configurationbut rather closures of Ω and Ωe. Likewise, we consider AT , discretization ofsurface or area A bounding volume V

AT =⋃e

Γe (10.5)

in which Γe is the surface or boundary of Ωe and

ΩT = ΩT ∪AT =(⋃

e

Ωe)∪(⋃

e

Γe)

(10.6)

We can write (10.1) over ΩT

∫ΩT

δφhTρF b dV +

∫AT

δφT(dAdA

)p dA =

∫ΩT

δεT σ dV (10.7)

or∑e

∫Ωe

δφehTρF b dV +∑e

∫Γe

δφehT(dAdA

)p dA =

∑e

∫Ωe

δεT σ dV

(10.8)φh is approximation of φ over discretization ΩT and φeh is local ap-proximation of φ over Ωe.

Consider local approximation φeh over Ωe

φeh =

uehvehweh

= [N ]δe and δφeh = [N ]δδe (10.9)

in which φeh is the local approximation of displacements φT = [u v w]over Ωe. In (10.35), [N ] and δe are defined by

[N ] =

[N˜ ] [0] [0]

[0] [N˜ ] [0]

[0] [0] [N˜ ]

; δe =

uevewe

(10.10)


[N ] is the local approximation matrix and [N˜ ] are local approximation func-

tions for ueh, veh and weh. ue, ve and we are nodal degrees of freedomfor ueh, veh and weh. δe are the total degrees of freedom for element e inthe local approximation (10.9). We also need δε i.e. variation of Green’sstrain in (10.7) or (10.8).

The Green’s strain ε can be written as

ε =

εxxεyyεzz2εyz2εzx2εxy

=

∂ueh∂x∂veh∂y∂weh∂z

∂veh∂z +

∂weh∂y

∂weh∂x +

∂ueh∂z

∂ueh∂y +

∂veh∂x

+

12

((∂ueh∂x

)2+(∂veh∂x

)2+(∂weh∂x

)2)12

((∂ueh∂y

)2+(∂veh∂y

)2+(∂weh∂y

)2)12

((∂ueh∂z

)2+(∂veh∂z

)2+(∂weh∂z

)2)∂ueh∂y

∂ueh∂z +

∂veh∂y

∂veh∂z +

∂weh∂y

∂w∂z

∂ueh∂x

∂ueh∂z +

∂veh∂x

∂veh∂z +

∂weh∂x

∂w∂z

∂ueh∂x

∂ueh∂y +

∂veh∂x

∂veh∂y +

∂weh∂x

∂w∂y

(10.11)

or

ε = εl+ εn (10.12)

Let

θxT =

[∂ueh∂x

,∂veh∂x

,∂weh∂x

]θyT =

[∂ueh∂y

,∂veh∂y

,∂weh∂y

]θzT =

[∂ueh∂z

,∂veh∂z

,∂weh∂z

] (10.13)

and

θT =[θxT , θyT , θzT

](10.14)

Using (10.14) we can write

ε = εl+ εn = [H]θ+1

2[Aθ]θ (10.15)

in which

[H] =

1 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 00 0 0 0 0 0 0 0 10 0 0 0 0 1 0 1 00 0 1 0 0 0 1 0 00 1 0 1 0 0 0 0 0

(10.16)


and

[Aθ] =

θxT 0T 0T

0T θyT 0T

0T 0T θzT

0T θzT θyT

θzT 0T θxT

θyT θxT 0T

(10.17)

In (10.15) εl and εn are linear and nonlinear strains.Consider

εl = [H]θ (10.18)

We define

θ = [G]δe =

[Gx]

[Gy]

[Gz]

uevewe

(10.19)

in which

[Gx] =

∂[N˜ ]

∂x [0] [0]

[0]∂[N˜ ]

∂x [0]

[0] [0]∂[N˜ ]

∂x

(10.20)

[Gy] =

∂[N˜ ]

∂y [0] [0]

[0]∂[N˜ ]

∂y [0]

[0] [0]∂[N˜ ]

∂y

(10.21)

[Gz] =

∂[N˜ ]

∂z [0] [0]

[0]∂[N˜ ]

∂z [0]

[0] [0]∂[N˜ ]

∂z

(10.22)


εl = [H][G]δe = [Bl]δe (10.23)

Clearly [Bl] in (10.23) is given by

[Bl] =

∂[N˜ ]

∂x [0] [0]

[0]∂[N˜ ]

∂y [0]

[0] [0]∂[N˜ ]

∂z∂[N˜ ]

∂z [0]∂[N˜ ]

∂x∂[N˜ ]

∂y

∂[N˜ ]

∂x [0]

(10.24)


Using (10.19) in (10.15), εn can be written as

εn =1

2[Aθ]θ =

1

2[Aθ][G]δe =

1

2[Bn]δe (10.25)

where [Bn] = [Aθ][G]. Therefore we have the following for ε

ε = εl+ εn =[[Bl] +

1

2[Bn]

]δe (10.26)

Using (10.19) we can write

δθ = [G]δδe (10.27)

and

δεl = [Bl]δδe (10.28)

δεn =1

2δ[Aθ]θ+

1

2[Aθ]δθ (10.29)

=1

2[Aθ]δθ+

1

2[Aθ]δθ (10.30)

= [Aθ]δθ = [Aθ][G]δδe = [Bn]δδe (10.31)

Thereforeδε = δεl+ δεn =

[[Bl] + [Bn]

]δδe (10.32)

orδε = [B]δδe ; [B] = [Bl] + [Bn] (10.33)

Equations (10.26) and (10.32) are fundamental relationships. Now since wehave all the variations, we return to (10.8) for ΩT resulting from the principleof virtual work. Substituting from (10.9) and (10.33) in (10.8)∑

e

∫Ωe

δδeT [N ]TρF b dV +∑e

∫Γe

δδeT [N ]T(dAdA

)p dA

=∑e

∫Ωe

δδeT [B]T σ dV (10.34)

or∑e

δδeT(∫

Ωe

[N ]TρF b dV +

∫Γe

[N ]T(dAdA

)p dA−

∫Ωe

[B]T σ dV)

= 0

(10.35)or

δδT∑e

(∫Ωe

[N ]TρF b dV +

∫Γe

[N ]T(dAdA

)p dA−

∫Ωe

[B]T σ dV)

= 0

(10.36)


whereδδ =

⋃e

δδe (10.37)

δ are the total degrees of freedom for discretization ΩT . Since δ andδδ in (10.36) are arbitrary, the following must hold.∑

e

∫Ωe

[N ]TρF b dV +∑e

∫Γe

[N ]T(dAdA

)p dA =

∑e

∫Ωe

[B]T σ dV

(10.38)Let

Reb =

∫Ωe

[N ]TρF b dV (10.39)

Rep =

∫Γe

[N ]T(dAdA

)p dA (10.40)

P e =

∫Ωe

[B]T σ dV (10.41)

where Reb, Rep are nodal load vectors due to body forces and surfaceloads and P e is the vector of internal forces due to stress field. Using(10.39) - (10.41) in (10.38) we can write∑

e

Reb+∑e

Rep =∑e

P e (10.42)

orRb+ Rp = P (10.43)

orR = P (10.44)

orψ = R − P (10.45)

For nonlinear problems (such as finite deformations, plasticity) both Rand P can be functions of δ, hence we must find a solution δ thatsatisfies the residual vector ψ = 0 iteratively. It is preferable to use thefollowing form for (10.45).

ψ = R −∑e

∫Ωe

[B]T σ dV (10.46)

Thus, for an element e of ΩT , we need only to determine Reb, Rep andP e defined by (10.39) - (10.41). Reb and Rep pose no particular problemas these are loads, hence are vectors containing numbers. We consider detailsof P e in (10.41) in the following.


10.3.1 Stiffness matrix

Consider P e in (10.41). Let us assume that the total stress σ isrelated to the total strain ε through

σ = [Ds]ε (10.47)

Substituting for ε from (10.25)

σ = [Ds][[Bl] +

1

2[Bn]

]δe. (10.48)

Substituting from (10.34) and (10.48) in (10.41)

P e =

∫Ωe

[[Bl] + [Bn]

]T[Ds]

[[Bl] +

1

2[Bn]

]δe dV (10.49)

or

P e =

[∫Ωe

[[Bl] + [Bn]

]T[Ds]

[[Bl] +

1

2[Bn]

]dV

]δe (10.50)

orP e = [Ke

s ]δe (10.51)

[Kes ] is called the element secant stiffness matrix first derived by Oden [8,

9]. [Kes ] is non-symmetric even if [Ds] is symmetric. Use of [Ke

s ] requiresnon-symmetric algebraic equations solver and it is restrictive due to thestress-strain relationship (10.47). Instead of using the total stress-total strainrelationship, we can consider an incremental form

δσ = [DT ]δε (10.52)

Use of (10.52) in (10.41) requires that we derive an incremental form of(10.41), i.e we need to consider δP e.

δP e = δ

(∫Ωe

[B]T σ dV)

(10.53)

or

δP e =

∫Ωe

(δ[B]T σ+ [B]T δσ

)dV (10.54)

where∫Ωe

[B]T δσ dV =

∫Ωe

[B]T [DT ] δε dV =

∫Ωe

[B]T [DT ][B] δδe dV

=

[∫Ωe

[B]T [DT ][B] dV

]δδe = [Ke] δδe (10.55)


and ∫Ωe

δ[B]T σ dV =

∫Ωe

δ(

[Bl]T + [Bn]T)σ dV

=

∫Ωe

δ[Bn]T σ dV

=

∫Ωe

δ[[Aθ][G]

]T σ dV=

∫Ωe

[GT ] δ[Aθ]T σ dV (10.56)

We can show that

δ[Aθ]T σ = [S] δθ = [S][G] δδe (10.57)

where

[S] =

σxx[I] σxy[I] σxz[I]

σyy[I] σyz[I]

σzz[I]

(10.58)

Therefore∫Ωe

δ[B]T σ dV =

∫Ωe

[G]T [S][G] δδe dV

=

[∫Ωe

[G]T [S][G] dV

]δδe = [Ke

σ] δδe (10.59)

Substituting from (10.55) and (10.59) into (10.54)

δP e =[[Ke] + [Ke

σ]]δδe (10.60)

Furthermore

[Ke] =

∫Ωe

[[Bl] + [Bn]

]T[DT ]

[[Bl] + [Bn]

]dV (10.61)

= [Kel ] + [Ke

n] (10.62)

in which

[Kel ] =

∫Ωe

[Bl]T [DT ][Bl] dV (10.63)

10.4. SOLUTION METHOD 635

and

[Ken] =

∫Ωe

[[Bl]T [DT ][Bn] + [Bn]T [DT ][Bl] + [Bn]T [DT ][Bn]

]dV (10.64)

[Kel ] is the usual small displacement stiffness matrix. [Ke

n] is termed as initialdisplacement matrix by Marcal [10]. Using (10.62) in (10.60) we finally have

δP e =

[[Ke

l ] + [Ken] + [Ke

σ]

]δδe = [Ke

T ] δδe (10.65)

Matrix [Keσ] is influence of the stress field on the element stiffness. The

derivation given above for finite deformation and finite strain has also beenpresented in references [6, 11–18]. [Ke

T ] is called element tangent stiffnessmatrix. [Ke

T ] is obviously symmetric if [DT ] is symmetric.

10.4 Solution method

We note that we are faced with finding a solution δ, nodal degrees offreedom for discretization ΩT that satisfies

ψ(δ)

=R(δ)

−P (δ)

= 0 (10.66)

in which ψ(·) is a nonlinear function of δ. Thus, we must use (10.66) tofind a solution δ iteratively. At this stage many strategies are possible. Inthe following we only discuss Newton’s linear method or Newton–Raphsonmethod.

Let δ0 be a known solution, a guess or generally a null vector or estab-lished using linear solution, then

ψ (δ0)6= 0 (10.67)

Let ∆δ be correction to δ0 such thatψ(δ0+ ∆δ)

= 0 (10.68)

Expand ψ(·) in (10.68) in Taylor series about δ0 and retain only up tolinear terms in ∆δ (Newton’s linear method).

ψ(δ0+ ∆δ)

∼= ψ(δ0)

+∂ψ∂δ

∣∣∣∣δ0∆δ = 0 (10.69)

or

∆δ = −[∂ψ∂δ

]−1

δ0

ψ(δ0)

(10.70)


New solution δ is obtained using

δ = δ0+ ∆δ (10.71)

Recall thatψ = R − P (10.72)

If we assume that R is not a function of δ i.e the loads are conservative(maintain their direction during deformation process), then

∂ψ∂δ

= −∂P∂δ

= −[KT ] = −∑e

[KeT ] (10.73)

Using (10.73) in (10.70) and then (10.70) in (10.71)

δ = δ0 −[∑

e

[KeT

]]−1

δ0

ψ(δ0

)(10.74)

Convergence of the iterative process is checked by examiningψ(δ)

forproximity to null vector.

10.4.1 Summary of solution procedure

1. Assume δ0.2. Calculate [Ke

T ], Reb, Rep, P e.3. Assemble

R =∑e

Reb+ Rep and P =∑e

P e

4. Formψ = R − P

5. Check for convergence

(a) |ψi| ≤ ∆1, i = 1, . . . , n∗, n∗ = total number of degrees of freedom

(b) ∆1: preset tolerance for zero

(c) If converged: stop

(d) If not converged then follow steps 6-8

6. Calculate ∆δ and δ using

∆δ = −[∑

e

[KeT

]]−1

δ0

ψ(δ0

)δ = δ0+ ∆δ

10.5. FINITE ELEMENT FORMULATION FOR 2D SOLID CONTINUA 637

7. Calculate [KeT ], Reb, Rep, P e using δ.

8. Form

ψ = R − P

Set δ0 to δ and go to step 5.

10.5 Finite element formulation for 2Dsolid continua

In this section we present details of the finite element formulation forisotropic, homogeneous 2D continua (plane stress or plane strain) in La-grangian description for finite deformation. We follow the general derivationpresented in earlier sections that holds in R1, R2 and R3 and for any kine-matic description. In this section we are seeking to establish specific forms of[N ], [Bl], [Bn], hence [B], θ, [Aθ], [G], [Ke

l ], [Ken], [Ke

σ], hence [KeT ] and [S]

for 2D continua. Consider a discretization ΩT =⋃e Ωe of a two-dimensional

domain Ω. Let a typical element e with domain Ωe = Ωe⋃

Γe be a nine-nodep-version element with distorted faces in xy space [see Fig. 10.1(a)]. A mapΩξη of this element (master element) in the natural coordinate space ξη ina two-unit square is shown in Fig. 10.1(b). Mapping of Ωξη, master elementin ξη space into Ωe in xy space can be described using the following (or anyother suitable form):

x(ξ, η) =9∑i=1

Ni(ξ, η)xi

y(ξ, η) =9∑i=1

Ni(ξ, η) yi

(10.75)

Ni(ξ, η) are Lagrange family 2D shape functions of class C00(Ωe). Let ueh, veh

be the local approximations of u and v over Ωξη:

ueh = [N˜ (ξ, η)]ue

veh = [N˜ (ξ, η)]ve(10.76)


23

4

567

8

1 1 2 3

567

8 99

42

2

(a) (b)

η

ξ

y

x

Map of Ωe in the natural coordi-nate space ξ, η (i.e. Ωξη)

Quadrilateral element in xy-space

ΩeΩξη

Figure 10.1: A quadrilateral element Ωe and its map Ωξη in natural coordinate spaceξ, η

where [N˜ ] is (1 × n) approximation function matrix in which n = (pξ +

1)(pn + 1); pξ and pη being p-levels in ξ and η directions. ue and veare degrees of freedom for ueh and veh. Equations (10.76) can be combined inmatrix form as

φeh =

uhvh

=

[[N˜ ] [0]

[0] [N˜ ]

](2×2n)

δe

(2n×1)=

[[N˜ ] [0]

[0] [N˜ ]

]ueve

= [N ]δe

(10.77)where

δe =

ueve

(10.78)

We note that δφeh = [N ]δδe. Let

θ =

θxθy

=

∂ueh∂x∂veh∂x

∂ueh

∂y∂veh∂y

= [G][δe] =

[[Gx][Gy]

]ueve

(10.79)

in which

[Gx] =

∂[N˜ ]

∂x [0]

[0]∂[N˜ ]

∂x

and [Gy] =

∂[N˜ ]

∂y [0]

[0]∂[N˜ ]

∂y

(10.80)

ε =

εxxεyy2εxy

=

∂ueh∂x∂veh∂y

∂ueh∂y +

∂veh∂x

+

12

((∂ueh∂x

)2+(∂veh∂x

)2)

12

((∂ueh∂x

)2+(∂veh∂y

)2)

∂ueh∂x

∂ueh∂y +

∂veh∂x

∂veh∂y

(10.81)


or

ε = εl+ εn (10.82)

εl = [H]θ = [H][G]δe = [Bl]δe (10.83)

in which

[H] =

1 0 0 00 0 0 10 1 1 0

(10.84)

Clearly

[Bl] =

∂[N˜ ]

∂x [0]

0∂[N˜ ]

∂y∂[N˜ ]

∂y

∂[N˜ ]

∂x

(10.85)

Let

[Aθ] =

θxT 0T0T θyTθyT θxT

(10.86)

Then, following the derivation in section 10.3 we can write

εn =1

2[Aθ]θ =

1

2[Aθ][G]δe =

1

2[Bn]δe (10.87)

where

[Bn] = [Aθ][G] (10.88)

δεl = [Bl]δδe (10.89)

δεn = [Bn]δδe (10.90)

Therefore

δε = [B]δδe =[[Bl] + [Bn]

]δδe (10.91)

Consider incremented form of the constitutive equations

δσ = [DT ] δε (10.92)

or

δ

σxxσyyσxy

=

D11 D12 0D12 D22 0

0 0 D33

δεxεyγxy

, γxy = 2εxy (10.93)

If we only consider linear elastic material, then Dij for plane stress and planestrain are given by [1, 2]


Plane Stress:

D11 = D22 =E

1− ν2

D12 =νE

1− ν2

D33 =E

2(1 + ν)= G

(10.94)

Plane Strain:

D11 = D22 =E(1− ν)

(1 + ν)(1− 2ν)

D12 =ν

1− νD11

D33 =E

2(1 + ν)= G

(10.95)

In which E is modulus of elasticity, ν is Poisson’s ratio, and G is shearmodulus. Now we can determine [Ke

T ], tangent stiffness matrix

[KeT ] =

∫Ωe

[B]T [DT ][B] dV + [Keσ] (10.96)

or

[KeT ] =

∫Ωe

[[Bl]T + [Bn]T

][DT ]

[[Bl] + [Bn]

]dV + [Ke

σ] (10.97)

We use the element map in the natural coordinate system ξ, η [see Fig. 10.2(b)].See chapter 8 for details.

dV = t(ξ, η) det[J ] dξ dη (10.98)

[J ] =

[x, y

ξ, η

]=

[∂x∂ξ

∂x∂η

∂y∂ξ

∂y∂η

](10.99)

∂N˜i∂x∂N˜i∂y

= [JT ]−1

∂N˜i∂ξ∂N˜i∂η

, i = 1, 2, . . . , n (10.100)

t(ξ, η) is thickness of the plate for plane strain case. In case of plane straint(ξ, η) = 1. Now we can write [Ke

T ] in (10.96) as

[KeT ] =

1∫−1

1∫−1

[[Bl]T + [Bn]T

][DT ]

[[Bl] + [Bn]

]t(ξ, η) det[J ] dξ dη + [Ke

σ]

(10.101)


[Bl, [Bn] and [DT ] are defined in (10.85), (10.88) and (10.94), (10.95). Wekeep in mind that [Ke

T ] is to be calculated at δe0, assumed solution, thus[Aθ] used in (10.101) is defined. Calculations of load vectors due to bodyforces and conservative surface loads is rather straight forward. We also notethat [Ke

T ] can be written as

[KeT ] = [Ke

l ] + [Ken] + [Ke

σ] (10.102)

in which

[Kel ] =

1∫−1

[Bl]T [DT ][Bl] t(ξ, η) det[J ] dξ dη (10.103)

[Ken] =

1∫−1

([Bl]T [DT ][Bn]+[Bn]T [DT ][Bl]

+[Bn]T [DT ][Bn] t(ξ, η) det[J ] dξ dη (10.104)

[Keσ] =

1∫−1

[G]T [S][G] t(ξ, η) det[J ] dξ dη (10.105)

and

[S] =

[σxx[I2] σxy[I2]σxy[I2] σyy[I2]

], [I2] =

[1 00 1

](10.106)

10.6 Finite element formulation for 3Dsolid continua

The details of the finite element formulation presented in section 10.4clearly holds for 3D solid continua for finite deformation as well. Basedon the choice of element shape, i.e. hexahedron, tetrahedron, etc, we canaddress the remaining details. Consider a 27-node distorted hexahedronelement in x, y, z space [see Fig. 10.2(a)] mapped into a two-unit cube inξ, η, ζ natural coordinate space with the origin of ξ, η, ζ coordinate systemat the center of the element [see Fig. 10.2(b)]. Let

φeh =

uehvehweh

= [N˜ (ξ, η, ζ)]δe = [N˜ ]

uevewe

; δφeh = [N˜ ]δδe

(10.107)be the local approximations of u, v, w over an element Ωe with its map innatural coordinate space. See chapter 8 for details of [N˜ (ξ, η, ζ)] functions.


Following the details of chapter 8 we have the following mapping of pointsbetween element mapped in x, y, z and ξ, η, ζ spaces.

x(ξ, η, ζ) =

27∑i=1

Ni(ξ, η, ζ)xi

y(ξ, η, ζ) =27∑i=1

Ni(ξ, η, ζ) yi

z(ξ, η, ζ) =27∑i=1

Ni(ξ, η, ζ) zi

(10.108)

where Ni(ξ, η, ζ) are Lagrange family of 3D shape functions, and [N˜ i] in[N˜ ] can be any suitable choice such as p-version hierarchical. Details of [G],

[Aθ], [Bl], [Bn] etc. have already been presented in section 10.3. Calculationof element tangent matrix [Ke

T ] requires constitutive equations. Considerincremental form of the constitutive relations.

δσ = [DT ]δε or δε = [CT ]δσ (10.109)

ζ

ξ

η

x

y

zA 27-node element in x, y, z space(a) (b) Element map in natural coor-

dinate space in ξ, η, ζ

Figure 10.2: A 27-node three-dimensional hexahedron element

The components of ε and σ are arranged as in (10.11). The materialmatrix [CT ] for isotropic, homogeneous elastic matter can be written as

[CT ] =

C11 C12 C13 0 0 0C21 C22 C23 0 0 0C31 C32 C33 0 0 00 0 0 C44 0 00 0 0 0 C55 00 0 0 0 0 C44

(10.110)


in which C21 = C12, C31 = C13, C32 = C23 and

C11 = C22 = C33 =1

E

C12 = C13 = C23 =−νE

(10.111)

and [DT ] = [CT ]−1. The element tangent stiffness matrix [KeT ] is given by

[KeT ] =

∫Ωe

[B]T [DT ][B] dV + [Kσ] (10.112)

Considering the element map in ξ, η, ζ coordinate space

dV = |J | dξ dη dζ

and

[J ] =

[x, y, z

ξ, η, ζ

]=

xξ xη xζyξ yη yζzξ zη zζ

(10.113)

In (10.113) the subscripts ξ, η, ζ indicate differentiations with respect tothese and since Ni = Ni(ξ, η, ζ) ; i = 1, 2, . . . , n we have

∂N˜i∂x∂N˜i∂y∂N˜i∂z

= [JT ]−1

∂N˜i∂ξ∂N˜i∂η∂N˜i∂ζ

, i = 1, 2, . . . , n (10.114)

Finally, we have the following for the tangent stiffness matrix for element e,

[KeT ] =

1∫−1

1∫−1

1∫−1

[B]T [DT ][B] |J | dξ dη dζ + [Kσ] (10.115)

in which

[B] = [Bl] + [Bn] (10.116)

Also if we substitute (10.116) in (10.115), we obtain

[KeT ] = [Ke

l ] + [Ken] + [Ke

σ] (10.117)

The matrices in (10.116) and (10.117) are defined in section 10.3. Using the


element map in ξ, η, ζ coordinate space we can define these as

[Kel ] =

1∫−1

1∫−1

1∫−1

[Bl]T [DT ][Bl] |J | dξ dη dζ (10.118)

[Ken] =

1∫−1

1∫−1

1∫−1

([Bl]T [DT ][Bn] + [Bn]T [DT ][Bl] + [Bn]T [DT ][Bn]

)|J | dξ dη dζ

(10.119)

[Keσ] =

1∫−1

1∫−1

1∫−1

[G]T [S][G] |J | dξ dη dζ (10.120)

(10.121)

in which

[S] =

σxx[I] σxy[I] σxz[I]σxy[I] σyy[I] σyz[I]σxz[I] σyz[I] σzz[I]

(10.122)

where [I] is a (3× 3) identity matrix.

10.7 Axisymmetric solid finite elements

These finite element formulations can be used to study axisymmetricdeformation in bodies of revolution due to axisymmetric loads. The defor-mation field in such cases is independent of the circumferential coordinateθ. In the following we choose x to be radial direction r and y to be axialdirection z of a (r, θ, z) cylindrical coordinate system. Figure 10.3 shows anine-node p-version element in x, y space. Let u, v be the displacements inx, y directions at a point P (x, y) in Ωe or P (ξ, η) in Ωξη. In the following weconsider finite element formulation of axisymmetric finite elements for finitedeformation in Lagrangian description. Mapping of geometry from x, y toξ, η space is as usual.

xy

=

9∑i=1

N(ξ, η)

xiyi

(10.123)

N(ξ, η) are standard 2D Lagrange interpolation functions. Consider an ele-ment e with domain Ωe = Ωe∪Γe in which Γe is the boundary of the element.The local approximation φeh of the displacement field φT = [u v] overΩe (or Ωξη) can be written using p-version hierarchical local approximation(see chapter 8).

φeh =

uehveh

= [N˜ ]δe = [N˜ ]

ueve

=

[[N˜ ] [0][0] [N˜ ]

]ueve

(10.124)

10.7. AXISYMMETRIC SOLID FINITE ELEMENTS 645

and

δφeh = [N˜ ]δδe

[N˜ ] is the approximation function matrix and [N˜ i] are local approximationfunctions (N˜ i(ξ, η); i = 1, 2, . . . , n) used in defining sub-matrices of [N˜ ].The Jacobian of mapping is given by

[J ] =[x,yξ,η

]=

[xξ xηyξ yη

](10.125)

and ∂N˜i∂x∂N˜i∂y

= [JT ]−1


, i = 1, 2, . . . , n (10.126)

(radial r)x

y

Ωe

ηv

u

ξ

P (x, y)P (ξ, η)

(axial z)

Figure 10.3: A nine-node p-version element

The Green’s strain tensor ε can be written as

ε =

εxxεyyγxyεθ

=

∂ueh∂x∂veh∂y

∂ueh∂x +

∂veh∂y

uehx

+

12

((∂ueh∂x

)2+(∂ueh∂y

)2)12

((∂veh∂x

)2+(∂veh∂y

)2)∂ueh∂x

∂veh∂x +

∂ueh∂y

∂veh∂y

12

(uehx

)2

=εl+ εn

(10.127)in which εθ is the circumferential strain. Let σT = [σxx, σyy, σxy, σθ] bethe second Piola-Kirchhoff stress tensor. We consider incremental stress-strain relations between σ and ε

δσ = [DT ]δε (10.128)


where

[DT ] =

D11 D12 0 D14

D21 D22 0 D24

0 0 D33 0D41 D22 0 D44

(10.129)

in which D21 = D12, D41 = D14, D42 = D24.

[DT ] =E(1− ν)

(1 + ν)(1− 2ν)

1 ν

1−ν 0 ν1−ν

1 0 ν1−ν

sym.1−2ν

2(1−ν) 0

1

(10.130)

Let

θ =

θxθyuehx

; θx =

∂ueh∂x∂veh∂x

; θy =

∂ueh∂y∂veh∂y

(10.131)

and

θ = [G]δe =

[Gx][Gy][G3]

δe (10.132)

in which

[Gx] =

∂[N˜ ]

∂x [0]

[0]∂[N˜ ]

∂x

(10.133)

[Gy] =

∂[N˜ ]

∂y [0]

[0]∂[N˜ ]

∂y

(10.134)

[G3] =[

1x [N˜ ] [0]

](10.135)

Using (10.127) and (10.132) we can write εl as

εl = [H]θ = [H][G]δe = [Bl]δe (10.136)

[H] =

1 0 0 0 00 0 0 1 00 1 1 0 00 0 0 0 1

(10.137)

Clearly [Bl] is given by

[Bl] =

∂[N˜ ]

∂x [0]

[0]∂[N˜ ]

∂y∂[N˜ ]

∂y

∂[N˜ ]

∂x1x [N˜ ] [0]

(10.138)

10.7. AXISYMMETRIC SOLID FINITE ELEMENTS 647

Let

[Aθ] =

θxT 0T 00T θyT 0

0 0uehx

(10.139)

Following the derivation in section 10.3 we can write

εn =1

2[Aθ]θ =

1

2[Aθ][G]δe =

1

2[Bn]δe (10.140)

where

[Bn] = [Aθ][G] (10.141)

and

δεl = [Bl] δδe (10.142)

δεn = [Bn] δδe (10.143)

Therefore

δε = [B] δδe =[[Bl] + [Bn]

]δδe (10.144)

Element tangent stiffness matrix [KeT ] is given by

[KeT ] =

∫Ωe

[B]T [DT ][B] dV + [Keσ] (10.145)

We note that for axisymmetric solids

dV = 2πx dx dy = 2πx|J | dξ dη (10.146)

where

[J ] =

[x, y

ξ, η

]=

[xξ xηyξ yη

](10.147)

∂N˜i∂x∂N˜i∂y

= [JT ]−1


; i = 1, 2, . . . , n (10.148)

If we use [B] = [Bl] + [Bn] in (10.145), then we can write

[KeT ] = [Ke

l ] + [Ken] + [Ke

σ] (10.149)


in which

[Kel ] =

1∫−1

1∫−1

[Bl]T [DT ][Bl] 2π x(ξ, η) |J | dξ dη (10.150)

[Ken] =

1∫−1

1∫−1

([Bl]T [DT ][Bn]+[Bn]T [DT ][Bl]

+[Bn]T [DT ][Bn])

2π x(ξ, η) |J | dξ dη (10.151)

[Keσ] =

1∫−1

1∫−1

[G]T [S][G] 2π x(ξ, η) |J | dξ dη (10.152)

in which

[S] =

σxx[I2] σxy[I2] 0

σxy[I2] σyy[I2] 0

0 0 σθ

; [I2] =

[1 00 1

](10.153)

10.8 Summary

The finite element formulations are derived in this chapter using princi-ple of virtual work. These formulations are valid for solid matter undergoingfinite deformation and finite rotations. Incremental form of the constitutivetheory between second Piola-Kirchhoff stress and the Green’s strain tensoryields symmetric tangent stiffness matrix when Newton’s linear method isemployed for obtaining solutions of non-linear algebraic equations. Eventhough the derivation is presented for elastic material, its extension to plas-ticity is quite straight forward. This eventually results in modification of[DT ]. In case of small deformation ε is replaced by εl and the secondPiola-Kirchhoff stress is simply Cauchy stress, hence [Ke

T ] only consists of[Ke

l ] if the influence of stress field on the stiffness defined by [Keσ] is neglected

as done in linear elasticity. References [6,11–17] provide additional materialfor further reading on various finite element formulations such as beams, ax-isymmetric shells, curved shells, etc. for finite deformation, finite rotationsand finite strains. These references also contain many model problems andtheir numerical solutions using the finite element formulations presented inthem that are parallel to the ones considered here. The finite element formu-lations that incorporate large deformation, finite rotations and finite strainsand are based on second Piola-Kirchhoff stress and Green’s strain tensor withincremental elastic constitutive equations (like the ones presented here) aregenerally referred to as geometrically nonlinear finite element formulationsin the published literature.


References for additional reading[1] J. N. Reddy. An Introduction to Continuum Mechanics. Cambridge University Press,

2nd edition, 2013.





[6] J. N. Reddy. An Introduction to Nonlinear Finite Element Analysis. Oxford Univer-sity Press, 2nd edition, 2015.

[7] J. N. Reddy. Energy Principles and Variational Methods in Applied Mechanics. JohnWiley, New York, (3rd in print) 2nd edition, 2002.

[8] J. T. Oden. Numerical solution of nonlinear plasticity problems. J. Struct. Div.,93:235–255, 1967.

[9] J. T. Oden. Finite plane stress of incommpressible elastic solids by finite elementmethod. The Aero. Quart., 19:254–264, 1968.

[10] P. V. Marcal. The effect of initial displacements on problems of large deflection andstability. Technical Report ARPA E54, Brown University, November 1967.

[11] K. S. Surana. Geometrically non-linear formulations for the axi-symmetric shell ele-ments. Int. J. Num. Meth. Eng., 18:447–502, 1982.

[12] K. S. Surana. Geometrically non-linear formulations for the axi-symmetric transitionfinite elements. Comp. Struct., 17(2):243–255, 1983.

[13] K. S. Surana. Geometrically non-linear formulations for the three-dimensional solid-shell transition finite elements. Comp. Struct., 15(5):549–566, 1982.

[14] K. S. Surana. Geometrically non-linear formulations for the two-dimensional curvedbeam elements. Comp. Struct., 17(1):105–114, 1983.

[15] K. S. Surana. Geometrically non-linear formulations for the curved shell elements.Int. J. Num. Meth. Eng., 19:581–615, 1983.

[16] K. S. Surana. Geometrically non-linear formulations with large rotations for finiteelements with rotational degrees of freedom. Comp. Struct., 23(2):279–289, 1986.

[17] K. S. Surana. Geometrically non-linear formulations for the three-dimensional curvedbeam elements with large rotations. Int. J. Num. Meth. Eng., 1987.

[18] K. S. Surana and R. M. Sorem. Three dimensional curved beam elements with largetranslations and rotations. Presented at 12th Canadian Congress of Applied Mechan-ics, May 1989.

11

Additional Topics in LinearStructural Mechanics

11.1 Introduction

In this chapter we consider various finite element formulations for linearstructural mechanics in R1, R2, and R3. The mathematical descriptions forstructural members are momentum equations in Lagrangian description, of-ten referred to as equations of equilibrium as these are statement of forcebalance in R1, R2, and R3. The constitutive theories describing relationsbetween Cauchy stress tensor and infinitesimal strain are generally basedon generalized Hooke’s law. In case of linear structural mechanics the dif-ferential operators appearing in the mathematical descriptions are alwaysself-adjoint, hence Galerkin method with weak form, principle of minimumpotential energy, or Ritz method including principle of virtual work are allideally suited to derive finite element formulations for such applications. Theoutcomes from each one of these methods are identically the same in termsof the resulting finite element formulations and the associated element equa-tions. Generally it is a matter of choice and preference as to which methodone chooses. In this chapter we explore applications of some of these methodsin deriving finite element formulations i.e. element equations for structuralelements in R1, R2, and R3 and consider some simple model problems.

11.2 1D axial spar or rod element in R1 (1D space)

In this section we consider finite element formulations for purely one di-mensional deformation of bars or rods in R1. The resulting finite elementsare called bar, rod, or spar elements in R1. The governing differential equa-tions describing stationary 1D deformation is the momentum equation in theaxial direction. Considering spatial direction x, the following BVP describesthe axial deformation of a bar or a rod in R1 in the absence of body forces:

d

dx

(Edu(x)

dx

)− f(x) = 0 (11.1)

In which E(x) is modulus of elasticity, u(x) is displacement, and f(x) isapplied external load along the length of the rod.

651

652 ADDITIONAL TOPICS IN LINEAR STRUCTURAL MECHANICS

Consider a 1D rod of length L subdivided into four two-node elements[see Fig. 11.1(a)]. The element equations can be derived using GM/WF,minimization of total potential energy (quadratic functional), or principleof virtual work, the final outcome is the same. Consider an element e [seeFig. 11.1(b)] showing nodal displacements ue1 and ue2, nodal internal forces(secondary variables) P e1 and P e2 at local node numbers 1 and 2.

1 2 3 4 5

f(x)

y

1 2 3 4

Px

(a) Finite element discretization of 1D rod

f(x)

y

e

xe+1

P e22

ue2

he

xe

P e1 1

ue1

x

(b) A two node 1D element

Figure 11.1: Discretization of a 1D rod and a typical 2 node element e

We derive the element equations using principle of minimum potentialenergy (or minimization of quadratic functional). In the derivations we con-sider a local coordinate system x, y with its origin at node 1 and the x axispointing from node 1 to node 2 (see Fig. 11.2).

y

x1 2

y

P e2

ue1 ue2

e

P e1 f(x)

he

x

Figure 11.2: Element local coordinate system xy and nodal variables

Figure 11.2 shows an element e with local node numbers 1 and 2 withnodal displacements ue1 and ue2 and internal forces P e1 and P e2 . The localapproximation ueh(x) describing u at a point x in the element can be written

11.2. 1D AXIAL SPAR OR ROD ELEMENT IN R1 (1D SPACE) 653

as

ueh =

2∑i=1

Ni(x)uei = [N ]

ue1ue2

(11.2)

where Ni(x) (i = 1, 2) are nodal local approximation functions and are givenby (see Chapter 8)

N1(x) =

(1− x

he

), N2(x) =

x

he(11.3)

We note thatN1(x1) = 1, N1(x2) = 0

N2(x1) = 0, N2(x2) = 1(11.4)

That is

Ni(xj) =

1, j = i

0, j 6= i, i, j = 1, 2 (11.5)

and [N ] is local approximation function matrix.

11.2.1 Stresses and strains

For purely one dimensional deformation of the element we have

εxx =du

dx, σxx = Eεxx (11.6)

In matrix and vector notation

ε = εxx =

du

dx

=

[d

dx

]u = [L]u

σ = σxx = [D]ε ; [D] = [E]

(11.7)

where E is modulus of elasticity.

11.2.2 Total potential energy: Πe

The total potential energy Πe of an element e is the sum of strain energyand the potential energy of loads, i.e.

Πe = Πe1 + Πe

2 (11.8)

where Πe1 is the strain energy and Πe

2 is the potential energy of loads:

Πe1 =

1

2

∫Ωe

εT σ dΩ =1

2

he∫0

εT σAdx (11.9)


in which A is area of the cross-section of the rod and Ωe = [0, he], the domainof an element e,

Πe2 = work done by P e1 , P

e2 , and f(x)

or

Πe2 = −ue1P e1 − ue2P e2 −

he∫0

u(x)f(x) dx (11.10)

Using (11.9) and (11.10) in (11.8)

Πe =1

2

he∫0

εT σAdx− ue1P e1 − ue2P e2 −he∫

0

u(x)f(x) dx (11.11)

Substituting for ε and σ from (11.7) and replacing u(x) with ueh (localapproximation for element e)

Πe =1

2

he∫0

duehdx

TE

duehdx

Adx− ue1P e1 − ue2P e2 −

he∫0

u(x)f(x) dx (11.12)

Substituting ueh(x) from (11.3)

Πe =1

2

he∫0

[dN1

dx

dN2

dx

]ue1ue2

TE

[dN1

dx

dN2

dx

]ue1ue2

Adx

− ue1P e1 − ue2P e2 −he∫

0

(N1ue1 + N2u

e2)f(x) dx (11.13)

or

Πe =1

2

he∫0

[ ue1 ue2 ]

dN1

dx

dN1

dx

dN1

dx

dN2

dx

dN2

dx

dN1

dx

dN2

dx

dN2

dx

ue1ue2EAdx


0

(N1ue1 + N2u

e2)f(x) dx (11.14)


or

Πe = [ ue1 ue2 ]

1

2

he∫0

dN1

dx

dN1

dx

dN1

dx

dN2

dx

dN2

dx

dN1

dx

dN2

dx

dN2

dx

EAdx ue1ue2


0

(N1ue1 + N2u

e2)f(x) dx (11.15)

Since Πe = Πe(ue1, ue2), minimization of total potential energy implies

∂Πe

∂ue1= 0 and

∂Πe

∂ue2= 0 (11.16)

Using (11.16) and (11.15) and writing the results in the matrix and vectorform, we obtain

he∫0

EA

dN1

dx

dN1

dx

dN1

dx

dN2

dx

dN2

dx

dN1

dx

dN2

dx

dN2

dx

dx ue1ue2

=

he∫0

N1(x)f(x) dx

he∫0

N2(x)f(x) dx

+

P e1P e2

(11.17)

Equations (11.17) are the force equilibrium equations for element e. By

substituting N1(x), N2(x) anddN1

dx= − 1

he,dN2

dx=

1

heinto (11.17) and

carrying out the integration, we obtain the following equations for elemente (assuming E and A are constant over the length of the element and are Eeand Ae):

AeEehe

[1 −1−1 1

]ue1ue2

=

fe1fe2

+

P e1P e2

(11.18)

We note that the local coordinate system x, y is due to pure translation ofthe x, y coordinate system hence ue1, ue2, P e1 , P e2 , fe1 , fe2 in the x, y coordinatesystem are same as ue1, ue2, P e1 , P e2 , fe1 , fe2 in the x, y coordinate system, hence(11.18) in the x, y coordinate system (fixed global cartesian frame) can bewritten as

AeEehe

[1 −1−1 1

]ue1ue2

=

fe1fe2

+

P e1P e2

(11.19)

This is same as (5.62) in (5.59) using Galerkin method with weak form.Numerical studies using (11.19) for a rod fixed at one end are presented insection 5.2 (example 5.2.1), hence are not repeated here.



In this section we consider 1D axial rod or spar elements in R2 or in2D space. Figure 11.3 shows a one dimensional rod or spar element in twodimensional space x, y.

v

ux

x

e

1

2

θ

y

y

ue1P e1fe1

fe2

P e2

ue2

Figure 11.3: 1D spar in R2 (2D space)

In the local coordinate system x, y the element of Fig. 11.3 is purely axial(following section 11.2), hence we can write the following:

[Ke]ue = fe+ P e (11.20)

or [Ke

11 Ke12

Ke21 K

e22

]ue1ue2

=

fe1fe2

+

P e1P e2

(11.21)

or

AeEehe

[1 −1−1 1

]ue1ue2

=

fe1fe2

+

P e1P e2

(11.22)

11.3.1 Coordinate transformation

Consider x, y coordinate system obtained by rotating x, y coordinate sys-tem through an angle θ (see Fig. 11.4). From simple geometry, coordinatex, y of a point P can be expressed in terms of its coordinates x, y in thexy-frame.

11.3. 1D AXIAL SPAR OR ROD ELEMENT IN R2 657

y

x

x

y

P (x, y)

y x

θx cos θ

x sin θ

θ y cos θ

y sin θ

Figure 11.4: Coordinate transformation

x = x cos θ − y sin θ

y = x sin θ + y cos θ(11.23)

In matrix and vector form we can write (11.23) asxy

=

[cos θ − sin θsin θ cos θ

]xy

= [Re]

xy

or

xy

= [Re]T

xy

(11.24)

We note that [Re]−1 = [Re]T since [Re] is orthogonal.

Using (11.24), we can transform displacements and forces from xy-frameto xy-frame.

uv

= [Re]

uv

fxfy

= [Re]

fxfy

(11.25)

Let ue1, ve1 and ue2, ve2 be the displacements of nodes 1 and 2 in the x, ycoordinate system. Let P e1x, P e1y and P e2x, P e2y be secondary variables atnodes 1 and 2 in the xy-frame and fe1x, fe1y and fe2x, fe2y be the forces atnodes 1 and 2 due to f(x) acting along the length of the element. If (x1, y1)and (x2, y2) are the coordinates of the element nodes in the x, y coordinatesystem, then

he =√

(x2 − x1)2 + (y2 − y1)2 (11.26)


If l,m are the direction cosines of the line segment constituting element ei.e. x axis, then

l = cos θ =x2 − x1

he, m = sin θ =

y2 − y1

he(11.27)

Thus for an element e we can write [Re] as

[Re] =

[cos θ − sin θsin θ cos θ

]=

[l −mm l

]=

x2 − x1

he−y2 − y1

hey2 − y1

he

x2 − x1

he

(11.28)

Hence, using global coordinates (x1, y1) and (x2, y2) of the two nodes of theelement the coordinate transformation matrix [Re] is defined. Using [Re]and (11.25) we can transform the displacements and the forces at nodes 1and 2 from x, y local coordinate system to x, y global coordinate system.

At node 1:

δe1 =

ue1ve1

= [Re]

ue10

=

lm

ue1 = teue1

P e1 =

P e1xP e1y

= [Re]

P e10

=

lm

P e1 = teP e1

fe1 =

fe1xfe1y

= [Re]

fe10

=

lm

fe1 = tefe1

(11.29)

Similarly, at node 2:

δe2 =

ue2ve2

= teue2

P e2 =

P e2xP e2y

= teP e2

fe2 =

fe2xfe2y

= tefe2

(11.30)

From (11.29) and (11.30) we can write the following using the property that[te]T [te] = 1:

ue1 = teT δe1, P e1 = teT P e1 , fe1 = teT fe1ue2 = teT δe2, P e2 = teT P e2 , fe2 = teT fe2

(11.31)

Returning to the element equations (11.21) (in symbolic form) in xy coordi-nate system, we can write

Ke11u

e1 + Ke

12ue2 = fe1 + P e1

Ke21u

e1 + Ke

22ue2 = fe2 + P e2

(11.32)


Substituting for ue1 and ue2 from (11.31) in (11.32)

Ke11teT δe1+ Ke

12teT δe2 = fe1 + P e1

Ke21teT δe1+ Ke

22teT δe2 = fe2 + P e2(11.33)

Premultiply (11.33) by te

teKe11teT δe1+ teKe

12teT δe2 = tefe1 + teP e1teKe

21teT δe1+ teKe22teT δe2 = tefe2 + teP e2

(11.34)

Using (11.29) and (11.30) for the secondary variables in (11.34) and com-bining both equations in (11.34) in the matrix and vector form,[teKe

11teT teKe12teT

teKe21teT teKe

22teT]δe1δe2

=

fe1fe2

+

P e1 P e2

(11.35)

or[Ke]δe = fe+ P e (11.36)

where [Ke] is the stiffness matrix of a 1D spar in xy-space in R2, δeT =[δe1T δe2T ] = [ue1 v

e1 u

e2 v

e2] is a vector of nodal displacements in global

xy coordinate system, fe is a vector of equivalent nodal forces, P e is avector of secondary variables (internal nodal forces) at the element nodes,and [Ke] is often called global element stiffnes matrix:

feT = [fe1T fe2T ] = [teT fe1 teT fe2 ] = [fe1x fe1y f

e2x f

e2y]

likewise

P eT = [P e1 T P e2 T ] = [teT P e1 teT P e2 ] = [P e1x Pe1y P

e2x P

e2y]

Remarks.

(1) The two dimensional spar elements in their own local coordinate systemx, y can only experience purely axial deformation and loads.

(2) When such elements are members of two dimensional pin connectedstructures, then due to application of external loads these members ro-tate relative to each other such that in the final equilibrium state theloads in the members are purely axial.

11.3.2 A two member truss

Example 11.1. Consider a two member truss shown in Fig. 11.5. All mem-bers are pin connected. Grid points 1 and 2 are constrained from translationsin x and y directions but the members connected to them are free to rotate


and the grid point 3 is free to move in x and y directions. The members atgrid point 3 are free to rotate relative to each other. A load of 10 tons isapplied at grid point 3 in the negative y-direction. The areas of cross-sectionA1 and A2 of the members are 2 and 4 square inches and the modulus ofelasticity E of both members is 30 × 106 psi. The dimensions of the trussare also shown in Fig. 11.5(a).

1

3

2

10 tonsA2

8’

8’A1

8’

(a) Schematic of the two membertruss

2

1

y

y2 x1

x

x2

y1

(b) Local coordinate systems

Figure 11.5: A two member truss and member local coordinate systems

Figure 11.5(b) shows local coordinate systems x1, y1 and x2, y2 of thetwo members of the truss. Figure 11.6 shows the displacements and theforces at the two nodes of each member in their local coordinate system aswell as in the xy global coordinate system. Our objective is to determinedeflection u3 and v3 of grid point 3, reactions at grid points 1 and 2, and theaxial forces in the two members of the truss. Figure 11.6(a) shows the twomembers of the truss with nodal displacements and forces in their own localcoordinate systems (x1, y1) and (x2, y2). Figure 11.6(b) shows the forces anddisplacements at the nodes of the members in global xy coordinate system.

11.3.2.1 Computations

The coordinates of the grid points are

(x1, y1) = (0, 4), (x2, y2) = (0,−4), (x3, y3) = (√

48, 0)

The direction cosines l1,m1 of element 1 and l2,m2 of element 2 are

l1 =x3 − x2

h1=

√48− 0

8=

√48

8, m1 =

y3 − y2

h1=

0− 4

8= −1

2

l2 =x3 − x1

h2=

√48− 0

8=

√48

8, m2 =

y3 − y1

h2=

0 + 4

8=

1

2

(11.37)


1

2y2

x2

x1

y1

P 11

u12P 12

P 22

u22

P 21

u11

u21

(a) Displacements and forces inthe element local coordinate sys-tems

1v3, P 1

3y, f13y

v2, P 12y, f12y

u3, P 13x, f13x

u2, P 12x, f12x

u1, P 21x, f21x

2

v1, P 21y, f21y

u3, P 23x, f23x

v3, P 23y, f23y

(b) Displacements and forces inx, y global coordinate system

Figure 11.6: Element local and global displacements and forces

Therefore

t1 =

l1m1

=

√

488

−12

, t2 =

l2m2

=

√

488

12

(11.38)

A1E1

h1= 7.5× 106 lb/ft,

A2E2

h2= 15× 106 lb/ft (11.39)

Element equations for elements 1 and 2 in their own local coordinate systems:

A1E1

h1[K1]

u1

1

u12

=

P 1

1

P 12

A2E2

h2[K2]

u2

1

u22

=

P 2

1

P 22

(11.40)

Using (11.39) and [K1] and [K2] from (11.18)

7.5× 106

[1 −1−1 1

]u1

1

u12

=

P 1

1

P 12

; element 1

15× 106

[1 −1−1 1

]u2

1

u22

=

P 2

1

P 22

; element 2

(11.41)

The element equations in the global coordinate system can be obtained using(11.35). Since Ke

11 = Ke22 = 1 and Ke

12 = Ke21 = −1 for e = 1 and 2, the


element equations (11.35) reduce to

7.5× 106

[t1t1T −t1t1T−t1t1T t1t1T

]u2

v2

u3

v3

=

P 1

2x

P 12y

P 13x

P 13y

, element 1 (11.42)

and

15× 106

[t2t2T −t2t2T−t2t2T t2t2T

]u1

v1

u3

v3

=

P 2

1x

P 21y

P 23x

P 23y

, element 2 (11.43)

t1t1T =

34 −

√3

4

−√

34

14

, t2t2T =

34

√3

4√

34

14

(11.44)

Using (11.44) in (11.42) and (11.43) we have

7.5× 106

34 −

√3

4 −34

√3

4

−√

34

14

√3

4 −14

−34

√3

434 −

√3

4√3

4 −14 −

√3

414

u2

v2

u3

v3

=

P 1

2x

P 12y

P 13x

P 13y

; element 1 (11.45)

and

15× 106

34

√3

4 −34 −

√3

4√3

414 −

√3

4 −14

−34 −

√3

434

√3

4

−√

34 −1

4

√3

414

u1

v1

u3

v3

=

P 2

1x

P 21y

P 23x

P 23y

, element 2 (11.46)

Let δT = [u1 v1 u2 v2 u3 v3] be the arrangement of the degrees offreedom at grid points 1, 2, and 3, then the element equations (11.45) and(11.46) can be assembled into the following system of linear simultaneous


equations in δ.

7.5× 106

32

√3

2 0 0... −3

2 −√

32

√3

212 0 0

... −√

32 −1

2

0 0 34 −

√3

4

... −34

√3

4

0 0 −√

34

14

...√

34 −1

4

. . . . . . . . . . . .... . . . . . .

−32 −

√32 −3

4

√3

4

... 34 + 3

2 −√

34 +

√3

2

−√

32 −1

2

√3

4 −14

... −√

34 +

√3

214 + 1

2

u1

v1

u2

v2

. . .u3

v3

=

P 21x

P 21y

P 12x

P 12y

. . .P 1

3x + P 23x

P 13y + P 2

3y

(11.47)

Boundary Conditions:

(a) Essential BCs

u1 = 0, v1 = 0

u2 = 0, v2 = 0(11.48)

(b) Natural BCs

P 11x, P

11y, P

22x, P

22y : unknown

P 13x + P 2

3x = 0 : no external load at grid point 3 in x direction

P 13y + P 2

3y = −20000lbs : external load

(11.49)

Partition equations (11.47). Let

δ1T = [u1 v1 u2 v2], δ2T = [u3 v3]

P1T = [P 11x P

11y P

22x P

22y], P2T = [P 1

3x + P 23x, P 1

3y + P 23y]

(11.50)

Then, (11.47) can be written as[[K11] [K12][K21] [K22]

]δ1δ2

=

P1P2

(11.51)


Using (11.48) and (11.49) in (11.50) and then using (11.50) in (11.47) weobtain

[K11]δ1+ [K12]δ2 = P1 (11.52)

[K21]δ1+ [K22]δ2 = P2 (11.53)

In which (from (11.47))

[K11] = 7.5× 106

32

√3

2 0 0√

32

12 0 0

0 0 34 −

√3

4

0 0 −√

34

14

, [K12] = 7.5× 106

−3

2 −√

32

−√

32 −1

2

−34

√3

4√3

4 −14

[K21] = [K12]T = 7.5× 106

−32 −

√3

2 −34

√3

4

−√

32 −1

2

√3

4 −14

[K22] = 7.5× 106

34 + 3

2 −√

34 +

√3

2

−√

34 +

√3

214 + 1

2

(11.54)

From (11.53)δ2 = [K22]−1

(P2 − [K21]δ1

)(11.55)

Thereforeu3

v3

= 7.5×10−6

[2.25 0.433023

0.433023 0.75

]−1(0

−20000

−

00

)(11.56)

∴

u3

v3

=

0.0007698′

−0.004′

=

0.0092376”−0.048”

(11.57)

and

P1 =

P 1

1x

P 11y

P 22x

P 22y

= [K11]δ1+ [K12]δ2 = 0+ [K12]δ12 (11.58)

P1 = [K12]

0.0007698′

−0.004′

(11.59)

Using [K12] from (11.56), we obtain

P1 =

P 1

1x

P 11y

P 22x

P 22y

=

17320.510000.0−17320.5

10000.0

(11.60)


where P1 are the reactions in x and y directions at nodes 1 and 2. The com-puted results can be compactly represented graphically as shown in Fig. 11.7.

x

y

1

2

17320.5 lbs

10000.0 lbs

20000 lbs

17320.5 lbs

10000.0 lbs

(u3, v3) = (0.0092376”,−0.048”)

Figure 11.7: Reactions, loading, and displacements (example 11.1)

11.3.2.2 Post-processing

First we convert grid point displacements to element nodal displacementsin the element local coordinate systems.Element 1:

u12 = t1T

u3

v3

=

[√48

8− 1

2

]0.0092376”−0.048”

(11.61)

u12 = 0.032”

u1h(x) = N1(x)u1

1 + N2(x)u12, N(x) =

x

he=

x

96

u1h(x) =

x

96(0.032”) =

(0.001

3

)x (11.62)

ε1xx =

∂u1h(x)

∂x=

0.001

3= 0.0003333 (axial strain)

σ1xx = Eε1

xx = 30× 106(0.0003333) = 10000 psi (axial stress)

(11.63)

Element 2:

u22 = t2T

u3

v3

=

[√48

8

1

2

]0.0092376”−0.048”

(11.64)


u22 = −0.016”

u2h(x) = N1(x)u2

1 + N2(x)u22 ; N(x) =

x

he=

x

96

u2h(x) =

x

96(−0.016”) =

(−0.001

6

)x (11.65)

ε2xx =

∂u2h(x)

∂x=−0.001

6= −0.0001666 (axial strain)

σ2xx = Eε2

xx = 30× 106(−0.0001666) = −5000 psi (axial stress)

(11.66)


The transformations derived in section 11.3 for 1D spars in R2 (2D space)can be extended for 1D spars in R3 or 3D space. Figure 11.8 shows a 1Dspar element in its own local coordinate system x, y, z in R3 (i.e. xyz-space).

The element nodal displacements, equivalent nodal forces, and the sec-ondary variables in the element local coordinate system (x, y, z) are shownin Fig. 11.8. In the element local coordinate system the element equationsremain the same as derived for 1D space:[

Ke11 K

e12

Ke21 K

e22

]ue1ue2

=

fe1fe2

+

P e1P e2

(11.67)

Figure 11.9 shows displacements, equivalent nodal loads, and the secondaryvariables at the nodes of element e in the global x, y, z coordinate system.

x

y

fe2

P e2

ue1

fe1

e

z

(x2, y2, z2)2

y

xu

z

w

P e1

ue2

1

v

(x1, y1, z1)

Figure 11.8: 1D spar element in R3


e

2

y

x

z

1

we1, fe1z, P

e1z

ue1, fe1x, P

e1x

we2, fe2z, P

e2z

ue2, fe2x, P

e2xve1, f

e1y, P

e1y

ve2, fe2y, P

e2y

Figure 11.9: Nodal forces and displacements in the global x, y, z coordinate system

Using the coordinates (x1, y1, z1) and (x2, y2, z2) of nodes 1 and 2 of theelement e, its length he, and its direction cosines l, m, and n in the xyz-framecan be determined.

he =√

(x2 − x1)2 + (y2 − y1)2 + (x2 − x1)2

l =x2 − x1

he, m =

y2 − y1

he, n =

z2 − z1

he

(11.68)

Let

te =

lmn

(11.69)

Then xyz

= tex (11.70)

At node 1:

δe1 =

ue1ve1we1

= teue1, fe1 =

fe1xfe1yfe1z

= tefe1 ,

P e1 =

P e1xP e1yP e1z

= teP e1 (11.71)


At node 2:

δe2 =

ue2ve2we2

= teue2, fe2 =

fe2xfe2yfe2z

= tefe2 ,

P e2 =

P e2xP e2yP e2z

= teP e2 (11.72)

From (11.71) and (11.72) we obtain

ue1 = teT δe1, fe1 = teT fe1, P e1 = teT P e1 ue2 = teT δe2, fe2 = teT fe2, P e2 = teT P e2

(11.73)

Expanding (11.67)

Ke11u

e1 + Ke

12ue2 = fe1 + P e1

Ke21u

e1 + Ke

22ue2 = fe2 + P e2

(11.74)

Substituting for ue1 and ue2 from (11.73) and pre-multiplying both equationsby te and writing the resulting equations in the matrix and vector form[teKe

11teT teKe12teT

teKe21teT teKe

22teT]δe1δe2

=

tefe1tefe2

+

teP e1teP e2

(11.75)

or[Ke]δe = fe+ P e (11.76)

[Ke] is the global stiffness matrix of 1D spar in R3 and

δeT =[δe1T δe2T

]= [ue1 v

e1 w

e1 u

e2 v

e2 w

e2]

feT =[fe1T fe2T

]= [fe1x f

e1y f

e1z f

e2x f

e2y f

e2z]

P eT =[P e1 T P e2 T

]= [P e1x P

e1y P

e1z P

e2x P

e2y P

e2z]

(11.77)

are nodal displacements, nodal loads, and the nodal secondary variables(internal nodal forces) at the two nodes of element e.

11.5 The Euler–Bernoulli beam element

The Euler-Bernoulli beam theory is based on the assumption that theplane cross-sections perpendicular to the beam axis remain planar and per-pendicular to the axis after deformation. The schematic in Fig. 11.10 showsa cantilever beam of length L along x-axis subjected to distributed load q(x)along its length and a moment ML and shear force FL at x = L. The endof the beam is completely clamped.

11.5. THE EULER–BERNOULLI BEAM ELEMENT 669

z

w

xu

x = 0

x = L

q(x)FL

ML

Figure 11.10: Schematic of a beam along x-axis

Let w be the transverse displacement of the beam. The differential equa-tion describing the bending behavior of the beam is given by

d2

dx2

(bd2w

dx2

)= q(x) ∀x ∈ (0, L) ⊂ R1 (11.78)

This is a fourth order ordinary differential equation in displacement w, hencerequires four boundary conditions. For the schematic shown in Fig. 11.10,the boundary conditions would be

w(0) = 0,dw

dx

∣∣∣∣x=0

= 0(bd2w

dx2

)∣∣∣∣x=L

= ML,d

dx

(bd2w

dx2

)∣∣∣∣x=L

= FL

(11.79)

In (11.78) b = EI; E is modulus of elasticity and I is bending moment ofinertia of the beam cross-section. Figure 11.11 shows an m element dis-cretization of the center line of the beam using two node beam elements.

q(x) FL

ML

e1 2 m

z

x = 0

x

q(x)

xe

21

he

xe+1e

Figure 11.11: Discretization and typical two node beam element


Isolate an element e with domain Ωe = [xe, xe+1] and of length he =xe+1−xe with local node numbers 1 and 2. We derive element equations forthis beam element using the differential equation (11.78) and the Galerkinmethod with weak form as in this case the differential operator

A =d2

dx2

(bd2

dx2

)(11.80)

is linear and we can also show that A∗ = A i.e. the adjoint of the operatoris same as the operator, hence GM/WF will yield VC integral form. Thisensures that the coefficient matrix in the element algebraic equations will besymmetric.

11.5.1 Derivation of the element equations (GM/WF)

Let v = δw be the test function, then the following integral form is validbased on fundamental lemma of calculus of variations.

(Aw − q, v)Ωe =

xe+1∫xe

(d2

dx2

(bd2w

dx2

)− q)v dx (11.81)

We transfer two orders of differentiation from w to v using integration byparts

(Aw − q, v)Ωe =

xe+1∫xe

(d2v

dx2bd2w

dx2− qv

)dx

+

(d

dx

(bd2w

dx2

)v

)∣∣∣∣xe+1

xe

−((

bd2w

dx2

)dv

dx

)∣∣∣∣xe+1

xe

(11.82)

From the boundary terms (concomitant) we conclude that w, dwdx are primary

variables, ddx

(bd

2wdx2

)and bd

2wdx2

are secondary variables, hence w = w and

dwdx = θ on some boundary Γ are essential boundary conditions and thesecondary variables taking fixed values on some boundary Γ are naturalboundary conditions. Γ of course consists of the end points of the element.Let us introduce the following notations for the secondary variables:

d

dx

(bd2w

dx2

)∣∣∣∣xe

= Qe1 ; − d

dx

(bd2w

dx2

)∣∣∣∣xe+1

= Qe2 ; shear forces

bd2w

dx2

∣∣∣∣xe

= M e1 ; − bd

2w

dx2

∣∣∣∣xe+1

= M e2 ; moments

(11.83)


Expanding boundary terms in (11.82) and then using (11.83)

(Aw − q, v)Ωe =

xe+1∫xe

bd2w

dx2

d2v

dx2dx−

xe+1∫xe

qv dx

− v(xe)Qe1 − v(xe+1)Qe2 −

(−dvdx

)∣∣∣∣xe

M e1 −

(−dvdx

)∣∣∣∣xe+1

M e2 (11.84)

or(Aw − q, v)Ωe = Be(w, v)− le(v) (11.85)

where

Be(w, v) =

xe+1∫xe

bd2w

dx2

d2v

dx2dx (11.86)

le(v) =

xe+1∫xe

qv dx+ v(xe)Qe1 + v(xe+1)Qe2

+

(−dvdx

)∣∣∣∣xe

M e1 +

(−dvdx

)∣∣∣∣xe+1

M e2 (11.87)

Be(·, ·) is bilinear and symmetric and le(·) is linear, a direct consequence oflinearity of A and the fact that A∗ = A.

The quadratic functional representing the total potential energy can beobtained using

Ie(w) =1

2Be(w, v)− le(w) (11.88)

in which the first term represents elastic strain energy and the second termis the potential energy of loads i.e. work done by f(x) and also due tosecondary variables at the nodes of the element.

11.5.2 Local approximation

We note that the differential equation describing the bending behavior ofa beam is a fourth order differential equation in displacement w. Let weh bethe local approximation of w over an element domain Ωe of the descretizationΩT = ∪

eΩe, then the approximation wh of w over ΩT is given by wh = ∪

eweh.

Consider the integral form over ΩT based on fundamental lemma used as astarting step in the GM/WF:∫

ΩT

(Awh − q)v dΩ =

∫ΩT

(d2

dx2

(bd2whdx2

)− q)v dΩ = 0 (11.89)


This integral form is valid if the integrand is continuous. This requires that

wh ∈Vh ⊂ H5(ΩT )

f ∈H1

v ∈V ⊂ H1

(11.90)

That is, wh and hence weh must be of class C4 over ΩT and hence over Ωe.This is generally not entertained in the conventionally used finite elementformulations for bending of beams even though this is precisely what needsto be done. Here we only present the approach used currently. The integralform (11.85) over Ωe implies that for the discretization ΩT the followingholds:

(Awh − q, v)ΩT =∑e

(Be(weh, v)− le(v)

)= 0 ; v = δweh (11.91)

i.e. for (Awh − q)v to be continuous over ΩT , wh must be at least of classC2(ΩT ), i.e. wh ∈ Vh ⊂ H3(ΩT ). However, if we accept integral over ΩT inthe Lebesgue sense, then wh and hence weh of class C1 over ΩT and henceover Ωe can be used in the integral form resulting from the GM/WF. This iswhat is done commonly. We present details of this approach in the following.

Consider wh of class C1(ΩT ). This requires that we establish weh suchthat wh = ∪

eweh is of class C1(ΩT ). We refer to such local approximation weh

of class C1 as it yields wh of class C1(ΩT ). In simple terms this requiresthat at the inter-element boundaries w and −dw

dx = θ (rotation) must becontinuous. Thus, if we choose w and θ as nodal degrees of freedom ateach of the two nodes of the element then the resulting local approximationwould yield wh of class C1(ΩT ). Two degrees of freedom at each node of theelement (total of four degrees of freedom for the element) suggest that wecan begin with

weh = c0 + c1x+ c2x2 + c3x

3 (11.92)

The constants c0, c1, c2, and c3 can be evaluated using

weh(xe) = we1, θe1 =

(−dwehdx

)∣∣∣∣xe

,

(11.93)

weh(xe+1) = we2, θe2 =

(−dwehdx

)∣∣∣∣xe+1

Using (11.92) and (11.93) we obtain

we1 = c0 + c1xe + c2x2e + c3x

3e

θe1 = −c1 − 2c2xe − 3c3x2e

we2 = c0 + c1xe+1 + c2x2e+1 + c3x

3e+1

θe2 = −c1 − 2c2xe+1 − 3c3x2e+1

(11.94)


or we1θe1we2θe2

=

1 xe x2

e x3e

0 −1 −2xe −3x2e

1 xe+1 x2e+1 x3

e+1

0 −1 −2xe+1 −3x2e+1

c0

c1

c2

c3

(11.95)

Solving for ci; i = 0, 1, . . . , 3 from (11.95) and substituting in (11.92), weobtain

weh = Nw1 (x)we1 +N θ

1 (x)θe1 +Nw2 (x)we2 +N θ

2 (x)θe2 (11.96)

in which

Nw1 (x) = 1− 3

(x− xehe

)2

+ 2

(x− xehe

)3

N θ1 (x) = −(x− xe)

(1− x− xe

he

)2

Nw2 (x) = 3

(x− xehe

)2

− 2

(x− xehe

)3

N θ2 (x) = −(x− xe)

((x− xehe

)2

− x− xehe

)(11.97)

These are called Hermite cubic interpolation functions or simply C1 localapproximation functions. The local approximation functions can also beexpressed in an element local coordinate system. Consider a local coordinatesystem x with its origin at node 1 and x axis pointing from node 1 to node2. The x and x coordinates are related by x = x − xe, which when usedin (11.97), yields the local approximation functions in the local coordinatesystem:

Nw1 (x) = 1− 3

(x

he

)2

+ 2

(x

he

)3

N θ1 (x) = −x

(1− x

he

)2

Nw2 (x) = 3

(x

he

)2

− 2

(x

he

)3

N θ2 (x) = −x

((x

he

)2

− x

he

)(11.98)

From (11.97) we note that the local approximation functions have the fol-


lowing properties:

Nw1 (xe) = 1, Nw

1 (xe+1) = 0,dNw

1

dx

∣∣∣∣xe

= 0,dNw

1

dx

∣∣∣∣xe+1

= 0

Nw2 (xe) = 0, Nw

2 (xe+1) = 1,dNw

2

dx

∣∣∣∣xe

= 0,dNw

2

dx

∣∣∣∣xe+1

= 0

N θ1 (xe) = 0, N θ

1 (xe+1) = 0, − dN θ1

dx

∣∣∣∣xe

= 1,dN θ

1

dx

∣∣∣∣xe+1

= 0

N θ2 (xe) = 0; N θ

2 (xe+1) = 0,dN θ

2

dx

∣∣∣∣xe

= 0, − dN θ2

dx

∣∣∣∣xe+1

= 1

(11.99)

These properties of the local approximation functions ensure that condition(11.93) are satisfied by weh. Substituting local approximation weh in (11.84)and using

v = δweh = Nw1 , Nw

2 , N θ1 , and N θ

2 (11.100)

we obtain (using (11.84))

(Aweh − q, v)ΩT = [Ke]δe − F e − P e (11.101)

If we let Nw1 = N1, N θ

1 = N2, Nw2 = N3, and N θ

2 = N4, then v = δweh =Nj ; j = 1, 2, . . . , 4 and [Ke] can be represented more compactly as (using befor b)

[Ke] =

xe+1∫xe

bed2Ni

dx2

d2Nj

dx2dx ; i, j = 1, 2, . . . , 4

δeT = [we1 θe1 w

e2 θ

e2]

F ei =

xe+1∫xe

f(x)Ni(x) dx

P eT = [Qe1 Me1 Q

e2 M

e2 ]

(11.102)

If we choose q(x) = qe, a constant value over element e, then we obtain

[Ke] =2beh3e

6 −3he −6 −3he−3he 2h2

e 3he h2e

−6 3he 6 3he−3he h2

e 3he 2h2e

F e =qehe12

6−he

6he

(11.103)

in which be = EeIe; I being bending moment of inertia.

11.6. EULER-BERNOULLI FRAME ELEMENTS IN R2 675

11.6 Euler-Bernoulli frame elements in R2

In this section we consider finite element formulations of structural mem-bers in R2 that can be used to study frames in which the structural membersare welded or reveted to each other. In such applications the member cansupport axial load as well as bending behavior. Thus, the finite elementformulation for such structural members for linear elastic behavior can bederived by superposition of a rod element in R2 and a pure bending element(of section 11.5) in R2.

x

e2

z

x

θe2

ue2

ue1

we2

β1

θe1

z

we1

Figure 11.12: A frame element in R2

Let (x1, z1), (x2, z2) be the coordinates of the two nodes of the elementin x, z frame, then the direction cosines of the line segment 1-2 are

l =x2 − x1

he= cosβ ; n =

z2 − z1

he= sinβ

xz

=

[cosβ − sinβsinβ cosβ

]xz

(11.104)

or xz

=

[cosβ sinβ− sinβ cosβ

]xz

(11.105)

and xyz

=

cosβ 0 sinβ0 1 0

− sinβ 0 cosβ

xyz

(11.106)

For bending we can writeue

we

θe

=

cosβ sinβ 0− sinβ cosβ 0

0 0 1

ue

we

θe

(11.107)


Thus,

ue1we1θe1ue2we2θe2

=

cosβ sinβ 0 0 0 0− sinβ cosβ 0 0 0 0

0 0 1 0 0 00 0 0 cosβ sinβ 00 0 0 − sinβ cosβ 00 0 0 0 0 1


(11.108)

or

δe = [T e]δe (11.109)

For axial deformation (1D rod in R2) we have

AeEehe

1 0 −1 00 0 0 0−1 0 1 0

0 0 0 0

ue1we1ue2we2

= F e+ P e (11.110)

After introducing θ1 and θ2 in (11.110) we cam write the following for axialrods.

AeEehe

1 0 0 −1 0 00 0 0 0 0 00 0 0 0 0 0−1 0 0 1 0 0

0 0 0 0 0 00 0 0 0 0 0


=

fehe200

fehe200

+

P e1x00P e2x00

(11.111)

The beam element equation (11.101) can also be written as

2EeIeh3e

0 0 0 0 0 00 6 −3he 0 −6 −3he0 −3he 2h2

e 0 3he h2e

0 0 0 0 0 00 −6 3he 0 6 3he0 −3he h2

e 0 3he 2h2e


=

0Qe1M e

1

0Qe2M e

2

+

0qehe

2qeh2e120

qehe2

qeh2e12

(11.112)

A frame element in R2 is simply superposition of (11.111) and (11.112). Sinceboth (11.111) and (11.112) have same degrees of freedom their superpositionis simply their addition and we have the following for the frame element inthe element local coordinate system xz:

[Ke]δe = F e+ P e (11.113)

11.7. THE TIMOSHENKO BEAM ELEMENTS 677

in which [Ke], δe, F e, and P e are given by

[Ke] =2EeIeh3e

Aeh2e2Ie

0 0 −Aeh2e2Ie

0 0

0 6 −3he 0 −6 −3he0 −3he 2h2

e 0 3he h2e

−Aeh2e2Ie

0 0 Aeh2e2Ie

0 0

0 −6 3he 0 6 3he0 −3he h2

e 0 3he 2h2e

(11.114)

δeT = [ue1 we1 θ

e1 u

e2 w

e2 θ

e2]

P eT = [P e1x Qe1 M

e1 P

e2x Q

e2 M

e2 ]

F eT = [1

2fehe

1

2qehe −

1

12qeh

3e

1

2fehe −

1

2qehe

1

12qeh

3e]

(11.115)

Using δe = [T e]δe in (11.113) and premultiplying (11.113) by [T e]T , weobtain [

[T e]T [Ke][T e]]δe = [T e]T F e+ [T e]T P e (11.116)

or[Ke]δe = F e+ P e (11.117)

Equations (11.117) are the element equations for the frame element of Fig. 11.12in the global coordinate system xz. Explicit expressions for [Ke] can be ob-tained.

11.7 The Timoshenko beam elements

The Euler-Bernoulli beam theory is based on the assumption that theplane sections remain planar and normal to the longitudinal axis of the beamafter deformation. A consequence of this assumption is that transverse shearstrain εxz is zero. This is obviously non-physical in non-slender beams. Ifwe assume that the plane sections remain planar but not necessarily normalto the longitudianl axis of the beam, then the shear strain εxz is not zero.For this case the rotation of cross-sections (transverse normal planes to thelongitudinal axis) is not equal to −dw

dx . The beam bending theory basedon this assumption is a shear deformation theory most commonly referredto as Timoshenko beam theory. In this theory since the rotation of thecross-section about y-axis is not described by −dw

dx , we need to denote thisby an independent parameter ψ(x). The balance of linear momenta for thestationary case i.e. equations of equilibrium are given by

d

dx

(GAks

(ψ +

dw

dx

))+ q = 0 (11.118)


d

dx

(EA

dψ

dx

)−GAks

(ψ +

dw

dx

)= 0 (11.119)

In which G is shear modulus and ks is shear correction factor to correctthe constant shear stress across the cross-section in this theory compared toactual parabolic distribution. The Euler-Bernoulli beam theory is recoveredfrom (11.118) and (11.119) by substituting GAks

(ψ + dw

dx

)from (11.119)

into (11.118) and by letting ψ be −dwdx .

From (11.118) and (11.119) we observe that both contain up to secondorder derivatives of w and ψ thus GM/WF appears to be a viable option.Alternatively, since we are considering thermoelastic behavior, a reversibleprocess, the strain energy suggests existence of a quadratic functional, thetotal potential energy, all of which suggest GM/WF for constructing finiteelement formulations for (11.118) and (11.119).

11.7.1 Element equations: GM/WF

Let p1 and p2 be the test functions representing variations of w and ψi.e. p1 = δw and p2 = δψ, then for Ωe, the domain [xe, xe+1] of an elemente, we can write (let GAks = α)

xe+1∫xe

− d

dx

(α

(ψ +

dw

dx

)+ q

)p1 dx = 0 (11.120)

xe+1∫xe

−(d

dx

(EI

dψ

dx

)− α

(ψ +

dw

dx

))p2 dx = 0 (11.121)

Integrating first term in (11.120) and (11.121) by parts once

xe+1∫xe

(dp1

dxα

(ψ +

dw

dx

)− qp1

)dx−

(p1α

(ψ +

dw

dx

))∣∣∣∣xe+1

xe

= 0 (11.122)

xe+1∫xe

(dp2

dxEI

dψ

dx+ α

(ψ +

dw

dx

)p2

)dx−

(p2EI

dψ

dx

)∣∣∣∣xe+1

xe

= 0 (11.123)

xe+1∫xe

αdp1

dx

dw

dxdx+

xe+1∫xe

αdp1

dxψ dx =

xe+1∫xe

qp1 dx+

(p1α

(ψ +

dw

dx

))∣∣∣∣xe+1

xe

(11.124)xe+1∫xe

(EI

dp2

dx

dψ

dx+ αp2ψ

)dx+

xe+1∫xe

αp2dw

dxdx =

(p2EI

dψ

dx

)∣∣∣∣xe+1

xe

(11.125)

11.7. THE TIMOSHENKO BEAM ELEMENTS 679

Let (α

(ψ +

dw

dx

))∣∣∣∣xe

= ve1

−(α

(ψ +

dw

dx

))∣∣∣∣xe+1

= ve2

EIdψ

dx

∣∣∣∣xe

= M e1

− EI dψdx

∣∣∣∣xe+1

= M e2

(11.126)

Using (11.126) in (11.124) and (11.125)

xe+1∫xe

αdp1

dx

dw

dxdx+

xe+1∫xe

αdp1

dxψ dx =

xe+1∫xe

qp1 dx+ p1(xe)ve1 + p1(xe+1)ve2

(11.127)

xe+1∫xe

(EI

dp2

dx

dψ

dx+ αp2ψ

)dx+

xe+1∫xe

αp2dw

dxdx = p2(xe)M

e1 + p2(xe+1)M e

2

(11.128)Let weh and ψeh be local approximations for w and ψ over Ωe = [xe, xe+1].Since ψ is derivative of w with respect to x, it suggests that if weh is analgebraic polynomial of degree p then ψeh must be an algebraic polynomial ofdegree p−1 so that weh and ψeh are consistent. The integral forms in (11.127)and (11.128) require weh and ψeh of class C1(Ωe) if the integrals over ΩT areto be Riemann. If we accept Lebesgue integrals over ΩT , then weh and ψeh ofcan be of class C0(Ωe). We choose this option here. Let

weh =nw∑i=1Nwi w

ei = [Nw]we

ψeh =nψ∑i=1Nψi ψ

ei = [Nψ]ψe

(11.129)

in which Nwi (x) and Nψ

i (x) are local approximation functions for weh and ψehand wei and ψei are nodal degrees of freedom for weh and ψeh:

p1 = δweh = Nwj (x), j = 1, 2, . . . , nw

p2 = δψeh = Nψj (x), j = 1, 2, . . . , nψ

(11.130)


Substituting (11.129) and (11.130) into (11.127) and (11.128)

xe+1∫xe

αdNw

j

dx

(nw∑i=1

dNwi

dxwei

)dx+

xe+1∫xe

αdNw

j

dx

(nψ∑i=1Nψi ψ

ei

)dx

=

xe+1∫xe

qNwj dx+Nw

j (xe)ve1 +Nw

j (xe+1)ve2, j = 1, 2, . . . , nw (11.131)

xe+1∫xe

(EI

dNψj

dx

(nψ∑i=1

dNψi

dxψei

)+ αNψ

j

(nψ∑i=1Nψi ψ

ei

))dx

+

xe+1∫xe

αNψj

(nw∑i=1

dNwi

dxwei

)dx = Nψ

j (xe)Me1 +Nψ

j (xe+1)M e2 (11.132)

nw∑j=1

11Keijw

ej +

nψ∑j=1

12Keijψ

ej = 1F ei + 1P ei , i = 1, 2, . . . , nw (11.133)

nw∑j=1

21Keijw

ej +

nψ∑j=1

22Keijψ

ej = 2F ei + 2P ei , i = 1, 2, . . . , nψ (11.134)

or [[11Ke] [12Ke][21Ke] [22Ke]

]weψe

=

1F e2F e

+

1P e2P e

(11.135)

or

[Ke]δe = F e+ P e (11.136)

in which

11Keij =

xe+1∫xe

αdNw

i

dx

dNwj

dxdx ; i, j = 1, 2, . . . , nw

12Keij =

xe+1∫xe

αdNw

i

dxNψj dx ;

i = 1, 2, . . . , nwj = 1, 2, . . . , nψ

21Keij =

xe+1∫xe

αNψi

dNwj

dxdx ;

i = 1, 2, . . . , nψj = 1, 2, . . . , nw

22Keij =

xe+1∫xe

(EI

dNψi

dx

dNψj

dx+ αNψ

i Nψj

)dx, i, j = 1, 2, . . . , nψ

(11.137)

11.8. FINITE ELEMENT FORMULATIONS IN R2 AND R3 681

δeT = [weT ψeT ]

1F ei =

xe+1∫xe

qNwi dx ; i = 1, 2, . . . , nw

2F ei = 0 ; i = 1, 2, . . . , nψ

1P eT = [V e1 0 0 . . . 0 V e

2 ]

2P eT = [M e1 0 0 . . . 0 M e

2 ]

(11.138)

Numerical values of the coefficients of [ke] and the vector F e can becalculated using Gauss quadrature.

11.8 Finite element formulations in R2 and R3

Finite element formulations for plane stress, plane strain, and axisym-metric solids in R2 and deformation in R3 have already been presented in:(1) chapter 5 using differential mathematical models and GM/WF, LSP and(2) chapter 9 using principle of minimum potential energy in which the totalpotential energy statement is derived directly using physics of deformationas opposed to differential model.

11.9 Summary

The main objective of the finite element formulations presented in thischapter for linear structural mechanics is to illustrate the concept of localcoordinate system, 1D elements in 2D and 3D space, and to present somerepresentative finite element formulations for bending of beams. The formu-lations presented in chapters 5 and 9 for linear solid and structural mechanicsare not repeated here for the sake of brevity.

12

Convergence, ErrorEstimation, and Adaptivity

12.1 Introduction

In this chapter basic concepts of the convergence of a computed finiteelement solution, rates of convergence, a priori and a posteriori error esti-mations, computations of error in the finite element solutions, adaptive andself-adaptive finite element processes are considered. There are three mainsources of error in the finite element computational processes that are basedon variationally consistent integral forms: due to approximation of bound-aries, due to inadequate precision in computation and faulty or inadequatealgorithms, and due to local approximations of the theoretical solution. Thefirst two sources can be almost completely eliminated while the error due tolocal approximations remains intrinsic in all finite element computations. Inthis chapter and elsewhere in the book the error implies error due to localapproximation.

In finite element computation processes as more degrees of freedom areadded to a discretization the accuracy of the computed solution improveswhen the theoretical solution of the BVP is analytic or regular (i.e. smooth)and when the integral forms are VC. The characteristic length h of the dis-cretization, degree of local approximations p, and the order k of the approx-imation space (that defines the global smoothness of the approximation overthe whole discretization of order k − 1) are three independent parametersin all finite element computational processes, thus hpk framework. Param-eters h, p, k control the achievable accuracy of a finite element solution. Wedefine convergence of a finite element solution as the process through whichthe computed solution approaches theoretical solution as more degrees offreedom are added to the discretization. Our objective of course is to ap-proach the theoretical solution for least number of degrees of freedom in thediscretization. A finite element process that requires the addition of fewestdegrees of freedom in some progression is said to have the fastest convergencerate.

We discuss details of the convergence and convergence rates of the fi-nite element processes for BVPs in this chapter. A priori error estimates

683

684 CONVERGENCE, ERROR ESTIMATION, AND ADAPTIVITY

are estimates of the error established prior to the computations that re-veal specific functional dependence of the error in the finite element solutionmeasured in some norm based on h, p, and k so that prudent judgmentscan be made regarding their choices to achieve the fastest convergence rate.A posteriori error estimates are those that are derived or determined usingthe currently computed solution and are often used to guide how and wheremore degrees of freedom need to be added to improve the accuracy of thecurrent solution. Adaptivity or adaptive finite element processes help indetermination of specific changes in h, p, and k using current solution inspecific regions of the discretization for improving accuracy of the computedsolution. A self-adaptive finite element process initiates computation withsome discretization (h, p, k) and continuously modifies h, p, and k througha built-in or intrinsic mechanism in the most prudent manner (least degreesof freedom) to achieve desired accuracy. We consider specific details of vari-ous aspects of convergence, convergence rates, errors, and adaptivity in thefollowing sections.

We only consider those finite element processes in which the integralforms are variationally consistent (VC), hence ensuring symmetric, positive-definite coefficient matrices in the resulting algebraic systems, thus uncon-ditionally stable computational processes. Variational consistency of theintegral form requires that for self-adjoint differential operators we considerGM/WF or LSP based on residual functional and for non-self adjoint dif-ferential operators (linear but A∗ 6= A) and nonlinear differential operatorswe only consider LSP based on residual functional. VC integral forms areof paramount importance in determining the convergence rates of the finiteelement processes for all three classes of differential operators (self adjoint,non-self adjoint, and non-linear), as shown in later sections of this chapter.

12.2 h-, p-, k-versions of FEM and theirconvergence

Consider a BVPAφ− f = 0 in Ω (12.1)

Let Ω be closure of Ω such that Ω = Ω ∪ Γ, Γ being closed boundary of Ω.Let ΩT be discretization of Ω such that

ΩT =⋃e

Ωe (12.2)

Ωe being a finite element e of the discretization ΩT . Let φh be approximationof φ over ΩT and φeh be local approximation of φ over an element e withdomain Ωe such that

φh =⋃eφeh (12.3)

12.2. H-, P -, K-VERSIONS OF FEM AND THEIR CONVERGENCE 685

Let he be the characteristic length of an element e with domain Ωe, then wedefine characteristic length h of the discretization ΩT by

h = maxe

(he) (12.4)

Let pe be the degree of local approximations of φeh over Ωe, then we definep, the p-level for the discretization ΩT by

p = mine

(pe) (12.5)

12.2.1 h-version of FEM and h-convergence

When additional degrees of freedom are added to a discretization by re-fining the discretization i.e. subdividing existing elements thereby reducingh but holding p and k constant, then we have h-version of the finite elementmethod. If we define error in some norm and monitor reduction in this errornorm as a function of the deegrees of freedom, then we have h-convergenceof the finite element method. The h-refinement and the corresponding h-convergence may be of the following types.

Uniform h-refinementWhen the mesh refinement is done uniformly over the discretization

(i.e. h is reduced to h/2 and then to h/4 and so on) we have uniformh-convergence). Uniform h-refinement may be wasteful as in this process wewill be subdividing all elements in the discretization, hence adding additionaldegrees of freedom in portions of the discretization where the computed so-lution is already sufficiently converged, hence has desired accuracy. This isobviously wasteful.

Selective h-refinementIn this process uniform mesh refinement is done selectively in the desired

portions of the discretization. This obviously requires additional informationto determine locations of such regions. Element-wise error indicators, eitherestimated or computed, are needed to accomplish this. This is discussedfurther in later sections.

Graded h-refinementWhen element error indicators are either not available or are not reliable,

this approach is useful. In this method we consider a sequence of discretiza-tions in which the element characteristic lengths are in some fixed geometricratio. Smaller element sizes are intentionally biased towards the known lo-cations of high solution gradients. The increase in the number of elementsin this progressive sequence of discretizations can be by one or more at atime as deemed necessary.


12.2.2 p-version of FEM and p-convergence

When additional degrees of freedom are added to a discretization byincreasing the degree of approximation pe of the elements and thereby in-creasing p for the whole discretization but holding h and k constant, thenwe have p-version of the finite element method. If we define error in somenorm and monitor reduction in this error norm as a function of the degreesof freedom then we have p-convergence of the finite element method. Thep-refinement and the corresponding p-convergence may be of the followingtypes.

Uniform p-refinement

When the degree of local approximation (p-level) is increased uniformlyfor each element of the discretization we have uniform p-refinement. As incase of uniform h-refinement, this also can be wasteful due to addition ofextra degrees of freedom in the portions of the discretization where the so-lution is sufficiently converged or has desired accuracy.

Selective p-refinement

In this process p-level increase is initiated selectively in the desired por-tions of the discretization. This obviously requires additional informationto determine locations of such regions. Element-wise error indicators eitherestimated or computed are necessary to accomplish this. In the elementsor regions requiring additional degrees of freedom the choice of selective h-or p- refinement also requires a criterion through which the choice can beautomated. In selective p-refinement we may encounter interelement bound-aries with dissimilar p-levels. Such boundaries require special considerationto ensure that the solution behavior remains physical at these boundaries.As a general rule the additional degrees of freedom at a common boundarydue to the higher p-level element must be constrained such that the commonboundary behavior is in conformity as dictated by the lower p-level element.

12.2.3 hp-version of FEM and hp-convergence

In this process h and p are changed simultaneously in the entire dis-cretization, leading to uniform hp-refinement, or selectively in the desiredportions of the subdomains, resulting in selective hp-refinement. Changes inh and p simultaneously in the selected desired portions of the discretizationrequires prudent indicators, either estimated or computed, that guide thehp-refinement process.

12.2. H-, P -, K-VERSIONS OF FEM AND THEIR CONVERGENCE 687

12.2.4 k-version of FEM and k-convergence

Surana et al. [1–4] have shown the order k of the approximation space tobe an independent parameter in all finite element computational processesin addition to h and p, hence k-version of finite element method in additionto h- and p-versions. The order k of the approximation space ensures globaldifferentiability of order k−1 over the whole discretization. The appropriatechoice of k is essential in ensuring that (1) the desired physics is preservedin the computational process and (2) the integrals are Riemann in the entirefinite element process so that the equivalence of BVP with the integral form ispreserved and the errors in the calculated solution can be computed correctlywithout knowledge of the theoretical solution. We elaborate more on someof these aspects in the following.

If the differential operator contains highest order derivatives of the de-pendent variables of orders 2m, then the approximation of the solutions ofthe BVP must at least be of class C2m (i.e. of global differentiability oforder 2m in order for this approximation to be admissible in the BVP inthe pointwise sense). This requires that order k of the approximation spacemust at least be 2m+1; that is, k = 2m+1 is minimally conforming order ofthe approximation space. Clearly the order k of the minimally conformingspace is determined by the highest order of the derivatives of the dependentvariable(s) in the BVP. When k ≥ 2m+1 all integrals over the discretizationΩT remain Riemann. When k = 2m, the integrals over ΩT are in Lebesguesense and the corresponding approximation φh of the solution φ over ΩT

is not admissible in the BVP Aφ − f = 0 in the pointwise sense. Whenk ≤ 2m − 1, the approximation φh of φ over ΩT is not admissible at all inthe BVP. Choosing k > 2m + 1 may be beneficial if the theoretical solu-tion φ of the BVP is of higher order global differentiability than 2m as thischoice incorporates higher order global differentiability aspects of φ in thecomputational process.

Let us define residual function E over ΩT

E(φh) = Aφh − f over ΩT ∀φh ∈ Vh ⊂ H(ΩT ) (12.6)

or

E =∑eAφeh − f =

∑eEe ∀φeh ∈ Vh ⊂ H(Ωe) (12.7)

and the residual functional I(φh) over ΩT

I(φh) = (E,E)ΩT (12.8)

or

I(φh) =∑e

(Ee, Ee)Ωe (12.9)


whereEe = Aφeh − f (12.10)

For theoretical solution φ of Aφ− f = 0 we have

I(φ) = (E,E)Ω = 0 (12.11)

asE(φ) = Aφ− f = 0 over Ω (12.12)

Consider k ≥ 2m + 1. For this choice of k all integrals over ΩT areRiemann, hence when I(φh) → 0, then E(φh) → 0 and Ee(φeh) → 0 in thepointwise sense (i.e. φh → φ in the pointwise sense over ΩT or Ω). That isthe converged finite element solution φh is same as the theoretical solutionin all aspects up to the derivative of order k− 1. Thus proximity of I(φh) tozero is a measure of the error in the computed solution φh. This is obviouslydeterministic without the knowledge of the theoretical solution of the BVP.Furthermore, since

I(φh) =∑eIe(φeh) (12.13)

Ie(φeh) is a measure of error in the approximation φeh for an element e. ThusIe(φeh) can be computed for each element using current solution and canbe used for refinement (adaptivity). Residual functional values Ie over Ωe

provide built-in measure of error in the computed solution for each elementof the discretization and adaptivity. Choice of h-, p-, or hp-refinement isnot clear at this stage but will be considered in more details in a followingsection. In any case, the elements with values of Ie greater than some averagethreshold value for ΩT become candidates for addition of more degrees offreedom.

Remarks.

(a) Significance of the choice of minimally conforming k or greater thanminimally conforming (≥ 2m + 1) is quite clear. Without this choicedetails and conclusions in section 12.2.4 do not hold in the precise sense.

(b) When k ≥ 2m + 1, I and Ie are measures of error in the computedsolution over ΩT or Ωe as I and Ie are zero for the theoretical solution.

(c) When k ≥ 2m + 1, computations of Ie serve as error indicators andprovide a rational mechanism for adaptivity that is built into the com-putational process in specific regions of ΩT where Ie are greater than athreshold value chosen for ΩT .

(d) When k ≥ 2m + 1 and when I → 0 (≤ 10−8), φh → φ in the pointwisesense. That is the numerically computed solution is undoubtedly sameas theoretical solution within the limitations of chosen k. This claim is

12.3. CONVERGENCE AND CONVERGENCE RATE 689

only possible within hpk framework. Without higher values of k (k > 1),that is, with just hp framework with solutions of class C0 this claim cannot be made.

(e) The most significant aspect of k and the choice k ≥ 2m + 1 is that thephysics of the BVP that is intrinsic in the mathematical description ispreserved in the computational process.

(f) The choice k ≥ 2m + 1 permits design of finite element computationalprocesses in which desired features of the theoretical solution can beincorporated in the computational process with absolute assurance ofunconditional stability if the integral forms are variationally consistent.

(g) In k-convergence we keep h and p constant and progressively increase k.As explained in earlier chapters for fixed h and p, increasing k results indecrease in the degrees of freedom. Thus, for fixed degrees of freedomincrease in k permits increase in p so that the same number of degreesof freedom are maintained. This results in substantial improvement inthe solution accuracy (see chapters 5, 6, and 7).

(h) Equations (12.6) – (12.12) should not be misunderstood to imply thatthese only hold for finite element processes based on residual functional.This is not the case. Computations of Ee and Ie only require spacesin which k ≥ 2m + 1, but the integral form could be from any desiredmethods of approximation.

12.3 Convergence and convergence rate

Convergence of a finite element solution implies behavior of the error inthe finite element solution (measured in some norm) as a function of thedegrees of freedom or the characteristic length of the discretization. Whenthe theoretical solution is known, the error in the finite element solutionin some norm (L2-norm, H1-norm, etc.) can be computed and thereforewe can study its behavior as a function of the degrees of freedom. Whenthe theoretical solution is not known, perhaps estimating the error in somenorm in the computed solution is a viable option. However, we shall seein a later section that this option only works in a restricted range of thebehavior of error norm versus dofs. The third option is that if we are usingminimally conforming spaces Vh ⊂ H then residual functional I(φh) can becomputed precisely as for minimally conforming spaces all integrals over ΩT

are Riemann. Proximity of I(φh) to zero is a measure of error due to thefact that when φh ' φ, I(φh) ' I(φ) = 0. Thus, I(φh) is in fact errormeasure in the solution φh over ΩT . This option can always be used for anyapplications as it does not require theoretical solution but necessitates theapproximation φh to be in a space of order k ≥ 2m+ 1. In what follows we


can use I(φh) as a measure of error over ΩT , hence convergence of the com-puted solution φh to φ implies studying I(φh) versus dofs as more degrees offreedom are added to the discretization. When I(φh) ≤ ∆, a predeterminedtolerance of computed zero, we consider the finite element solution φh to beconverged to the theoretical solution φ. We consider

√I(φh) versus dofs or√

I =√

(E,E) = ||E||L2, L2-norm of residual E. We study

√I versus dofs

using log-log scale, or more precisely we study log||E||L2versus log(dofs).

log||E||L2and log(dofs) or log-log scale are necessary as the range of I could

be O(101) - O(10−20) and the range of dof could be O(101) - O(106) orhigher.

12.3.1 Convergence behavior of computations

The material presented in this section is based on ||E||L2versus dof be-

havior, but the same concepts hold true for any other measure of error norm(i.e. ||E||L2

can be replaced with any other error norm without affectingthe basic behavior of the convergence graph). A typical convergence behav-ior of log(

√I) or log(||E||L2

) versus log(dof) is shown in Fig. 12.1. Thisgraph is generated using 1D convection-diffusion equation (a second orderODE) with Pe = 1000 and least squares finite element formulation based onresidual functional. The progressively graded discretizations are generatedbeginning with two elements using a constant geometric ratio of 1.5. Thesmallest element is located at x = 1.0. k = 3 is used as it corresponds tothe minimally conforming space. Minimum p-level of 5 (needed for k = 3) isconsidered for each progressively refined discretization. From Fig. 12.1 weobserve five distinct zones. In each one of these zones the behavior of

√I

versus dofs is unique and distinct.

Pre-asymptotic range (AB): The range AB is called pre-asymptoticrange. In this range as we move from location A toward location B additionaldegrees of freedom are added to the discretization but there is virtually nomeasurable reduction in the L2-norm of E. The accuracy of the computedsolution in this range is very poor (due to ||E||L2

of the O(1)) due to poor

accuracy of the solution φh, hence√I and the Ie values for the elements are

poor as well, therefore these can not be used to guide any form of adaptiverefinement process. A posteriori error estimations in this range are notpossible either as these require some regularity in the computed solutionwhich is absent in φh in range AB. Thus, in this range adaptive processesare not possible as reliable indicators (either estimated or computed) basedon φh are not possible.

12.3. CONVERGENCE AND CONVERGENCE RATE 691

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2

log(√

I =

|| E

|| L

2)

log(dofs)

AB : Pre-asymptotic range

BC : Onset of asymptotic range

CD : Asymptotic range

DE : Onset of post-asymptotic range

EF : Post-asymptotic range

A B

C

D

E F

Figure 12.1: Typical convergence behavior of a finite element solution

Onset of asymptotic range (BC): The range BC is called onset ofasymptotic range. In this range addition of degrees of freedom to the dis-cretization results in measurable reduction in ||E||L2

reflecting progressiveimprovement in accuracy of the computed solution φh from B to C. Inthis range Ie values or any other possible element error indicators are moreaccurate than range AB. In this range adaptive processes in h, p, or hpcan be utilized keeping in mind that as we move closer to C, the values ofIe (or other indicators) for the elements of the discretization become moreaccurate, hence can be more effective in the adaptive process.

Asymptotic range (CD): In this range as more dofs are added to the dis-cretization the improvement (reduction) in ||E||L2

is most significant. Thisrange on log-log scale is nearly linear, hence constant slope. Adaptive re-finements in this range are most effective in reducing ||E||L2

. We observethat between C and D there are several orders of magnitude reduction inthe value of ||E||L2

. Slope of the error norm versus dof graph in this rangeis called the asymptotic convergence rate of the finite element solution.

Onset of post-asymptotic range (DE): This range is almost reverse ofthe onset of asymptotic range. In this range reduction in ||E||L2

progressivelydiminishes with the addition of degrees of freedom to the discretization indi-cating that substantial achievable reduction in ||E||L2

has taken place up topoint D. Computations in this range result in waste of significant resources(dofs) with very little gain in the objective of reducing ||E||L2

.


Post-asymptotic range (EF ): In this range in spite of the addition of dofsto the discretization no measurable reduction is observed in ||E||L2

. This isgenerally due to the fact that within the accuracy of the computations (i.e.the word size on the computer we have reached a limit), hence the accuracyremains limited to the same number of decimal places in ||E||L2

regardlessof the increase in dofs.

12.3.2 Convergence rates

In an abstract sense the convergence rate of a finite element computa-tional process is the rate at which the computed solution φh is approachingthe theoretical solution φ as more degrees of freedom are added to the dis-cretization through refining h or increasing p or changing k. That is it is therate at which the error norm is approaching zero as more degrees of freedomare added. Thus, a measure of convergence rate of the finite element solutioncould be the slope of

√I (or ||E||L2

) versus dof behavior. Since dofs can beadded through h, p, and k, the convergence rate of a finite element solutioncan be a function of h, p, k, and the smoothness of the theoretical solutionat this stage of the discussion.

In range AB, the slope is almost zero. From B to C the slope increasesas more dofs are added to the discretization thereby progressively increasingconvergence rate from B to C. From C to D, the asymptotic range, theslope of ||E||L2

versus dofs is almost constant and the reduction in ||E||L2is

most significant as more dofs are added. Thus, in the asymptotic range theconvergence rate is the highest (due to highest slope of log(||E||L2

) versuslog(dof)) and is constant. In the onset of post-asymptotic range DE theconvergence rate decreases and eventually becomes almost zero in the post-asymptotic range EF .

Remarks.

(i) Behavior of ||E||L2versus dofs shown in Fig. 12.1 is typical of other

error norms as well, hence the discussion and conclusions related toFig. 12.1 are applicable in the convergence behavior study using anyother desired error norm.

(ii) Pre-asymptotic range AB, onset of post-asymptotic range DE, andpost-asymptotic range EF should be avoided as in these ranges solutionaccuracy improvement is poor.

(iii) In range AB Ie values (or other measures) are not accurate enough toguide an adaptive process of any kind.

(iv) Adaptive processes (h, p, k) can be initiated in the range BC as Ie

values in this range are reasonable measure of error. Adaptive processes

12.4. ERROR ESTIMATION AND ERROR COMPUTATION 693

become more and more effective when we initiate them as we approachfrom B to C, i.e. closer to C. In the range BC the slope of ||E||L2

versus dof increases from B to C indicating improving convergence rateand eventually achieves the highest convergence rate value at C whichremains almost constant in the asymptotic range CD.

(v) A priori and a posteriori error estimates are only valid in the asymptoticrange due to the fact it is only in this range that computed φh hasdesired regularity and the convergence rate is the highest, hence worthestimating a priori. The error estimates (a priori and a posteriori) canneither be derived accurately nor can be used meaningfully in regionsother than BC.

12.4 Error estimation and error computation

There are two types of error estimations generally considered: a priorierror estimation and a posteriori error estimation. A priori error estimationrefers to establishing dependence of some error norm on h, p, k, and theregularity of the theoretical solution before the computations are performedso that we have knowledge of the precise nature of the functional dependenceof error norm on h, p, k, and the regularity of the theoretical solution. Aposteriori error estimation refers to error estimates derived using a computedsolution with specific choices of h, p, and k. The sole purpose of a posteriorierror estimation is to use current finite element solution to derive elementindicators that can perhaps be used to guide an adaptive process. Both ofthe error estimations require some regularity of the computed solution whichonly exists in the asymptotic range (range CD, Fig. 12.1). This is a verysignificant restriction on the use of these estimates. For example, a priorierror estimate can not be used to predict convergence rate in the ranges AB,BC, DE, and EF as this is specifically derived using the regularity of φhthat only exists in the asymptotic range. Likewise a posteriori estimate cannot be used for adaptivity in any ranges except CD.

Another point to note is that a posteriori error estimates are generallyderived such that they quantify the weakness(es) in the finite element globalapproximation φh = ∪

eφeh. Their derivations are largely based on C0 local

approximations which result in interelement discontinuity of the first deriva-tives normal to the interelement boundaries. This may be quantified byestablishing bounds that can be used for adaptivity. However if we use φehof class C1 thereby φh of class C1, then such bounds are meaningless. Ink-version of finite element methods enabling higher order global differentia-bility approximations, majority of the a posteriori error estimates based oninterelement discontinuity of the derivatives are not meaningful. With theuse of higher order approximations φh, the integrals can be maintained Rie-


mann, ||E||L2and ||Ee||L2

are true measures of the error in the finite element

solution for ΩT and Ωe and can indeed be used in adaptive processes. Theseaspects are discussed in more details in later sections.

12.5 A priori error estimation: derivation of errorestimate and convergence rates

The published literature on a priori error estimation is almost exclusivelyfor BVPs described by self adjoint differential operators in which the integralform, hence the finite element formulation, is constructed using Galerkinmethod with weak form. The finite element solution using GM/WF forBVPs described by self adjoint differential operators has best approximationproperty in B-norm. This property has been viewed as essential in derivingthe a priori error estimates. Thus, we begin with boundary value problemsdescribed by self adjoint differential operators.

12.5.1 Galerkin method with weak form (GM/WF):self-adjoint operators

In this section we revisit main steps of GM/WF for self-adjoint operators.Let

Aφ− f = 0 in Ω (12.14)

be a boundary value problem in which the differential operator A is symmet-ric and its adjoint A∗ = A (i.e. the differential operator A is self adjoint).Based on fundamental lemma (chapter 2) we can write the following integralform:

(Aφ− f, v)Ω =

∫Ω

(Aφ− f)v dΩ =

∫Ω

(Aφ)v dΩ−∫Ω

fv dΩ = 0 (12.15)

in which v = 0 on Γ∗ if φ = φ0 (given) on Γ∗. v is called test function, hencev = δφ is admissible in (12.15). When v = δφ in (12.15), the integral form(12.15) is called integral form in Galerkin method. Since A is self adjoint,the BVP (12.14) only contains even order derivatives of φ. We transfer halfof the differentiation from φ to v using integration by parts in the first termin (12.15) and collect those terms that contain both φ and v and define themcollectively as B(φ, v) and those that contain only v and define them as l(v),hence we can write the following.

B(φ, v) = l(v) (12.16)

12.5. A PRIORI ERROR ESTIMATION 695

Each term in B(φ, v) contains both φ and v but more importantly the ordersof derivatives of φ and v in each term is same (i.e. B(φ, v) is symmetric),thus

B(φ, v) = B(v, φ) (12.17)

and since A is linear, B(φ, v) is bilinear in φ, v and l(v) is linear in v. Hencein this case quadratic functional I(φ) is possible and is given by

I(φ) =1

2B(φ, φ)− l(φ) (12.18)

The integral form (12.16) is called weak form of (12.14). Due to the factthat (12.15) is integral form in Galerkin method, the weak form (12.16) iscalled integral form in Galerkin method with weak form (GM/WF). Thequadratic functional I(φ) has physical significance as explained in chapters2 and 5. If (12.14) represents a BVP associated with linear elasticity in solidmechanics, then 1

2B(φ, φ) is strain energy, l(φ) is potential energy of loadsand I(φ) is the total potential energy of the system described by (12.14).

Theorem 12.1. The weak form B(φh, v) = l(v) resulting from GM/WFfor self adjoint differential operator A in Aφ − f = 0 in which B(·, ·) issymmetric is variationally consistent.

Proof. Variational consistency of the weak form B(φh, v) = l(v) requires thatthere exist a functional I(φh) such that δI(φh) = 0 gives the weak form, theEuler’s equation resulting from δI(φh) = 0 is the BVP, and δ2I(φh) yieldsunique extremum principle. Following section 12.5.1 the existence of thefunctional I(φh) is by construction (equation (12.18))

I(φh) =1

2B(φh, φh)− l(φh)

If I(φh) is differentiable in φh, then δI(φh) = 0 is a necessary condition foran extremum of I(φh). Using δφh = v (due to GM/WF),

δI(φh) =1

2B(v, φh) +

1

2B(φh, v)− l(v) = 0

Since B(·, ·) is symmetric, we obtain

δI(φh) = B(φh, v)− v) = 0

orB(φh, v) = l(v), the weak form

The unique extremum principle (or sufficient condition) is given by

δ2I(φh) = δ(B(φh, v)− l(v)

)= B(v, v) > 0 ∀v ∈ Vh ⊂ H


Hence, a unique extremum principle.To show that the Euler’s equation resulting from the weak form is in fact

the BVP, we just have to transfer differentiation back to φ (or φh) from vin the weak form using integration by parts. This is rather straightforward.Thus, the weak form B(φh, v) resulting from the GM/WF is variationallyconsistent. δ2I(φh) = B(v, v) > 0 implies that a φh from the weak formminimizes I(v),∀v ∈ Vh,

I(φh) ≤ I(v) ∀v ∈ Vh

Theorem 12.2. Let Aφ−f = 0 be a BVP in which A is self adjoint and letB(φh, v) = l(v) be weak form resulting from GM/WF in which B(φh, v) =B(v, φh) and φh, v ∈ Vh ⊂ H, then φh has best approximation property inB(·, ·)-norm. That is, if e = φ− φh, φ ∈ H being theoretical solution, then

(a) B(e, v) = 0 ∀v ∈ V(b) B(e, e) ≤ B(φ− w, φ− w), ∀w ∈ V

Proof.

(a)B(φh, v) = l(v)

B(φ, v1) = l(v1), v1 ∈ H

Choosing v1 = v ∈ V ⊂ H

B(φ, v) = l(v)

Hence,B(φ− φh, v) = 0

orB(e, v) = 0

This implies that no element of V is a better approximation of φ thanφh, the solution for the weak form when measured in B(·, ·) as e is B(·, ·)-orthogonal to every element v of V . This is called the best approximationproperty of GM/WF for self adjoint operators.

(b) For any v ∈ Vh

B(e+ v, e+ v) = B(e, e) + 2B(e, v) +B(v, v)

But B(e, v) = 0, hence

B(e+ v, e+ v) = B(e, e) +B(v, v)


Since B(v, v) > 0 we have

B(e, e) ≤ B(e+ v, e+ v)

e+ v = φ− φh + v = φ− (φh − v)

φh, v ∈ V ; hence φh − v = w ∈ VhThus,

B(e, e) ≤ B(φ− w, φ− w) ∀w ∈ Vh ⊂ H

orB(φ− φh, φ− φh) ≤ B(φ− w, φ− w)

or||φ− φh||B ≤ ||φ− w||B

That is, error in φh in B-norm is the lowest compared to any othersolution w. This completes the proofs of (a) and (b).

12.5.2 GM/WF for non-self adjoint and non-linear operators

Theorem 12.3. Let Aφ − f = 0 in Ω be a BVP in which A is a non-selfadjoint differential operator. Let B(φ, v) − l(v) = 0 be all possible weakforms. Then all such integral forms are variationally inconsistent.

Proof. Let there exist a functional I(φ) such that δI(φ) = 0 yield the weakform B(φ, v) − l(v) = 0. Since A is non-self adjoint, B(φ, v) is bilinear butnot symmetric (i.e. B(φ, v) 6= B(v, φ)), hence

δ2I(φ) = δ(B(φ, v)− l(v))

= B(δφ, v)

= B(v, v)

> 0

= 0

< 0

∀v ∈ V

is not possible because B(·, ·) is not symmetric. Therefore, δ2I(φ) is not aunique extremum principle. Thus, the integral form B(φ, v)− l(v) = 0 withv = δφ is VIC when the differential operator is non-self adjoint.

Theorem 12.4. Let Aφ− f = 0 in Ω be a BVP in which A is a non-lineardifferential operator and let B(φ, v) − l(v) = be all possible weak forms ofAφ−f = 0 in Ω. Then all such integral forms or weak forms are variationallyinconsistent.


Proof. Let there exist a functional I(φ) such that δI(φ) yields the integralform B(φ, v) − l(v) = 0. Since the differential operator A is non-linear,B(φ, v) is linear in v but not linear in φ and l(v) is linear in v. Therefore,the second variation of I

δ2I(φ) = δ(B(φ, v)− l(v))

= δ(B(φ, v))

is a function of φ due to the fact that B(φ, v) is a non-linear function of φ.Thus, δ2I(φ) does not represent a unique extremum principle and, hence,the integral form or weak form B(φ, v)− l(v) = 0 is VIC.

12.5.3 Least-squares method based on residual functional:self-adjoint and non-self-adjoint operators

Theorem 12.5. The integral form in least-squares method based on resid-ual functional is variationally consistent when the BVP is described by selfadjoint differential operator.

Proof. Consider the BVP

Aφ− f = 0 ∀x ∈ Ω

in which A is self adjoint. Let φh ∈ Vh ⊂ H be an approximation of φ overdiscretization ΩT = ∪

eΩe of Ω. Let

E = Aφh − f ∀x ∈ ΩT

be residual function. We define residual functional

I(φh) = (E,E)

If I(φh) is differentiable in φh then the necessary condition is given byδI(φh) = 0.

δI(φh) = 2(E, δE) = 2(Aφh − f,Av) = 0, v = δφh ∈ Vh

or

(Aφh, Av) = (f,Av)

or

B(φh, v) = l(v)

B(φh, v) is bilinear and symmetric and l(v) is linear.

δ2I(φh) = (δE, δE) = (Av,Av) > 0 ∀v ∈ Vh


Hence, the integral form resulting from δI(φh) = 0 is variationally consistent.

Theorem 12.6. The integral form in least-squares method based on residualfunctional is variationally consistent when the BVP is described by non-selfadjoint operator.

Proof. Since non-self adjoint operators are linear the proof of this theoremis same as that for self adjoint operators (Theorem 12.5) which are alsolinear.

12.5.4 Least-squares method based on residual functional fornon-linear operators

Theorem 12.7. Let Aφ− f = 0 in Ω be a boundary value problem in whichA is a non-linear differential operator. Let φh be approximation of φ inΩT = ∪

eΩe, discretization of Ω and let Aφh − f = E be the residual function

in Ω. Then the integral form resulting from the first variation of the residualfunctional I(φh) = (E,E) set to zero is VC provided δ2I(φh) ∼= (δE , δE)and the system of non-linear algebraic equations resulting from δI(φh) = 0are solved using Newton-Raphson or Newton’s linear method.

Proof. Since A is non-linear, E is a non-linear function of φh, hence δE is afunction of φh.

I(φh) = (E,E) = (Aφh − f,Aφh − f) ; existence of I(φh)

If I(φh) is differentiable in φh, then

δI(φh) = 2(E, δE) = 2g(φh) = 0

Hence, g(φh) = 0 is a necessary condition.

Since δE = δ(Aφh − f) = δA(φh) +Av,

g(φh) = (Aφh − f, δA(φh) +Av) = 0

or (Aφh, Av + δA(φh)

)= (f, δA(φh) +Av)

orB(φh, v) = l(v)

Also

δ2I(φh) = 2(δE, δE) + 2(E, δ2E)

> 0

= 0

< 0

∀φh, v ∈ Vh ⊂ H


is not possible. Hence, we do not have a unique extremum principle. Atthis stage, the least-squares process is VIC. We rectify the situation in thefollowing.

We note that based on the necessary condition g(φh) = 0 must hold.Since g(φh) is a non-linear function of φh, we must find a φh iteratively thatsatisfies g(φh) = 0. Let φ0

h be an initial (or assumed) solution, then

g(φ0h) 6= 0

Let ∆φh be a change in φ0h such that

g(φ0h + ∆φh) = 0

Expanding g(φ0h + ∆φh) in Taylor series about φ0

h and retaining only up tolinear terms in ∆φh (Newton-Raphson or Newton’s linear method)

g(φ0h + ∆φh) ∼= g(φ0

h) +∂g(φh)

∂φh

∣∣∣∣φ0h

∆φh = 0

Therefore

∆φh = −

[∂g(φh)

∂φh

∣∣∣∣φ0h

]−1

g(φ0h)

But ∂g(φh)∂φh

= 12δ(δI(φh)) = 1

2δ2I(φh). Hence

∆φh = −1

2[δ2I(φh)]−1

φ0hg(φ0

h)

Thus, in order for the coefficient matrix [δ2I(φh)] to be positive-definite,

δ2I(φh) ∼= 2(δE, δE) > 0

This gives a unique extremum principle. The improved value of φh is givenby

φh = φ0h + α∗∆φh

We choose α∗ such that I(φh) ≤ I(φ0h). This is referred to as line search.

With this approximation of δ2I(φh), the integral form δI(φh) =(Aφh −

f,Av + δA(φh))

is variationally consistent.

Remarks.

(1) Justification for approximating δ2I(φh) is important to discuss.

(2) We note that

g(φh) = (E, δE)


Justification of δ2I(φh) ∼= 2(δE, δE) is only necessary in the asymptoticrange of convergence as the a priori error estimation only holds in thisrange, thus establishing best approximation property of LSM method insome norm is also only required in this range.

δ2I(φh) = (δE, δE) + (E, δ2E)

In the asymptotic range E → 0 in the pointwise sense if the approxima-tion spaces are minimally conforming to ensure that all integrals overΩT are Riemann. When E → 0 ∀x ∈ ΩT then (E, δ2E) → 0, henceδ2I(φh) ∼= (δE, δE) is valid. Further discussion on the validity of thisapproximation can be found in chapter 3.

Theorem 12.8. The integral form resulting from the least-squares methodbased on residual functional has best approximation property in L2-norm ofE.

Proof. From section 12.5.3, we have

(E, δE) = 0

or(Aφh − f,Av) = 0

For theoretical or exact solution φ, we have

Aφ− f = 0 ⇒ f = Aφ

Hence, (A(φh − φ), Av

)= (Ae,Av) = 0, e = φh − φ

Thus, A(φh−φ) or Ae is orthogonal to Av ∈ AVh (dual of Vh). We note that

||A(φh − φ)||L2= ||Ae||L2

= ||E||L2

That is L2-norm of E obtained using φh is lowest out of all v ∈ Vh. Hence,LSP has best approximation property in L2-norm of E or ||E||L2

.

Theorem 12.9. A variationally consistent integral form has a best approx-imation property in some associated norm. Conversely, if an integral formhas a best approximation property in some norm, then it is variationallyconsistent.

Proof. Proof of this theorem follows due to the fact that VC integral formin GM/WF has best approximation property in B-norm because B(·, ·) isbilinear and symmetric. The integral form in the LSP is also VC but LSPhas best approximation property in L2-norm of E. Both GM/WF and LSP


are VC but have best approximation property in different norms. In bothcases, VC integral form is not possible without best approximation propertyand the best approximation property is not possible without VC integralform. This is obviously due to the fact that they both require the functionalB(·, ·) to be bilinear and symmetric. As long as this holds, how B(·, ·) isderived is not important.

We note that

(1) Since the integral forms for non-self adjoint and non-linear differentialoperators are VIC in GM/WF, the approximation φh from GM/WFdoes not have best approximation property in B-norm (Theorem 12.9).

(2) Lack of best approximation property and lack of VC of the integral formresulting from GM/WF for non-self adjoint and non-linear differentialoperators are both obviously due to the fact that the functional B(·, ·)in the weak forms is not symmetric.

(3) In LSP for all classes of differential operator I(φh) = (E,E)ΩT is mini-mized, therefore φh has best approximation property in E-norm(||E||L2

= (E,E)12

).

(4) We note that variational consistency of the integral form holds for allchoices of h, p, and k whereas the best approximation property onlyholds in the asymptotic range.

12.5.5 Integral forms based on other methods ofapproximation

The integral forms used in finite element method based on Petrov-Galerkinmethod, Galerkin method, and weighted residual method are not consid-ered as these always yield integral forms that are variationally inconsistent.Hence, when using these integral forms computations may not even be pos-sible.

12.5.6 General remarks

(1) We have established that GM/WF yields VC integral form only for selfadjoint operators when the functional B(·, ·) in the integral form is sym-metric and this method has best approximation property in B-norm.

(2) LSP based on residual functional yields VC integral forms for self adjoint,non-self adjoint, and non-linear (in the asymptotic range) differentialoperators and has best approximation property in E-norm.

(3) VC integral form implies best approximation property in some norm andvice versa.


(4) Best approximation property is necessary in a priori error estimation (inthe asymptotic range), as shown in subsequent sections.

(5) In general, when using GM, PGM, WRM, etc. error estimation is notpossible as in these methods the approximation φh of φ does not havebest approximation property in any norm used at present.

12.5.7 A priori error estimates: GM/WF and LSP

We consider simple model problems to demonstrate the best approxi-mation properties of GM/WF for self adjoint operators and LSP for linearoperators and present derivations of the a priori error estimates and conver-gence rates when φh ∈ Vh ⊂ Hk,p(Ωe). These estimates are derived usingmodel problems (as illustrations) and are then generalized for all BVPs.

12.5.7.1 Model problem 1: GM/WF

Consider the following BVP:

−d2φ

dx2= f(x) or − φ′′ = f(x) ∀x ∈ Ωx = (0, L) (12.19)

BCs: φ(0) = φ(L) = 0 (12.20)

GM/WF for (12.19) with BCs (12.20) gives(φ′ , v′

)= (f, v) ∀v ∈ V ⊂ H (12.21)

Let φh ∈ Vh ⊂ H be the finite element approximation of φ, then we have(φ′h , v

′) = (f, v) ∀v ∈ Vh (12.22)

Using (12.21) and (12.22) and since Vh ⊂ V , v in (12.22) is also in V andwe have (

φ′ − φ′h , v′)

= 0 ∀v ∈ Vh (12.23)

Theorem 12.10. For any v ∈ Vh we have∣∣∣∣φ′ − φ′h∣∣∣∣ ≤ ∣∣∣∣φ′ − v′∣∣∣∣ ∀v ∈ VhProof. ∣∣∣∣φ′ − φ′h∣∣∣∣2 =

(φ′ − φ′h , φ′ − φ′h

)Since (

φ′ − φ′h , w′)

= 0 ∀w ∈ Vhwe can choose w = φh − v as both φh, v ∈ Vh, then(

φ′ − φ′h , φ′h − v′)

= 0 ∀v ∈ Vh


Hence, ∣∣∣∣φ′ − φ′h∣∣∣∣2L2=(φ′ − φ′h , φ′ − φ′h

)+(φ′ − φ′h , φ

′h − v′

)=(φ′ − φ′h , (φ′ − φ′h) + (φ′h − v′)

)=(φ′ − φ′h , φ′ − v′

)Using Cauchy-Schwarz inequality (see chapter 2)∣∣∣∣φ′ − φ′h∣∣∣∣2L2

≤∣∣∣∣φ′ − φ′h∣∣∣∣L2

∣∣∣∣φ′ − v′∣∣∣∣L2

or ∣∣∣∣φ′ − φ′h∣∣∣∣L2≤∣∣∣∣φ′ − v′∣∣∣∣

L2∀v ∈ V

or||φ− φh||B ≤ ||φ− v||B

That is, in this case for the model problem (12.19)–(12.20) the derivative ofφh has the best approximation property in L2-norm. Alternatively, φ − φhhas best approximation property in B-norm. This completes the proof.

12.5.7.2 Model problem 2: LSP

Consider the following BVP described by non-self adjoint differentialoperator.

φ′ = f ∀x ∈ (0, 1) = Ω ⊂ R1 (12.24)

BC: φ(0) = 0 (12.25)

LSP based on residual functional gives (for f = 0)(φ′h, v

′) = 0, v = δφh, φh, v ∈ Vh ⊂ H (12.26)

φh is approximation of φ over Ω. This integral form is VC. Also for theoret-ical solution (

φ′, v′t)

= 0, vt = δφ (12.27)

Setting vt = v in (12.27) (φ′, v′

)= 0 (12.28)

Subtracting (12.26) from (12.28)(φ′ − φ′h, v′

)=(e′, v′

)= 0, v ∈ Vh (12.29)

Using interpolant φI of φ (interpolant matches φ at end nodes); φI ∈ Vh, lete = φ− φI + φI − φh, then we have∣∣∣∣e′∣∣∣∣2

L2= (e′, e′) =

(e′, (φ′ − φ′I) + (φ′I − φ′h)

)=(e′, φ′ − φ′I

)+(e′, φ′I − φ′h

) (12.30)


We note that φI − φh = w ∈ Vh, hence(e′, φ′I − φ′h

)=(e′, w′

)= 0 (due to (12.29)) (12.31)

Thus, (12.30) reduces to ∣∣∣∣e′∣∣∣∣2L2

=(e′, φ′ − φ′I

)(12.32)

Using Cauchy-Schwarz inequality∣∣∣∣e′∣∣∣∣2L2≤∣∣∣∣e′∣∣∣∣

L2

∣∣∣∣φ′ − φ′I ∣∣∣∣L2(12.33)

Thus, ∣∣∣∣e′∣∣∣∣L2≤∣∣∣∣φ′ − φ′I ∣∣∣∣L2

(12.34)

That is L2-norm of the derivative of error φ−φh is bounded by the finite ele-ment interpolant. Using proposition 12.1 (shown subsequently) and (12.34),we can write ∣∣∣∣e′∣∣∣∣

L2≤ h |φ|2 (12.35)

L2-norm of e; that is, ||e||L2for LSP is derived using Aubin-Nitsche trick

(Oden and Carey [5] and Reddy [6]). We consider details in the following.

Consider the same BVP (for f = 0),

φ′ = f ∀x ∈ (0, 1) = Ω

φ(0) = 0(12.36)

Let e = φ−φh. Assume that w is the solution of the second order differentialequation

−w′′ = e ∀x ∈ (0, 1) = Ω

w(0) = 0

w′(1) = 0

(12.37)

The finite element interpolant wI (wI(0) = wI(1) = 0) satisfies∣∣∣∣w′ − w′I ∣∣∣∣L2≤ h |w|2 = h

∣∣∣∣w′′∣∣∣∣L2

(12.38)

≤ h ||e||L2(using (12.37)) (12.39)

Consider

(e, e) = −(e, w′′) (12.40)

Using integration by parts and the fact that e = 0 at x = 0 and w′ = 0 atx = 1 and (e′, w′I) = 0 (orthogonal property)

(e, e) = −(e, w′′) = (e′, w′) = (e′, w′ − w′I) (12.41)


Hence, (using Cauchy-Schwarz inequality)

||e||2L2≤∣∣∣∣e′∣∣∣∣

L2

∣∣∣∣w′ − w′I ∣∣∣∣L2≤(h∣∣∣∣φ′′∣∣∣∣

L2

) (h ||e||L2

)(12.42)

Dividing by ||e||L2

||e||L2≤ h2

∣∣∣∣φ′′∣∣∣∣ = h2 |φ|2 (12.43)

We make the following remarks.

(1) For a first order BVP, the rate of convergence of the L2-norm of theerror in the finite element solution is proportional to h2 and the rate ofconvergence of the L2-norm of the derivative of the error is proportionalto h.

(2) These estimates are same as those for a second order BVP when usingGM/WF in which the integral form is variationally consistent.

General Remarks

(1) The error estimates have been derived for a second order BVP usingGM/WF in which the integral form is VC and the local approximationis linear (p = 1) over an element. In case of LSP the BVP is first orderODE, the integral form is VC, and p = 1 for local approximation.

(2) We note that the integral forms in both cases are VC and contain onlyup to first order derivatives, hence the reason for same convergence ratesof ||φ− φh||L2

and ||φ′ − φ′h||L2even though in case of GM/WF the BVP

is a second order ODE and in case of LSP it is only a first order ODE.This is rather significant to note that VC of the integral form and thehighest order of the derivative in the integral form control the rates ofconvergence.

(3) We need to extend these estimates for higher degree local approximation(i.e. p-level of ‘p’).

(4) The order of approximation space k needs to be incorporated in the errorestimates.

12.5.7.3 Proposition and proof

Proposition 12.1. Let the theoretical solution φ of (12.19)–(12.20) be atleast of class C2[0, L] and let φh be approximation of φ over the discretizationΩT = ∪

eΩe of Ω = [0, 1] in which Ωe = [xi, xi+1] is an element e. Let h

be the characteristic length of ΩT such that h = maxehe. Let φI = ∪

eφeI

be interpolant of φ that agrees with φ at the nodes [i.e. φ(xi) = φI(xi),i = 0, 1, . . . ]. Then


(a) ∣∣∣E(x)∣∣∣ = |φ(x)− φI(x)| ≤ h2 max

[0,L]

∣∣φ′′∣∣ (12.44)

(b) ∣∣∣E′(x)∣∣∣ =

∣∣φ′(x)− φ′I(x)∣∣ ≤ hmax

[0,L]

∣∣φ′′∣∣ (12.45)

(c) When (12.44) and (12.45) hold, the following hold∣∣∣∣∣∣E′(x)∣∣∣∣∣∣L2

=∣∣∣∣φ′(x)− φ′I(x)

∣∣∣∣L2

= |φ(x)− φI(x)|H1

= |φ(x)− φI(x)|1 ≤ h∣∣∣∣φ′′∣∣∣∣

L2= h |φ|2 (12.46)

∣∣∣∣∣∣E(x)∣∣∣∣∣∣L2

= ||φ(x)− φI(x)||L2= |φ(x)− φI(x)|H0

= |φ(x)− φI(x)|0 ≤ h2∣∣∣∣φ′′∣∣∣∣

L2= h2 |φ|2 (12.47)

∣∣∣∣∣∣E(x)∣∣∣∣∣∣H1

=∣∣∣∣∣∣E(x)

∣∣∣∣∣∣1

= ||φ(x)− φI(x)||H1

= ||φ(x)− φI(x)||1 ≤ h∣∣∣∣φ′′∣∣∣∣

L2= Ch |φ|2 (12.48)

in which

∣∣∣∣φ′′∣∣∣∣L2

= |φ|H2 = |φ|2 =

1∫0

(φ′′(x))2dx

12

(12.49)

Proof. Consider linear φeh(x) and φeI(x) (i.e. p = 1).

For an element e let Ee(x) = φ(x) − φeI(x) ∀x ∈ [xi, xi+1] be the inter-

polation error between φ and interpolant φeI(x). Since Ee(x) vanishes at xiand xi+1 of an element e, by virtue of Rolle’s theorem there exists at least

one point β between xi and xi+1 at which(Ee(x)

)′= 0. Then for any x

(Ee(x)

)′=

x∫β

(Ee(x)

)′′dx,

∣∣∣∣(Ee(x))′∣∣∣∣ ≤

x∫β

∣∣∣∣(Ee(x))′′∣∣∣∣ dx (12.50)

Since φeI(x) is linear, Ee(x) = φ− φeI implies that(Ee(x)

)′′= φ′′(x)− (φeI(x))′′ = φ′′(x) (12.51)


Applying Cauchy-Schwarz inequality to (12.50) and using (12.51)

∣∣∣∣(Ee(x))′∣∣∣∣ ≤

( x∫β

(1)2dx

) 12( x∫β

∣∣φ′′(x)∣∣2 dx) 1

2

(12.52)

≤ (he)12

( x∫β

∣∣φ′′(x)∣∣2 dx) 1

2

(12.53)

≤ (he)12

xi+1∫xi

(max

Ωe

∣∣φ′′(x)∣∣)2

dx

12

(12.54)

≤ (he)12

(he

(max

Ωe

∣∣φ′′(x)∣∣)2) 1

2

(12.55)

or ∣∣∣∣(Ee(x))′∣∣∣∣ ≤ he max

Ωe

∣∣φ′′(x)∣∣ (12.56)

Let

h = maxehe

maxΩe

∣∣φ′′(x)∣∣ ≤ max

[0,1]

∣∣φ′′(x)∣∣ (12.57)

Hence for ΩT we can write∣∣∣∣(Ee(x))′∣∣∣∣ =

∣∣φ′(x)− φ′I(x)∣∣ ≤ hmax

[0,1]

∣∣φ′′(x)∣∣ (12.58)

This proves (12.45).

Likewise (since Ee(xi) = 0),

Ee(x) =

x∫xi

(Ee(x)

)′dx,

∣∣∣Ee(x)∣∣∣ ≤ x∫

xi

∣∣∣∣(Ee(x))′∣∣∣∣ dx (12.59)

Applying Cauchy-Schwarz inequality

∣∣∣Ee(x)∣∣∣ ≤

xi+1∫xi

(1)2dx

12 x∫xi

∣∣∣∣(Ee(x))′∣∣∣∣2 dx

12

(12.60)



∣∣∣Ee(x)∣∣∣ ≤ (he)

12

x∫xi

he xi+1∫xi

(max

Ωe

∣∣φ′′(x)∣∣)2

dx

dx

12

≤ (he)12

hehe xi+1∫xi

(max

Ωe

∣∣φ′′(x)∣∣)2

dx

12

= (he)32

xi+1∫xi

(max

Ωe

∣∣φ′′(x)∣∣)2

dx

12

(12.61)

Hence, ∣∣∣Ee(x)∣∣∣ ≤ h2

e maxΩe

∣∣φ′′(x)∣∣ (12.62)

Using (12.57), (12.62) reduces to∣∣∣E(x)∣∣∣ ≤ h2 max

[0,1]

∣∣φ′′(x)∣∣ (12.63)

This proves (12.44). Consider

∣∣∣∣∣∣E′(x)∣∣∣∣∣∣2L2

≤∑e

xi+1∫xi

∣∣∣∣(Ee(x))′∣∣∣∣2 dx (12.64)

Substituting

∣∣∣∣(Ee(x))′∣∣∣∣ from (12.53) into (12.64)

∣∣∣∣∣∣E′(x)∣∣∣∣∣∣2L2

≤∑e

xi+1∫xi

(he

x∫β

∣∣φ′′(x)∣∣2 dx) dx (12.65)

≤ h∑e

xi+1∫xi

( xi+1∫xi

∣∣φ′′(x)∣∣2 dx) dx (12.66)

≤ h∑e

xi+1∫xi

(∣∣∣∣φ′′(x)∣∣∣∣2L2

)Ωedx (12.67)

≤ h2∑e

(∣∣∣∣φ′′(x)∣∣∣∣2L2

)Ωe

(12.68)

Thus, ∣∣∣∣∣∣E′(x)∣∣∣∣∣∣2L2

≤ h2∣∣∣∣φ′′(x)

∣∣∣∣2L2

(12.69)


Hence,∣∣∣∣∣∣E′(x)∣∣∣∣∣∣L2

=∣∣∣E′(x)

∣∣∣H1

=∣∣∣E′(x)

∣∣∣1≤ h

∣∣∣∣φ′′(x)∣∣∣∣L2

= h |φ|H2 = h |φ|2(12.70)

This proves (12.46).Consider

Ee(x) =

x∫xi

(Ee(x)

)′dx (12.71)

or ∣∣∣Ee(x)∣∣∣ ≤ x∫

xi

∣∣∣∣(Ee(x))′∣∣∣∣ dx (12.72)

Using Cauchy-Schwarz inequality∣∣∣Ee(x)∣∣∣ ≤ ( xi+1∫

xi

(1)2dx

) 12( x∫xi

∣∣∣∣(Ee(x))′∣∣∣∣2 dx) 1

2

(12.73)

≤ (he)12

( x∫xi

∣∣∣∣(Ee(x))′∣∣∣∣2 dx) 1

2

(12.74)

Substituting from (12.53)

∣∣∣Ee(x)∣∣∣ ≤ (he)

12

x∫xi

(he

x∫β

∣∣φ′′(x)∣∣2 dx) dx

12

(12.75)

∣∣∣∣∣∣E(x)∣∣∣∣∣∣2L2

=∑e

xi+1∫xi

∣∣∣Ee(x)∣∣∣2 dx (12.76)

≤∑e

xi+1∫xi

he

x∫xi

(he

x∫β

∣∣φ′′(x)∣∣2 dx) dx

dx (12.77)

≤∑e

xi+1∫xi

he

x∫xi

(he

xi+1∫xi

∣∣φ′′(x)∣∣2 dx) dx

dx (12.78)

≤ h2∑e

xi+1∫xi

( xi+1∫xi

(∣∣∣∣φ′′(x)∣∣∣∣2L2

)Ωedx

)dx (12.79)

≤ h4∑e

(∣∣∣∣φ′′(x)∣∣∣∣2L2

)Ωe

= h4∣∣∣∣φ′′(x)

∣∣∣∣2L2

(12.80)


Hence∣∣∣∣∣∣E(x)∣∣∣∣∣∣L2

= ||φ(x)− φI ||L2= |φ− φI |H0 = |φ− φI |0

≤ h2∣∣∣∣φ′′∣∣∣∣

L2= h2 |φ|H2 = h2 |φ|2 (12.81)

Now

||φ− φI ||2H1 = ||φ− φI ||21 = ||φ− φI ||2L2+∣∣∣∣φ′ − φ′I ∣∣∣∣2L2

(12.82)

Using (12.69) and (12.81) we have

||φ− φI ||2H1 ≤(h2 |φ|22 + h4 |φ|22

)(12.83)

≤ |φ|22(h2 + h4

)(12.84)

< C2h2 |φ|22 ;(h4 ≤ h2

)(12.85)

Hence

||φ− φI ||H1 = ||φ− φI ||1 ≤ Ch |φ|2 (12.86)

This proves (12.48).

Remarks.

From theorem 12.10 we have∣∣∣∣φ′ − φ′h∣∣∣∣L2≤∣∣∣∣φ′ − φ′I ∣∣∣∣L2

=∣∣∣∣∣∣E′(x)

∣∣∣∣∣∣L2

(12.87)

and

||φ− φh||L2≤ ||φ− φI ||L2

=∣∣∣∣∣∣E(x)

∣∣∣∣∣∣L2

(12.88)

Hence using (12.87), (12.88), (12.46), and (12.47) we finally have∣∣∣∣φ′ − φ′h∣∣∣∣L2≤ h |φ|2 = h

∣∣∣∣φ′′∣∣∣∣L2

= h |φ|H2 (12.89)

and

||φ− φh||L2≤ h2 |φ|2 = h2

∣∣∣∣φ′′∣∣∣∣L2

= h2 |φ|H2 (12.90)

and likewise

||φ− φh||H1 ≤ Ch |φ|2 = Ch∣∣∣∣φ′′∣∣∣∣

L2= Ch |φ|H2 (12.91)



Proposition 12.2. The derivation of the error estimates in proposition 12.1are presented for model problem 1 using GM/WF in which the operatoris self adjoint, hence the weak form is VC. In model problem 2 (section12.5.7.2) the differential operator is non-self adjoint and the error estimatesare derived for LSP in which the integral form is also VC. In this sectionwe consider a more general approach of deriving a priori error estimates forarbitrary degree of approximation p only based on the assumption that theintegral form is VC.

If the integral form resulting from a method of approximation is VC,then the following hold.

||φ− φh||L2≤ C1h

p+1 |φ|p+1

|φ− φh|Hq ≤ C2hp+1−q |φ|p+1 , seminorm of order q

||φ− φh||Hq ≤ C3hp+1−q |φ|p+1

(12.92)

And ifE = Aφ− f, residual function (12.93)

then||E||L2

≤ C4hp+1−2m |φ|p+1 (12.94)

In (12.94), 2m is the highest order of the derivative in the differential op-erator A. The constants C1, C2, C3, and C4 do not depend upon h andp.

Proof. Consider one dimensional BVP:

Aφ− f = 0 ∀x ∈ (0, L) = Ω (12.95)

Let ΩT = ∪eΩe be discretization of Ω in which Ωe = [xi, xi+1] is an element e.

Let φh be finite element approximation of φ over ΩT such that φh = ∪eφeh in

which φeh is local approximation of φ over Ωe. Let φI and φeI be interpolantsof φ of class C0 over ΩT and Ωe such that at the nodes φI agrees with thetheoretical solution φ. Thus, error estimation reduces to estimating errorbetween φ and φI over an element Ωe of length he. When φ(x) is analytic,it can be expanded in Taylor series in he over Ωe about some point j.

φ(x) = φ(he) = φj +he∂φj∂x

+h2e

2!

∂2φj∂x2

+ · · ·+ hpep!

∂pφj∂xp

+hp+1e

(p+ 1)!

∂p+1φj∂xp+1

+ . . .

(12.96)Consider a φeh over Ωe of degree p resulting from a VC integral form (hence,ensuring well-behaved solution), then the local approximation φeh at the same


point j can also be written as (assuming φeh agrees with φ(x) up to degreeof p, can only be true in asymptotic range),

φeh(x) = φh(he) = φj + he∂φj∂x

+h2e

2!

∂2φj∂x2

+ · · ·+ hpep!

∂pφj∂xp

(12.97)

Subtracting (12.97) from (12.96) we obtain

|φ(x)− φeh(x)| ≤ O(hp+1e )

∣∣∣∣∂p+1φ

∂xp+1

∣∣∣∣ (12.98)

||φ(x)− φeh(x)||2L2≤

xi+1∫xi

C21 (hp+1

e )2

∣∣∣∣∂p+1φ

∂xp+1

∣∣∣∣2 dx (12.99)

||φ(x)− φh(x)||2L2≤∑e

xi+1∫xi

C21 (hp+1

e )2

∣∣∣∣∂p+1φ

∂xp+1

∣∣∣∣2 dx (12.100)

Leth = max

ehe (12.101)

Then

||φ(x)− φh(x)||2L2≤ C2

1 (hp+1)2∑e

xi+1∫xi

∣∣∣∣∂p+1φ

∂xp+1

∣∣∣∣2 dx

≤ C21 (hp+1)2 |φ|2p+1

(12.102)

Therefore||φ(x)− φh(x)||L2

≤ C1hp+1 |φ|p+1 (12.103)

Using (12.96)–(12.103), it is rather straightforward to establish∣∣∣∣φ′(x)− φ′h(x)∣∣∣∣L2≤ C2h

p |φ|p+1 (12.104)

and by induction∣∣∣∣φq(x)− φqh(x)∣∣∣∣L2

= |φ(x)− φh(x)|Hq ≤ C2hp+1−q |φ|p+1 (12.105)

Using (12.103) and (12.105) we can establish that

||φ(x)− φh(x)||Hq ≤ C3hp+1−q |φ|p+1 , (q = 0 implies L2-norm) (12.106)


Remarks.

(1) The estimates in (12.105) and (12.106) apply to VC integral forms re-gardless of the method of approximation. Thus, these estimates hold forGM/WF for self adjoint operators and also hold for LSP for all threeclasses of differential operators.

(2) The local approximations used are always of class C0.

(3) The constants C1, C2, and C3 do not depend on h and p.

(4) The estimates (12.105) and (12.106) apply to all finite element processesin which the integral form is variationally consistent.

(5) From (12.105) and (12.106) we note that progressively increasing orderof derivatives of the finite element solution converge progressively slower.That is

||φ(x)− φh(x)||L2∝ hp+1 (12.107)∣∣∣∣φ′(x)− φ′h(x)

∣∣∣∣L2∝ hp (12.108)

and so on. Likewise

||φ(x)− φh(x)||H0 ∝ hp+1 (12.109)

||φ(x)− φh(x)||H1 ∝ hp (12.110)

and so on. From (12.108) and (12.110) we note that convergence rate inH1-norm is controlled by the convergence rate of the seminorm | ··· |H1

(i.e. highest order derivative in ||···||H1). This property holds universallyfor all operators and integral forms as long as they are variationallyconsistent.

(6) When examining ||E||L2, if the highest order of derivative in E is 2m,

then we have

||E||L2' ||φ− φh||H2m ≤ C4h

p+1−2m|φ|p+1 (12.111)

12.5.7.5 Convergence rates

In this section we present details of the convergence rates of various errornorms for finite element solutions obtained using GM/WF for self adjointoperators when B(···, ···) is symmetric and LSP for all three classes of differen-tial operators. We recall that when the integral form has best approximationproperty in some norm, hence is variationally consistent, we have the follow-ing a priori error estimate (derived for 1D BVP, equation (12.106)) in theasymptotic range:

||e||Hq = ||φ(x)− φh(x)||Hq ≤ (C3|φ|p+1)hp+1−q (12.112)


Taking log of both sides

log ||e||Hq ≤ log (C3|φ|p+1) + (p+ 1− q) log h (12.113)

ory ≤ C˜ +mx (12.114)

in whichy = ||e||Hq

C˜ = log (C3|φ|p+1)

m = p+ 1− qx = log h

(12.115)

We note that (12.114) is the equation of a straight line (when we useequality) in xy-space in which m is the slope and C˜ is the y-intercept. Thatis, if we plot log h versus ||e||Hq on an xy-plot, then we obtain a straight linewhose slope is (p+ 1− q) and intercept is log (C3|φ|p+1). Slope (p+ 1− q) iscalled the rate of convergence of ||e||Hq . Higher values of (p + 1 − q) implyfaster convergence of φh to φ measured in ||e||Hq . Equation (12.114) can beexpressed in terms of total degrees of freedom which is perhaps more ap-pealing in applications as dofs are more easily accessible than characteristiclength or size ‘h’ of the discretization ΩT . As the discretization ΩT is refined,the characteristic length h reduces and the total dofs increase, thus dofs areinversely proportional to h,

h ∝ 1

dofs, h = O

(1

dofs

)(12.116)

Using h = 1dofs in (12.113) and since log(1) = 0 we obtain

log ||e||Hq ≤ log (C3|φ|p+1)− (p+ 1− q) log(dofs) (12.117)

We keep in mind that dofs in (12.117) are purely due to uniform mesh re-finement. Thus, in order to determine convergence rate of ||e||Hq for finiteelement processes with VC integral forms we need to plot log ||e||Hq versuslog(dofs) and determine the slope of this curve (p + 1 − q), which is theconvergence rate in the asymptotic range. For a sequence of fixed discretiza-tions, as p increases convergence rate increases linearly.

Remarks.

(I) We note that ||e||Hq requires knowledge of theoretical solution φ,which may not be possible to determine for a practical application.

(II) When the approximation space Vh ⊂ Hk,p(Ωe) is minimally conform-ing or of higher order (i.e. k ≥ 2m + 1 for integrals over ΩT to be


Riemann or k = 2m if the Lebesgue integrals over ΩT are acceptable),then

√I =

√(E,E)ΩT = ||E||L2

in which the residual function can

be computed using E = Aφh− f over ΩT and Ee = Aφeh− f over Ωe.

||E||L2≤ C4h

p+1−2m|φ|p+1 (12.118)

using h = 1dofs and taking log of both sides

log ||E||L2≤ log (C4|φ|p+1)− (p+ 1− 2m) log(dofs) (12.119)

The dofs in (12.119) are also due to uniform h-refinement. Since||E||L2

does not require theoretical solution φ, it can be computed

using φh = ∪eφeh. Equation (12.119) can be used for any application

without the knowledge of theoretical solution as long as the approx-imation space is minimally conforming or of higher order than mini-mally conforming.


Proposition 12.3. When local approximation φeh is of progressively higherorder global differentiability, that is, in Vh ⊂ Hk,p(Ωe) scalar product spaces,the accuracy of the finite element solution progressively improves. In thisproposition we answer two important questions:

(1) Dependence of the a priori error estimates derived so far for local ap-proximations of class C0 on the order of the space k; that is, if the localapproximations are in Vh ⊂ Hk,p(Ωe) space how do the a priori estimateschange and the influence of k on convergence rate.

(2) The influence of the order k of the approximation space on the accuracyof the finite element computations.

Of course (1) and (2) are interdependent because when we have determined(1), the assessment of accuracy may be inferred from it.

The following a priori error estimate derived for 1D BVPs using C0 p-version local approximation can be extended when the local approximationsare in Vh ⊂ Hk,p(Ωe) spaces using the following two important considerationsor properties of local approximations in Vh ⊂ Hk,p(Ωe) spaces:

||φ(x)− φh(x)||Hq = ||φ(x)− φh(x)||q ≤ C3hp+1−q|φ|p+1 (12.120)

Property I

We consider a simple illustration of a 1D discretization using three nodep-version hierarchical local approximation finite elements in Vh ⊂ Hk,p(Ωe)


space. Let M be the number of elements in the discretization, then the totaldegrees of freedom (dofs) are given by

dofs = (M + 1)k +M(p− 2k + 1) (12.121)

p is the degree of local approximation (assumed same for all elements of thediscretization). Let us choose a p-level, say nine (9) and a one hundred (100)element discretization, then using (12.121) we can determine total degreesof freedom for k = 1, 2, . . . , 5 corresponding to the local approximations ofclass C0, C1, . . . , C4.

Table 12.1: Total dofs for a 100 element discretization at p = 9 for different values of theorder of space k

Type of local dofs

approximation

C0 ; k = 1 901

C1 ; k = 2 802

C2 ; k = 3 703

C3 ; k = 4 604

C4 ; k = 5 505

From Table 12.1 we observe that as k increases (i.e. progressively higherorder local approximations) the total degrees of freedom are progressivelyreduced. This is a significant property of the higher order local approxima-tions. From Table 12.1 we note that for C0, 901 dofs are reduced to 802 inthe case of C1 without much effect on accuracy of the solution. The sameholds for progressively higher order local approximations C2, C3, and so on;that is, the dofs continue to reduce with progressively increasing order ofspace without much effect on the accuracy. This behavior of the solutionaccuracy (say in ||E||L2

) holds regardless of the type of differential operatorand regardless of the method of approximation used to oconstruct the inte-gral form as long as the integral form is variationally consistent. Figure 12.2shows typical plots of ||E||L2

versus dofs at p = 5 for solutions of classes C0,C1, and C2. Typical points A, B, C correspond to solutions of classes C0,C1, and C2 for the same discretization and p-level (i.e. fixed h and p), withalmost same value of ||E||L2

but progressively reducing degrees of freedom.In view of the a priori error estimate (12.120) we can conclude that if h andp are fixed, then the dependence of the a priori estimate on k lies in C3, C4

[i.e. C3 = C3(k) in (12.120) and C4 = C4(k) in (12.111)].Property II

If we choose φh ∈ Vh ⊂ Hk,p(Ωe) and if φI is the interpolant that agrees

with φ at the inter-element nodes of the discretization; that is, φ(xi),∂jφ(xi)∂xj


-12

-10

-8

-6

-4

-2

0

1 1.5 2 2.5 3 3.5 4 4.5

log (

√I)

log (dofs)

First order system (2m = 1)

AB

C

C0, p=5

C1, p=5

C2, p=5

Figure 12.2: Typical ||E||L2versus dofs behavior for k = 1, 2, and 3

j = 1, 2, . . . agree with φI(xi),∂jφI(xi)∂xj

; j = 1, 2, . . . corresponding to localapproximations of classes C1, C2, . . . respectively, then in the considerationof the a priori error estimates we only need to consider (xi, xi+1) (i.e. interiorof the element). This suggests that in a priori estimate in (12.111) (forexample) only the coefficient C4 depends on k. That is, (12.111) holds whenφh ∈ Vh ⊂ Hk,p except that C4 = C4(k).

Remarks.

(1) From properties I and II it is clear that when φh ∈ Vh ⊂ Hk,p(Ωe), inthe error estimate (12.111) the terms hp+1−2m|φ|p+1 and likewise theterms hp+1−q|φ|p+1 in (12.120) remain unaffected. Only the coefficientsC4 and C3 show mild dependence on k.

(2) In view or properties I and II, we conclude that if C0, C1, . . . solutionsof a BVP are to be computed for a fixed number of degrees of freedom,then progressively more degrees of freedom can be added to solutions ofclass C1, C2, . . . so that the total dofs in all classes of solutions are thesame. We recall that with same values of h and p in the solutions ofclass C0, C1, C2,. . . , ||E||L2

(Fig. 12.2) remains virtually the same forall classes, however the total dofs are progressively reduced.

The consequence of adding more dofs (through h-refinement) with pro-gressively increasing order of space so that in each case the dofs matchwith C0 solutions is clearly improved accuracy of φh reflected by pro-gressively reducing ||E||L2

. Clearly, in doing so the convergence rate(p + 1 − q) or (p + 1 − 2m) is not affected. Thus, log||E||L2

versus

12.6. MODEL PROBLEMS 719

log(dofs) graphs for solutions of classes C0, C1, . . . in the asymptoticrange are parallel to each other but with progressively lower values of||E||L2

as shown in Fig. 12.2. That is graph for C1 is below C0 and thatof C2 is below C1 and so on, but they are all almost parallel.

12.5.7.7 General Remarks

(1) The a priori error estimates are presented for one dimensional boundaryvalue problems. Their extensions to 2D and 3D require more elaboratederivations (see references) and new definitions of h and |φ|p+1, but theconvergence rates remain the same as (p+ 1− q) or (p+ 1−2m) derivedfor 1D BVPs.

(2) We remark again that the rates only hold in the asymptotic range.

(3) The integral forms must be VC so that the best approximation propertyof φh holds in some norm in order for these estimates to remain valid.The estimates derived here hold for: (a) GM/WF for self adjoint oper-ators when the bilinear functional B(···, ···) is symmetric and (b) for LSPbased on residual functional for all three classes of differential operator.

(4) In case of GM/WF for non-self adjoint and non-linear operators, thea priori estimates derived here do not hold. In case of such operatorsthe functional B(···, ···) generally consists of a symmetric part and a non-symmetric part. With sufficient mesh refinement if we can ensure thatthe behavior is dominated by the symmetric part, then the estimatesderived here hold in the range of calculations when asymptotic range isrealized. We illustrate this aspect through model problems presented ina later section.

12.6 Computations of a priori error estimates andconvergence rates

In this section we present numerical studies related to the computationof a priori error estimates and convergence rates for BVPs described by selfadjoint, non-self adjoint, and non-linear differential operators in which VCintegral forms are constructed using GM/WF for BVP described by selfadjoint differential operators and using LSP for BVPs described by all threeclasses of differential operators.


12.6.1 Model problem 1: Self-adjoint operator, 1D diffusionequation

We consider the same BVP as in section 5.2.2 and section 5.2.2.1 case(a),

− d

dx

(adφ

dx

)= q(x) ∀x ∈ (0, L) = Ω ⊂ R1 (12.122)

BC: φ(0) = 0, adφ

dx

∣∣∣∣x=L

= 0 (12.123)

If we choose a = 1, L = 1, q(x) = xn, n = 6, then the theoretical solution φor φt is given by

φt(x) = φ(x) =

(− 1

a(x+ 1)(x+ 2)

)xn+2 + x

(1

a(x+ 1)

)Ln+1 (12.124)

(a) GM/WF: The differential operator A = − ddx

(a ddx

)is linear and A∗ = A.

The integral form using GM/WF is given by (over Ω = [0, 1])(dφ

dx,dv

dx

)Ω

= (q(x), v)Ω, v = δφ ∀v ∈ Vh ⊂ Hk,p (12.125)

or

B(φ, v) = l(v) (12.126)

B(···, ···) is bilinear and symmetric and l(···) is linear. The integral form(weak form) is VC due to the fact that δ(B(φ, v)) =

(dvdx ,

dvdx

)Ω> 0 ∀v ∈

Vh hence a solution φ from (12.126) minimizes I(φ) = 12B(φ, φ)− l(φ).

(b) LSP based on residual functional

(I) LSP using higher order system (without auxiliary equation)Using (12.122), referred to as the higher order differential equationor system, if we let φh be approximation of φ over ΩT then

I(φh) = (E,E)ΩT ; E = Aφh − f =d2φhdx2

+ q(x) ∀x ∈ [0.1]

(12.127)

δI(φh) = 2(E, δE) = 0⇒ (Aφh , Av)ΩT + (q(x), Av)ΩT = 0(12.128)

or (d2φhdx2

,d2v

dx2

)ΩT

+

(q(x),

d2v

dx2

)ΩT

= 0 (12.129)

or

B(φh, v)− l(v) = 0 (12.130)


δ2I(φh) = B(v, v) =

(d2v

dx2,d2v

dx2

)ΩT

> 0 ∀v ∈ Vh ⊂ Hk,p

(12.131)Hence, the integral form (12.130) is variationally consistent.

(II) LSP using first order systemLet τ = dφ

dx , hence (12.122) can be written as a system of two firstorder equations.

dτ

dx+ q(x) = 0

τ − dφ

dx= 0

(12.132)

LSP for (12.132) follows standard procedure. Let φh and τh ∈Vh ⊂ Hk,p be approximations of φ and τ , then

I =

2∑i=1

(Ei, Ei)ΩT , E1 =dτhdx

+ q(x), E2 = τh −dφhdx

= 0

(12.133)

δI = 22∑i=1

(Ei, δEi)ΩT = 0 (12.134)

δ2I =

2∑i=1

(δEi, δEi) > 0 (12.135)

Hence the integral form (12.134) resulting from LSP is variation-ally consistent.

Remarks.

(i) All other methods of approximation yield VIC integral forms, henceare not considered as in such cases the a priori error estimates and theconvergence rates are not valid.

(ii) In the numerical studies we consider GM/WF and LSP for higher orderas well as first order system of differential equations describing BVPs.

12.6.1.1 GM/WF

In this section we present numerical studies for the integral form (12.125)for ΩT = ∪

eΩe, discretization of Ω = [0, 1]. We consider uniform discretiza-

tions employing three node p-version hierarchical 1D elements with localapproximations in scalar product space Vh ⊂ Hk,p(Ω). We begin with twoelement uniform discretization and perform uniform mesh refinement con-taining 4, 8, 16, . . . elements. Since in this model problem the theoreticalsolution φ is known, various error norms can be computed. We note from


the description of the BVP (12.122) that in this case 2m = 2 (highest orderof the derivative in the BVP) and the integral form resulting from GM/WFcontains only up to first order derivatives of the dependent variable and thetest function. We consider computations using solutions of classes C0, C1,and C2 at different p-levels with uniform mesh refinements. Computed re-sults for solution of class C0 are summarized in Table 12.2. The integralform is VC and φh, the computed solution, has best approximation propertyin B(···, ···)-norm.

In this case the following a priori error estimates hold (Proposition 12.2):

||φ− φh||Hq ≤ C3hp+1−q|φ|p+1

|φ− φh|Hq ≤ C2hp+1−q|φ|p+1

||Aφ− f ||L2= ||E||L2

≤ C4hp+1−2m

(12.136)

For this BVP, 2m = 2 and q depends on the type of norm. Table 12.2shows the theoretical values of the convergence rates of various error normsfor solutions of class C0 at p-levels of 2 and 5. Graphs of the log of er-ror norms versus log of dofs for these solutions are shown in Fig. 12.3. Wenote that due to smoothness of the theoretical solution even the two elementdiscretization yields the error norms in the asymptotic range; that is, pre-asymptotic and onset of asymptotic ranges in these solutions do not appearin Fig. 12.3. All computations are in the asymptotic range, hence onsetof post-asymptotic and post-asymptotic ranges are also absent. Calculatedconvergence rates (shown in Fig. 12.3 and Table 12.2) are in perfect agree-ment with the theoretical convergence rates calculated using (12.136). Wenote that in ||···||H1 and |··· |H1 error norms the integrals over ΩT are Lebesgue,but the norms are well-behaved due to smoothness of φ.

Table 12.2: Theoretical and computed convergence rates of various error norms.GM/WF, C0 solutions, p = 2 and 5

Type ofp-level

Solution Class;q

Convergence Rates

Norm k Theoretical Calculated

||···||H0 = ||···||L2

2C0; k = 1

0 3 3

5 0 6 6

||···||H1 ; | ··· |H1

2C0; k = 1

1 2 2

5 1 5 5

Figure 12.4 shows plots of log of various norms and seminorms versuslog of degrees of freedom at p = 3 and 5 for C1 solutions. A summary ofthe computed convergence rates of various error norms and comparison withthe theoretical convergence rates obtained using (12.136) are shown in table12.3. The agreement is perfect. Here we note that in computing ||···||H2 and


-15

-10

-5

0

0.5 1 1.5 2 2.5 3 3.5 4

log

(||φ

- φ

h|| H

q)

o

r

log

(|φ

- φ

h| H

q)

log (dofs)

GM/WF, self adjoint operator

Vh ⊂ H

1,p ; C

0 Solutions

Theoretical rate: p+1-q

Convergence Rate

( Calculated )

3

2

6

5

1

3

12

1

51

6

p=2 : ||•||H

0

p=2 : ||•||H

1 ; |•|H

1

p=5 : ||•||H

0

p=5 : ||•||H

1 ; |•|H

1

Figure 12.3: log||···||Hq versus log(dofs) for solutions of class C0 (GM/WF, model problem1, p = 2 and 5)

| ··· |H2 , the integrals over ΩT are Lebesgue but error norms are well-behaveddue to smoothness of φ. Also, nearly all computations shown in Fig. 12.4are in the asymptotic range, except for the last point for ||···||H0 at p = 3 and||···||H1 at p = 5.


Type ofp-level

Solution Class;q

Convergence Rates


||···||H0 = ||···||L2

3C1; k = 2

0 4 4

5 0 6 6

||···||H1 ; | ··· |H1

3C1; k = 2

1 3 3

5 1 5 5

||···||H2 ; | ··· |H2

3C1; k = 2

2 2 2

5 2 4 4

Log of various error norms and seminorms versus log of degrees of freedomare for solutions of class C2 at p = 5 and 7 are shown in Fig. 12.5. Sincek = 3 for Vh ⊂ Hk,p(Ωe), all integrals in all error norms are Riemann overthe discretization ΩT . Computed error norms using (12.136) and comparisonwith the computed convergence rates of error norm are shown in Table 12.4.


-15

-10

-5

0

5

0.5 1 1.5 2 2.5 3 3.5 4

log

(||φ

- φ

h|| H

q)

o

r

log

(|φ

- φ

h| H

q)

log (dofs)


Vh ⊂ H

2,p ; C

1 Solutions


Convergence Rate

( Calculated )

4

3

2

6

5

4

p=3 : ||•||H

0

p=3 : ||•||H

1 ; |•|H

1

p=3 : ||•||H

2 ; |•|H

2

p=5 : ||•||H

0

p=5 : ||•||H

1 ; |•|H

1

p=5 : ||•||H

2 ; |•|H

2


-15

-10

-5

0

1 1.5 2 2.5 3

log

(||φ

- φ

h|| H

q)

o

r

log

(|φ

- φ

h| H

q)

log (dofs)


Vh ⊂ H

3,p ; C

2 Solutions


Convergence Rate

( Calculated )

6

5

4

8

7

6

p=5 : ||•||H

0

p=5 : ||•||H

1 ; |•|H

1

p=5 : ||•||H

2 ; |•|H

2

p=7 : ||•||H

0

p=7 : ||•||H

1 ; |•|H

1

p=7 : ||•||H

2 ; |•|H

2



We observe perfect match between the theoretical values and the computedvalues. Except for the last point shown in Fig. 12.5 for ||···||H1 at p = 5, allother computed results are in the asymptotic range due to smoothness of φ.


Type ofp-level

Solution Class;q

Convergence Rates


||···||H0 = ||···||L2

5C2; k = 3

0 6 6

7 0 8 8

||···||H1 ; | ··· |H1

5C2; k = 3

1 5 5

7 1 7 7

||···||H2 ; | ··· |H2

5C2; k = 3

2 4 4

7 2 6 6

Figure 12.6 shows plots of log||···||H0 (or ||···||L2) versus log of dof for so-

lutions of class C0, C1, and C2 (k = 1, 2, 3) at p = 5. All three graphs oflog||···||H0 versus log of dofs for k = 1, 2, 3 are parallel, confirming that theconvergence rate of ||···||H0 is independent of the order k of the approxima-tion space. We note that graph for C1 appears below C0 and the graphfor C2 is below C1 confirming that for given dofs, as the order k of spaceis increased, the error in the computed solution φh (measured in H0-norm)decreases without affecting the convergence rate.

The BVP in this model problem is described by a second-order differentialoperator (2m = 2); hence, k = 3 corresponds to minimally conforming spaceVh ⊂ Hk,p(Ωe) for which the integrals are always Riemann. However, dueto smoothness of φ, when k = 2 (solutions of class C1) in which case theintegrals over ΩT are Lebesgue, the solution φh is expected to convergeweakly to class C2. Next we consider solutions of class C2 with p = 5(minimum). Numerical solutions are computed for uniform mesh refinementsbeginning with a two-element uniform discretization. For each discretizationwe calculate ||···||H2 and ||E||L2

= ||Aφh − f ||L2. Since the rate of convergence

of ||···||H2 is controlled by | ··· |H2 , we expect the convergence rates of ||···||H2

and ||E||L2to be nearly same. We clearly see this in Fig. 12.7. Graphs

for C1 and C2 are parallel, confirming the same convergence rates of ||···||H2

(or ||E||L2) for solutions of class C1 (k = 2) and class C2 (k = 3). The

convergence rate in the case of ||···||H2 is p+ 1− q = 5 + 1− 2 = 4, whereas inthe case of ||E||L2

is p+ 1− 2m = 5 + 1− 2 = 4. Plots in Fig. 12.7 confirmthat rate of convergence of error norms is independent of the order of theapproximation space.


-16

-14

-12

-10

-8

-6

-4

-2

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

log

(||φ

- φ

h|| H

0)

log (dofs)



Convergence Rate

( calculated )

6

6

6

p=5 : C0

p=5 : C1

p=5 : C2

Figure 12.6: log||···||H0 versus log(dofs) for solutions of classes C0, C1, and C2 at p = 5(GM/WF, model problem 1)

-12

-10

-8

-6

-4

-2

0

0.5 1 1.5 2 2.5 3 3.5

log

(||φ

- φ

h|| H

2)

o

r

log

(||E

|| L2

)

log (dofs)


Theoretical rate: p+1-q ; p+1-2m Convergence Rate

( Calculated )

4

4

p=5 : ||•||H

2 ; ||E||L

2

: C1

p=5 : ||•||H

2 ; ||E||L

2

: C2

Figure 12.7: log||···||H2 or log||E||L2versus log(dofs) for solutions of classes C1 and C2

at p = 5 (GM/WF, model problem 1)


12.6.1.2 LSP, higher-order system (no auxiliaryequation)

In this study we consider finite element formulation of model problem(12.122) using least-squares process based on residual functional. We con-sider solutions of class C1 as well as C2. In case of C1 solutions integralsover ΩT are Lebesgue whereas for solutions of class C2 the integrals areRiemann. Figure 12.8 shows plots of log||···||Hq and log| ··· |Hq versus log ofdofs for p-levels of 3 and 5 calculated using uniform mesh refinement. Cal-culated convergence rates of various error norms are also shown in Fig. 12.8.A summary of the details and calculated theoretical convergence rates ofvarious error norms and comparison with calculated convergence rates arealso shown in Table 12.5.

-10

-5

0

5

0.5 1 1.5 2 2.5 3 3.5

log

(||φ

- φ

h|| H

q)

o

r

log

(|φ

- φ

h| H

q)

log (dofs)

LSP, self adjoint operator

Vh ⊂ H

2,p ; C

1 Solutions

(Higher order system, 2m = 2)


Convergence Rate

( Calculated )

4

3

2

6

5

4

p=3 : ||•||H

0

p=3 : ||•||H

1 ; |•|H

1

p=3 : ||•||H

2 ; |•|H

2

p=5 : ||•||H

0

p=5 : ||•||H

1 ; |•|H

1

p=5 : ||•||H

2 ; |•|H

2

Figure 12.8: log||···||Hq versus log(dofs) for solutions of class C1 (LSP, model problem 1,p = 3 and 5)

Agreement between theoretical and calculated values is excellent. Herealso we observe absence of pre-asymptotic and onset of asymptotic rangesdue to smoothness of the theoretical solution. Some graphs for significant re-finement show appearance of post-asymptotic (or onset of post-asymptotic)range.

Similar studies for solutions of class C2 are shown in Fig. 12.9 for ||···||H0 ;||···||H1 , | ··· |H1 ; and ||···||H2 , | ··· |H1 norms at p-levels of 5 and 7. The computedconvergence rates of the error norms are in perfect agreement with theoretical


Table 12.5: Theoretical and computed convergence rates of various error norms. LSP,C1 solutions, p = 3 and 5

Type ofp-level

Solution Class;q

Convergence Rates


||···||H0 = ||···||L2

3C1; k = 2

0 4 4

5 0 6 6

||···||H1 ; | ··· |H1

3C1; k = 2

1 3 3

5 1 5 5

||···||H2 ; | ··· |H2

3C1; k = 2

2 2 2

5 2 4 4

rates calculated using (p+ 1− q), shown in Table 12.6.

-15

-10

-5

0

1 1.5 2 2.5 3

log

(||φ

- φ

h|| H

q)

o

r

log

(|φ

- φ

h| H

q)

log (dofs)


Vh ⊂ H

3,p ; C

2 Solutions



Convergence Rate

( Calculated )

6

5

4

8

7

6

p=5 : ||•||H

0

p=5 : ||•||H

1 ; |•|H

1

p=5 : ||•||H

2 ; |•|H

2

p=7 : ||•||H

0

p=7 : ||•||H

1 ; |•|H

1

p=7 : ||•||H

2 ; |•|H

2

Figure 12.9: log||···||Hq versus log(dofs) for solutions of class C2 (LSP, model problem 1,p = 5 and 7)

Graphs of log||···||H2 and log||E||L2(or√I) versus log of dofs for solutions

of class C1 and C2 obtained using uniform mesh refinement are shown inFig. 12.10. Calculated convergence rates are also shown in Fig. 12.10. For||···||H2 error norm the theoretical rate is (p + 1 − q) whereas for ||E||L2

itis (p+ 1− 2m). The theoretical convergence rates are in perfect agreementwith those calculated using graphs in Fig. 12.10. We note that C1 and C2


Table 12.6: Theoretical and computed convergence rates of various error norms. LSP,C2 solutions, p = 5 and 7

Type ofp-level

Solution Class;q

Convergence Rates


||···||H0 = ||···||L2

5C2; k = 3

0 6 6

7 0 8 8

||···||H1 ; | ··· |H1

5C2; k = 3

1 5 5

7 1 7 7

||···||H2 ; | ··· |H2

5C2; k = 3

2 4 4

7 2 6 6

graphs for same p-level (5) are parallel to each other and C2 graph is belowC1, confirming that the convergence rates for k = 2 and k = 3 are same (i.e.independent of k), the order of space, but for k = 3 the solution has betteraccuracy compared to k = 2.

-12

-10

-8

-6

-4

-2

0

2

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

log (

||φ -

φh|| H

2)

or

lo

g (

||E|| L

2

)

log (dofs)



Theoretical rates: p+1-q ; p+1-2m Convergence Rate

( Calculated )

4

4

p=5 : ||•||H

2 ; ||E||L

2

: C1

p=5 : ||•||H

2 ; ||E||L

2

: C2


at p = 5 (LSP, model problem 1)

Remarks. Numerical studies for LSP using auxiliary equation (i.e. a first-order system) are not presented for this model problem but will be presentedfor the next model problem, 1D convection-diffusion equation.


12.6.2 Model problem 2: Non-self-adjoint operator,1D convection-diffusion equation

We consider 1D convection-diffusion equation described by non-self ad-joint operator (same as in section 6.2) for computing a priori error estimatesand convergence rates and compare them with their theoretical values.

dφ

dx− 1

Pe

d2φ

dx2= f(x) ; ∀x ∈ (0, 1) = Ω (12.137)

BCs: φ(0) = 1, φ(1) = 0 (12.138)

We consider f(x) = 0. Theoretical solution of (12.137)–(12.138), finite ele-ment solution using GM/WF and LSP using higher order system (no aux-iliary variables) and using first order system (using auxiliary variables) is

given in chapter 6. For this BVP the operator A = ddx −

1Pe

d2

dx2is linear but

A∗ = − ddx −

1Pe

d2

dx26= A, hence the integral form from GM/WF is VIC, but

LSP for higher order as well as first order system of differential equations isVC.

(a) GM/WF: The integral form of (12.137)–(12.138) is given by (for f(x) =0)(

dφ

dx, v

)Ω

+1

Pe

(dφ

dx,dv

dx

)Ω

= 0, v = δφ ∀v ∈ V ⊂ Hk,p(Ω) (12.139)

B(φ, v) = l(v) ; l(v) = 0 (12.140)

B(φ, v) is bilinear but not symmetric and

δB(φ, v) =

(dv

dx, v

)Ω

+1

Pe

(dv

dx,dv

dx

)Ω

(12.141)

does not yield a unique extremum principle. Hence, the integral form(12.140) is VIC.

(b) LSP based on residual functional:

(I) Higher order system (without auxiliary equation)In this case we use (12.137) without introducing auxiliary equa-tion, that is without reducing (12.137) into a first order system ofequations. Let φh be approximation of φ over ΩT , then

I(φh) = (E,E)ΩT , E = Aφh − f =dφhdx− 1

Pe

d2φhdx2

(12.142)

δI(φh) = 2(E, δE) = 0 ⇒ (Aφh, Av) = 0 (12.143)


or (dφhdx− 1

Pe

d2φhdx2

,dv

dx− 1

Pe

d2v

dx2

)ΩT

= 0 (12.144)

or

B(φh, v) = 0 (12.145)

δ2I(φh) = B(v, v) = (Av,Av)ΩT

=

(dv

dx− 1

Pe

d2v

dx2,dv

dx− 1

Pe

d2v

dx2

)ΩT

> 0

∀v ∈ Vh ⊂ Hk,p(ΩT ) (12.146)

Hence, the integral form (12.145) is VC.(II) First order system

Let τ = dφdx , hence (12.137) can be written as

dφ

dx− 1

Pe

dτ

dx= 0

τ − dφ

dx= 0

(12.147)

LSP for (12.147) follows standard procedure (parallel to equations(12.132)–(12.135)). Details are straightforward. See chapter 6 forother similar model problems.

Remarks.

(1) Since GM/WF yields VIC integral form and does not have best ap-proximation property as the operator A is not self adjoint, hence the apriori error estimates derived in earlier sections using best approxima-tion property in B-norm do not hold in this case. Nonetheless we presentnumerical studies for GM/WF for this model problem to illustrate someimportant aspects of error norms in a later section.

(2) Integral form derived using LSP is VC and has best approximation prop-erty in E-norm or

√I, I being residual functional, hence the same a pri-

ori error estimates derived for LSP for self adjoint operators hold hereas well.

12.6.2.1 LSP: First order system

Domain Ω = [0, 1] is discretized using 3-node p-version 1D elements ofhigher order global differentiability into 2, 4, 6,. . . element uniform meshes.The solutions are computed using finite element formulation based on LSPfor first order system of equations. Solutions of classes C0, C1, and C2


are considered at different p-levels. For this problem the a priori estimates(12.136) hold as well with 2m = 1 due to the fact that it is a first ordersystem of equations. LSP has best approximation property in E-norm andthe integral form is VC,

||φ− φh||Hq ≤ C3hp+1−q|φ|p+1


||Aφ− f ||L2= ||E||L2

≤ C4hp+1−2m

(12.148)

First we consider solutions of class C0 at p = 2 and 5 and with Pe = 100.Due to C0 local approximation and the first order system, integrals over ΩT

are Lebesgue but due to smoothness of φ weak convergence of computedφh to C1 class is expected. Figure 12.11 shows plots of log of various errornorms versus log of the dofs at p = 2 and 5. Details of the studies are alsogiven in table 12.7. Theoretical convergence rates are in perfect agreementwith the calculated rates (also shown in Fig. 12.11). As p-level increasesfrom 2 to 5 convergence rates also show increase by 3 at p = 5 comparedto those at p = 2. We clearly observe pre-asymptotic, onset of asymptotic,and asymptotic ranges in all cases. For p = 5 also observe onset of post-asymptotic and post-asymptotic range. We note that even though LSP doesnot have best approximation property in B-norm but due to the fact thatthe integral form is VC, the convergence rate of LSP (12.148) are same asthose of GM/WF for self adjoint operators (12.136).

Table 12.7: Theoretical and computed convergence rates of various error norms. LSP,first order system, C0 solutions, p = 2 and 5, Pe = 100

Type ofp-level

Solution Class;q

Convergence Rates


||···||H0 = ||···||L2

2C0; k = 1

0 3 3

5 0 6 6

||···||H1 ; | ··· |H1

2C0; k = 1

1 2 2

5 1 5 5

As p-level is increased convergence rate increases proportionately. Deriva-tives converge more slowly than functions, hence convergence rate of ||···||H1

is one order lower than that of ||···||H0 or ||···||L2. Since the convergence rate

of ||···||H1 is dominated by the first derivative, ||···||H1 and | ··· |H1 have sameconvergence rates (also clear from (12.148)).

Solutions of class C1 at p = 3 and 5 are considered here. Results ob-tained using uniform mesh refinement are given in table 12.8. Plots of logof various error norms versus log of dofs and calculated convergence rates


-15

-10

-5

0

5

10

1 1.5 2 2.5 3 3.5 4 4.5

log

(||φ

- φ

h|| H

q)

o

r

log

(|φ

- φ

h| H

q)

log (dofs)

LSP, non-self adjoint operator

Vh ⊂ H

1,p ; C

0 Solutions

(First order system, 2m=1)

Pe=100


Convergence Rate

( Calculated )

3

2

6

5

p=2 : ||•||H

0

p=2 : ||•||H

1 ; |•|H

1

p=5 : ||•||H

0

p=5 : ||•||H

1 ; |•|H

1

Figure 12.11: log||···||Hq versus log(dofs) for solutions of class C0 (LSP, model problem2, first order system, p = 2 and 5, Pe = 100)

are shown in Fig. 12.12 and are compared with theoretical convergence ratesin Table 12.8. Calculated and theoretical convergence rates are in perfectagreement. Comparing results in Tables 12.7 and 12.8, we note that whenp = 5 the convergence rates of error norms are independent of k (i.e. atp = 5), solutions of class C0 and C1 have same convergence rates for thesame norm, confirming that convergence rates of the error norms are not afunction of k, the order of the approximation space. Pre-asymptotic, onsetof asymptotic, and asymptotic ranges are clearly observed in Fig. 12.12.

Solutions of class C2 at p = 5 and 7 are considered next. Results obtainedusing uniform mesh refinement are shown in Fig. 12.13 and are presented inTable 12.9 and are compared with the theoretical convergence rates obtainedusing (12.148). Once again the agreement is perfect. Again, we note fromTables 12.8 and 12.9 and Figs. 12.12 and 12.13 that at p = 5 the convergencerates are independent of k. In this case the integrals in the computations ofthe error norms are always Riemann.

Figure 12.14 shows plots of log||···||H2 and log√I versus log of dof for so-

lutions of class C1 and C2 at p = 5. Since the differential operator has thehighest derivative of order 2, the convergence rate of

√I is expected to be

same as that of ||···||H2 or |···|H2 for both classes of solutions. This is confirmedin Fig. 12.14. Convergence rate in case of C1 and C2 solutions are same (4 inthis case), but C2 solutions have better accuracy for a given dofs, confirming


Table 12.8: Theoretical and computed convergence rates of various error norms. LSP,first-order system, C1 solutions, p = 3 and 5, Pe = 100

Type ofp-level

Solution Class;q

Convergence Rates


||···||H0 = ||···||L2

3C1; k = 2

0 4 4

5 0 6 6

||···||H1 ; | ··· |H1

3C1; k = 2

1 3 3

5 1 5 5

||···||H2 ; | ··· |H2

3C1; k = 2

2 2 2

5 2 4 4

-10

-5

0

5

10

15

1 1.5 2 2.5 3 3.5 4 4.5

log

(||φ

- φ

h|| H

q)

o

r

log

(|φ

- φ

h| H

q)

log (dofs)


Vh ⊂ H

2,p ; C

1 Solutions


Pe=100


Convergence Rate

( Calculated )

4

3

2

6

5

4

p=3 : ||•||H

0

p=3 : ||•||H

1 ; |•|H

1

p=3 : ||•||H

2 ; |•|H

2

p=5 : ||•||H

0

p=5 : ||•||H

1 ; |•|H

1

p=5 : ||•||H

2 ; |•|H

2


again that convergence rates of error norms or residual functional are nota function of the order k of the approximation space. Calculated rates arein perfect agreement with the theoretical rates. Figure 12.15 shows plotsof log of ||···||H0 = ||···||L2

versus log of dofs for solution of classes C0, C1,and C2 at p = 5. We observe same convergence rates for k = 1, 2, and 3but better accuracy of the solution with progressively increasing k. Theserates for LSP match perfectly with GM/WF for self adjoint operators due


-15

-10

-5

0

5

10

15

1 1.5 2 2.5 3 3.5 4

log

(||φ

- φ

h|| H

q)

o

r

log

(|φ

- φ

h| H

q)

log (dofs)


Vh ⊂ H

3,p ; C

2 Solutions


Pe=100


Convergence Rate

( Calculated )

6

5

4

8

7

6

p=5 : ||•||H

0

p=5 : ||•||H

1 ; |•|H

1

p=5 : ||•||H

2 ; |•|H

2

p=7 : ||•||H

0

p=7 : ||•||H

1 ; |•|H

1

p=7 : ||•||H

2 ; |•|H

2


Table 12.9: Theoretical and computed convergence rates of various error norms. LSP,first-order system, C2 solutions, p = 5 and 7, Pe = 100

Type ofp-level

Solution Class;q

Convergence Rates


||···||H0 = ||···||L2

5C2; k = 3

0 6 6

7 0 8 8

||···||H1 ; | ··· |H1

5C2; k = 3

1 5 5

7 1 7 7

||···||H2 ; | ··· |H2

5C2; k = 3

2 4 4

7 2 6 6

to the fact that in both cases the integral forms are variationally consistent.This proves again that the best approximation property in B-norm is nota requirement for establishing convergence rate. It is the variational con-sistency of the integral form that matters. Clearly the LSP does not havebest approximation property in B-norm, yet has same convergence rates asGM/WF for self adjoint operators due to the fact that in both cases theintegral forms are variationally consistent.


-10

-5

0

5

1 1.5 2 2.5 3 3.5 4 4.5

log

(||φ

- φ

h|| H

0)

log (dofs)



Pe=100


Convergence Rate

( Calculated )

6

6

6

p=5 : C0

p=5 : C1

p=5 : C2

Figure 12.14: log||···||H0 versus log(dofs) for solutions of classes C0, C1, and C2 at p = 5(LSP, first order system, model problem 2)

-10

-5

0

5

1 1.5 2 2.5 3 3.5 4 4.5

log

(||φ

- φ

h|| H

1)

o

r

log

(√

I)

log (dofs)



Pe=100

Theoretical rates:

p+1-q ; p+1-2m

Convergence Rate

( Calculated )

5

5

5

p=5 : ||•||H

1 ; √I : C1

p=5 : ||•||H

1 ; √I : C1

p=5 : ||•||H

1 ; √I : C2


at p = 5 (LSP, first order system, model problem 2)


12.6.2.2 GM/WF

Since the differential operator is non-self adjoint the GM/WF will yieldVIC integral form in which B(···, ···) is nonsymmetric and we lose the bestapproximation property in B-norm. Nonetheless we conduct some numericalexperiments to monitor convergence rates of various error norms. First wenote that GM/WF in this model problem will yield the following elementequations for an element e (when p = 1 and f(x) 6= 0). See chapter 6:[

[Ke1 ] +

1

Pehe[Ke

2 ]

]δe = P e+ fe (12.149)

and the assembled equations for discretization ΩT = ∪eΩe are

[K]δ =

[[K1] +

1

Pehe[K2]

]δ = P+ f ; h = he (12.150)

in which δ = ∪eδe and [K1], [K2] are due to assembly of [Ke

1 ] and [Ke2 ]

for ΩT . As we have seen in chapter 6, [K1] is due to convection term (i.e.dφdx ) and [K2] is due to diffusion (i.e. d2φ

dx2); [K1] is nonsymmetric with zeros

on the diagonals after φ(0) = 1 and φ(1) = 0 BCs are imposed, thus if Peh islarge, the contribution of [K2] to [K] is almost insignificant compared to thecontribution of [K1] and the computations using (12.150) will fail. On theother hand if the discretization ΩT is sufficiently refined, the contributionof [K2] to [K] overshadows that of [K1] and the behavior will be dominated

by [K2] (i.e. d2φdx2

term in the differential operator). When this happensthe integral form from GM/WF will behave like a VC integral form as it is

primarily due to 1Pe

d2φdx2

term in the differential operator which is self adjoint,hence the convergence rates of various error norms will be similar to GM/WFfor self adjoint operator.

For numerical experiments we consider Pe = 100 and Pe = 1000. ForPe = 1000 the solution gradients are more isolated near x = 1 and are higherin magnitude compared to Pe = 100. We consider solutions of class C1 atp = 3 for both Peclet numbers. Progressively refined uniform discretizationsare used for computing solutions and error norms. Figure 12.16 shows er-ror norms versus dof plots for solutions of class C1, p = 3 for Pe = 100.We note that due to smoothness of the solutions, the asymptotic range inwhich [K2] dominates is quickly achieved, and the computations succeed formeshes of 16 elements or more. In this range calculated convergence ratesmatch perfectly with the theoretical rates for self adjoint operator. In this

range the BVP reduces to 1Pe

d2φdx2

= 0 as the contribution of dφdx term in this

range is insignificant. For meshes with 16 elements or fewer the calculatedsolution from (12.150) does not satisfy (12.150) when substituted in them,


implying lack of equilibrium due to spuriousness of the computed solution.Figure 12.17 shows similar graphs for Pe = 1000. The computations failfor discretizations resulting in log(dofs) ≤ 2.5 (meshes coarser than 256 ele-ments) where equilibrium is not achieved, that is, calculated solution from(12.150) does not satisfy (12.150) when substituted into the equations. Thisis due to VIC nature of the integral form resulting from GM/WF. Corre-spondingly, the values of the error norms for the failed discretizations growout of control. When log(dofs) ≥ 2.5 (discretization contains 256 elementsor more), [K1] contribution becomes insignificant and the BVP behaves like1Pe

d2φdx2

= 0, hence the asymptotic range is observed with calculated conver-gence rates of the indicated error norms of 3.7, 2.9, 2 are achieved comparedto their theoretical values of 4, 3, 2 for self adjoint operators, rather amaz-ingly good performance for VIC integral form.

When performing the error computations for Pe higher than 1000 withuniform mesh refinement of 2, 4, . . . elements failure of computations occurswhen [K1] dominates the total [K] as expected.

-10

-5

0

5

10

0.5 1 1.5 2 2.5 3 3.5 4

log

(||φ

- φ

h|| H

q)

o

r

log

(|φ

- φ

h| H

q)

log (dofs)

GM/WF, non-self adjoint operator (VIC)

Vh ⊂ H

2,p ; C

1 Solutions

(Pe = 100)

Convergence Rate

( Calculated )

4

3

2

p=3 : ||•||H

0

p=3 : ||•||H

1 ; |•|H

1

p=3 : ||•||H

2 ; |•|H

2

Figure 12.16: log||···||Hq versus log(dofs) for solutions of class C1 (GM/WF, model prob-lem 2, p = 3, Pe = 100)

12.6.2.3 LSP: Higher order system (without auxiliary equation)

In this study we consider 1D convection-diffusion equation (12.137) with-out converting it to a system of first order equations through the use of aux-


-10

-5

0

5

10

15

0.5 1 1.5 2 2.5 3 3.5 4

log

(||φ

- φ

h|| H

q)

o

r

log

(|φ

- φ

h| H

q)

log (dofs)

GM/WF, non-self adjoint operator (VIC)

Vh ⊂ H

2,p ; C

1 Solutions

(Pe = 1000)

Convergence Rate

( Calculated )

3.7

2.9

2.0

p=3 : ||•||H

0

p=3 : ||•||H

1 ; |•|H

1

p=3 : ||•||H

2 ; |•|H

2

Figure 12.17: log||···||Hq versus log(dofs) for solutions of class C1 (GM/WF, model prob-lem 2, p = 3, Pe = 1000)

iliary equation. In this case Vh ⊂ Hk,p(Ωe), k = 3 is minimally conformingapproximation space if the integrals over ΩT are to be Riemann. For k = 2the integrals over ΩT are Lebesgue and k = 1 (solutions of class C0) is notadmissible.

Error norms are computed for progressively refined uniform discretiza-tions for k = 2, 3 (solutions of classes C1 and C2) at p-levels of 3 and 5 fork = 2 and p = 5 and 7 for k = 3. Plots of error norms versus dof for solu-tions of classes C1 and C2 and the calculated convergence rates are shownin Figs. 12.18 and 12.19. The details of computations including theoreticaland calculated convergence rates of various error norms are presented in Ta-bles 12.10 and 12.11. We note that the highest order of the derivative inthe mathematical model (2m) in this case is 2 as the convection-diffusionequation is not reduced to a first order system using auxiliary equations.Theoretical convergence rates are overall in good agreement with the calcu-lated convergence rates confirming importance of the variational consistencyof the integral form. In solutions of both classes, the convergence rate of||···||H0 is higher than predicted for p = 5. In this case 2m = 2 whereas incase of first order system derived using auxiliary equation 2m = 1, thus thefirst order system has higher convergence rate of

√I in the LSP.

Figure 12.20 shows plots of ||···||H2 versus dofs and√I versus dofs for

solutions of classes C1 and C2 at p = 5. Since the highest order derivative is


-10

-5

0

5

10

15

0.5 1 1.5 2 2.5 3 3.5 4

log

(||φ

- φ

h|| H

q)

o

r

log

(|φ

- φ

h| H

q)

log (dofs)


Vh ⊂ H

2,p ; C

1 Solutions

(Higher order system, 2m=2 )

Pe=100


Convergence Rate

( Calculated )

3.9

3.3

2.0

7.6

5.0

4.0

p=3 : ||•||H

0

p=3 : ||•||H

1 ; |•|H

1

p=3 : ||•||H

2 ; |•|H

2

p=5 : ||•||H

0

p=5 : ||•||H

1 ; |•|H

1

p=5 : ||•||H

2 ; |•|H

2

Figure 12.18: log||···||Hq versus log(dofs) for solutions of class C1 (LSP, model problem2, higher order system, p = 3 and 5, Pe = 100)

Table 12.10: Theoretical and computed convergence rates of various error norms. LSP,higher order system, C1 solutions, p = 3 and 5, Pe = 100

Type ofp-level

Solution Class;q

Convergence Rates


||···||H0 = ||···||L2

3C1; k = 2

0 4 3.9

5 0 6 7.6

||···||H1 ; | ··· |H1

3C1; k = 2

1 3 3.3

5 1 5 5

||···||H2 ; | ··· |H2

3C1; k = 2

2 2 2

5 2 4 4

two in the differential operator, the convergence rate of ||···||H2 is same as thatof√I. C1 and C2 solutions have same convergence rates but C2 solutions

have better accuracy for a given dofs, confirming that the convergence ratesof error norm and residual functional are not a function of the order k of theapproximation space. Thus, for higher order system we also observe thatthe rates for LSP match with GM/WF for self adjoint operators due to thefact that in both the integral forms are VC even though the two methods ofapproximation have best approximation property in different norms.


-10

-5

0

5

10

15

0.5 1 1.5 2 2.5 3 3.5

log

(||φ

- φ

h|| H

q)

o

r

log

(|φ

- φ

h| H

q)

log (dofs)


Vh ⊂ H

3,p ; C

2 Solutions


Pe=100


Convergence Rate

( Calculated )

7.7

5.0

4.0

8.0

7.0

6.0

p=5 : ||•||H

0

p=5 : ||•||H

1 ; |•|H

1

p=5 : ||•||H

2 ; |•|H

2

p=7 : ||•||H

0

p=7 : ||•||H

1 ; |•|H

1

p=7 : ||•||H

2 ; |•|H

2

Figure 12.19: log||···||Hq versus log(dofs) for solutions of class C2 (LSP, model problem2, higher order system, p = 5 and 7, Pe = 100)

Table 12.11: Theoretical and computed convergence rates of various error norms. LSP,higher order system, C2 solutions, p = 5 and 7, Pe = 100

Type ofp-level

Solution Class;q

Convergence Rates


||···||H0 = ||···||L2

5C2; k = 3

0 6 7.7

7 0 8 8

||···||H1 ; | ··· |H1

5C2; k = 3

1 5 5

7 1 7 7

||···||H2 ; | ··· |H2

5C2; k = 3

2 4 4

7 2 6 6

12.6.3 Model problem 3: Non-linear operator, 1D Burgersequation

We consider 1D Burgers equation described by a non-linear operator (seeexample 7.2 in chapter 7) to compute a priori error estimates and conver-gence rates of various error norms and compare them with their theoreticalvalues.

φdφ

dx− 1

Re

d2φ

dx2= 0 ∀x ∈ (0, 1) = Ω ⊂ R1 (12.151)


-8

-6

-4

-2

0

2

4

6

0.5 1 1.5 2 2.5 3 3.5 4

log

(||φ

- φ

h|| H

2)

o

r

log

(√

I)

log (dofs)



Pe=100

Theoretical rates:

p+1-q ; p+1-2m

Convergence Rate

( Calculated )

4

4

p=5 : ||•||H

2 ; ||E||L

2

: C1

p=5 : ||•||H

2 ; ||E||L

2

: C2


at p = 5 (LSP, higher order system, model problem 2)

BCs: φ(0) = 1, φ(1) = 0 (12.152)

For the studies presented in the following sections, a value of Re = 100is used. Theoretical solution φ of (12.151) and (12.152) and finite elementsolution φh using GM/WF and LSP (higher order and first order systems)are given in chapter 7. Some details are given in the following as a review.It is shown in chapter 7 that in this case A = φ d

dx −1Re

d2

dx2which is a

function of φ, hence non-linear. The GM/WF yields VIC integral form. Theintegral form from the LSP is VC with minor adjustments (chapter 7) oflittle consequence but immense benefit as they yield variational consistencyof the integral form.

(a) GM/WF: The integral form of (12.151) and (12.152) over Ω is given by(φdφ

dx, v

)Ω

+1

Re

(dφ

dx,dv

dx

)Ω

= 0 ; v = δφ ∀v ∈ V ⊂ Hk,p(Ω)

(12.153)or

B(φ, v) = 0 (12.154)

Functional B(···, ···) is linear in v but not linear in φ and is obviously not


symmetric.

δB(φ, v) =

(vdφ

dx+ φ

dv

dx, v

)+

1

Re

(dv

dx,dv

dx

)(12.155)

is obviously not > 0, = 0, or < 0 ∀v ∈ V ⊂ Hk,p(Ω), hence the integralform (12.154) is VIC.

(b) LSP based on residual functional: These can be constructed in twoalternate ways, as a higher order system (12.151) or by recasting (12.151)as a system of first order equations.

(I) Higher order system

E = Aφh − f = φhdφhdx− 1

Re

d2φhdx2

∀x ∈ ΩT = ∪eΩe (12.156)

and residual functional I(φh) is given by

I(φh) = (E,E)Ω (12.157)

δI(φh) = 2(E, δE) = 2g = 0

δE = vdφhdx

+ φhdv

dx− 1

Re

d2v

dx2

(12.158)

δ2I(φh) ' 2(δE, δE) (12.159)

The necessary condition g = 0 is satisfied by calculating a solutionusing Newton’s linear method. See chapter 7 for full details. Theintegral form in this case is variationally consistent.

(II) First order systemLet τ = dφ

dx , then (12.151) reduces to

φdφ

dx− 1

Re

dτ

dx= 0

τ − dφ

dx= 0

(12.160)

LSP for (12.160) is described in detail in chapter 7 and is omittedhere. This integral form is also VC.

12.6.3.1 LSP: Higher-order system (without auxiliary equation)

For this model problem we only present studies related to convergencerates of various error norms using (12.151) (i.e. without recasting it as asystem of first order equations). As in other problems Ω = [0, 1] is discretizedusing uniform meshes of 2, 4, 8, . . . 3-node p-version higher order globaldifferentiability elements and the solutions are computed using finite element


formulations based on GM/WF and LSP. In case of LSP, since the integralform is VC the same convergence rate estimates hold as in (12.148):

||φ− φh||Hq ≤ C3hp+1−q|φ|p+1


||Aφ− f ||L2= ||E||L2

≤ C4hp+1−2m

(12.161)

In this BVP, 2m = 2. Since the differential operator has derivative of φup to second order, the minimally conforming space in this case is k = 3 forthe integrals over ΩT to be Riemann and the integrals are in Lebesgue sensewhen k = 2. k = 1 is not admissible.

Figures 12.21 and 12.22 show plots of various error norms versus dofsfor solutions of class C1 and C2 as well as calculated convergence rates.Table 12.12 (for C1 solution) and Table 12.13 (for C2 solution) summarizedetails of the numerical studies and a comparison between theoretical andcalculated convergence rates.

First we note from Figs. 12.21 and 12.22 large pre-asymptotic and onsetof asymptotic ranges. The asymptotic range is rather limited, due to whichaccurate computation of convergence rates is difficult. Nonetheless we ob-serve that for most error norms the theoretical and calculated convergencerates are in good agreement. Once again, we observe that due to VC inte-gral form in LSP for nonlinear operators the convergence rate estimates forGM/WF for self adjoint operators and the same for LSP for linear operatorshold here, again confirming the significance and importance of VC integralforms.

Table 12.12: Theoretical and computed convergence rates of various error norms. LSP,higher order system, C1 solutions, p = 3 and 5, Re = 100

Type ofp-level

Solution Class;q

Convergence Rates


||···||H0 = ||···||L2

3C1; k = 2

0 4 4.1

5 0 6 9.5

||···||H1 ; | ··· |H1

3C1; k = 2

1 3 3.6

5 1 5 5.1

||···||H2 ; | ··· |H2

3C1; k = 2

2 2 2

5 2 4 4

Figure 12.23 shows plots of ||···||H2 versus dof and√I versus dof for solu-

tions of class C1 and C2 at p = 5. Since the differential operator is secondorder operator, the convergence rate of ||···||H2 is same as that of

√I for both


-10

-5

0

5

10

15

0.5 1 1.5 2 2.5 3 3.5 4

log

(||φ

- φ

h|| H

q)

o

r

log

(|φ

- φ

h| H

q)

log (dofs)

LSP, non-linear operator

Vh ⊂ H

2,p ; C

1 Solutions


Re=100


Convergence Rate

( Calculated )

4.1

3.6

2

9.5

5.1

4

p=3 : ||•||H

0

p=3 : ||•||H

1 ; |•|H

1

p=3 : ||•||H

2 ; |•|H

2

p=5 : ||•||H

0

p=5 : ||•||H

1 ; |•|H

1

p=5 : ||•||H

2 ; |•|H

2

Figure 12.21: log||···||Hq versus log(dofs) for solutions of class C1 (LSP, model problem3, higher order system, p = 3 and 5, Re = 100)

-10

-5

0

5

10

15

0.5 1 1.5 2 2.5 3

log

(||φ

- φ

h|| H

q)

o

r

log

(|φ

- φ

h| H

q)

log (dofs)


Vh ⊂ H

3,p ; C

2 Solutions


Re=100


Convergence Rate

( Calculated )

10

5.2

4

11

7.1

6

p=5 : ||•||H

0

p=5 : ||•||H

1 ; |•|H

1

p=5 : ||•||H

2 ; |•|H

2

p=7 : ||•||H

0

p=7 : ||•||H

1 ; |•|H

1

p=7 : ||•||H

2 ; |•|H

2

Figure 12.22: log||···||Hq versus log(dofs) for solutions of class C2 (LSP, model problem3, higher order system, p = 5 and 7, Re = 100)


Table 12.13: Theoretical and computed convergence rates of various error norms. LSP,higher order system, C2 solutions, p = 5 and 7, Re = 100

Type ofp-level

Solution Class;q

Convergence Rates


||···||H0 = ||···||L2

5C2; k = 3

0 6 10

7 0 8 11

||···||H1 ; | ··· |H1

5C2; k = 3

1 5 5.2

7 1 7 7.1

||···||H2 ; | ··· |H2

5C2; k = 3

2 4 4

7 2 6 6

-8

-6

-4

-2

0

2

4

0.5 1 1.5 2 2.5 3 3.5 4

log

(||φ

- φ

h|| H

2)

o

r

log

(√

I)

log (dofs)



Re=100

Theoretical rates:

p+1-q ; p+1-2m

Convergence Rate

( Calculated )

4

4

p=5 : ||•||H

2 ; ||E||L

2

: C1

p=5 : ||•||H

2 ; ||E||L

2

: C2


at p = 5 (LSP, higher order system, model problem 3)

C1 and C2 local approximations. However, C2 solutions have better accu-racy for a given dofs. We clearly observe that the convergence rate is nota function of k, the order of approximation space. Calculated convergencerates of ||···||H2 and

√I are the same and are in exact agreement with the

theoretical convergence rates.

12.7. A POSTERIORI ERROR ESTIMATION AND COMPUTATION 747

12.6.3.2 GM/WF

Since the differential operator is non-linear the integral form from GM/WFis VIC. B(···, ···) is not bilinear and is not symmetric, hence we lose best ap-proximation property of the GM/WF in B-norm. GM/WF will yield thefollowing form of the assembled equations for ΩT (when p = 1 and f(x) 6= 0)assuming uniform discretization (h = he):

[K]δ =

[[K1] +

1

Reh[K2]

]δ = P+ F (12.162)

in which [K1] = [K1(δ)] and (K1)ij 6= (K1)ji. [K2] is symmetric. [K1]

is due to φdφdx and [K2] is due to 1Re

d2φdx2

term in the differential equation.Furthermore [K1] has zeros on the diagonal after φ(0) = 1 and φ(1) = 0boundary conditions are imposed, thus if Reh is large, the contribution of[K2] to [K] is almost insignificant and the computations using (12.162) willfail. On the other hand if the discretization ΩT is sufficiently refined thencontribution of [K2] to [K] overshadows that of [K1] and the solution be-

havior will be dominated by [K2] (i.e. d2φdx2

term in the differential equation).When this happens the integral form from GM/WF will behave like a VCintegral form and the convergence rates of various error norms will be sameas those of GM/WF for self adjoint operator.

For numerical studies we consider Re = 100. Uniform mesh refinementis carried out for solutions of class C1 at p = 3. Figure 12.24 shows plotsof error norms versus dofs. We note that for discretizations coarser than128 elements the error norms correspond to erroneous computed solutionsin which equilibrium condition is violated for the assembled equations. Forfiner discretizations (128 elements or more) asymptotic range is observed.In this range discretization is sufficiently refined so that the integral form isdominated by the diffusion term. Calculated convergence rates (of ||···||H0 ;||···||H1 , | ··· |H1 ; ||···||H2 , | ··· |H2) 3.7, 3, and 2 are in close agreement with thetheoretical convergence rates 4, 3, 2. In this study for Re = 100 computa-tions failed for discretizations coarser than 128 elements where equilibriumwas not achieved.

12.7 A posteriori error estimation and computation

12.7.1 A posteriori error estimation

A posteriori error estimation refers to estimation of errors in the com-puted solution. The primary purpose is to be able to devise some element-wise measures as well as for the whole discretization that quantify the errorsin the computed solution as well as provide some guidance on the portionsof the domain where the computed solution needs to be improvced. Based


-10

0

10

20

30

40

0.5 1 1.5 2 2.5 3 3.5

log

(||φ

- φ

h|| H

q)

o

r

log

(|φ

- φ

h| H

q)

log (dofs)

GM/WF, non-linear operator (VIC)

Vh ⊂ H

2,p ; C

1 Solutions

Re = 100

Convergence Rate

( Calculated )

3.7

3

2

p=3 : ||•||H

0

p=3 : ||•||H

1 ; |•|H

1

p=3 : ||•||H

2 ; |•|H

2

Figure 12.24: log||···||Hq versus log(dofs) for solutions of class C1 (GM/WF, model prob-lem 3, p = 3, Re = 100)

on these measures one could design mesh refinement, p-level change, etc.strategies that result in the desired accuracy of the computed solution. Thisprocess of changing h, p, and possibly k based on measures estimated usingthe computed solution is referred to as adaptive process (i.e. we adapt h, p,and k as dictated by the current state of the solution and a posteriori errorestimators or indicators).

During the development of finite element technology and even now, solu-tions of class C0 have been used predominantly. The local approximations ofclass C0 result in interelement discontinuity of the derivatives normal to theinterelement boundaries. When the solutions of the BVPs are smooth, theseinterelement jumps in the derivatives are reduced upon h, p refinements andwe say C0 solutions converge weakly to class C1. The a posteriori errorestimations largely exploit the interelement discontinuities of the derivativesinherent in C0 local approximations. We note the following.

(1) When the local approximations are considered in higher order spaces, thea posteriori error estimates used currently that are derived based on C0

local approximations are meaningless as for higher order global differen-tiability local approximations the interelement jumps in the derivativesof the solutions used currently do not exist.

(2) The C0 local approximations can only be used in a system of first order

12.7. A POSTERIORI ERROR ESTIMATION AND COMPUTATION 749

differential equations to calculate the residuals and residual functionalsover Ωe as well as over ΩT , but only in Lebesgue sense. For higher orderBVPs such computations are not possible with local approximations ofclass C0. Even though the residual functional over Ωe and ΩT are truemeasures of how well the local approximation satisfies the BVP, theemphasis has been largely on a posteriori error estimation, primarilydue to the insistence on the use of C0 local approximations.

(3) Our view is that in a finite element computational framework the physicsof the BVP must be preserved and in such a framework, once a finiteelement solution has been calculated, the computational framework mustpermit a posteriori computations of any desired measures otherwise thecomputational framework is deficient.

12.7.2 A posteriori error computation

As mentioned in section 12.7.1, the computational framework must bedesigned such that it permits a posteriori computations of all desired mea-sures that are necessary and meaningful in adaptivity. Minimally conformingspaces play a crucial role in accomplishing this. We present details in thefollowing. Let

AφAφAφ− fff = 0 over Ω (12.163)

be a boundary value problem in which the differential operator may be selfadjoint, non-self adjoint, or non-linear. Let 2m be the highest order of thederivative of φφφ in (12.163). Let φφφeh and φφφh be approximations of φφφ over Ωe

and ΩT . The approximation φφφh is assumed to be computed from any of themethods of approximation in which the integral forms may be VC or VIC.Let

φφφeh ∈ Vh ⊂ Hk,p(Ωe) ; k ≥ 2m+ 1 (12.164)

φφφh =⋃e

φφφeh (12.165)

The approximation space Vh is minimally conforming ensuring that the in-tegrals over ΩT are Riemann. Using (12.163) and (12.165) we can defineresidual functions Ei

Ei =∑j

Aij(φh)j − fi ; i = 1, 2, . . . , n˜ over ΩT (12.166)

where n˜ is the number of differential equations in (12.163). Let

Eei =∑j

Aij(φeh)j − fi ; i = 1, 2, . . . , n˜ over Ωe (12.167)


We define residual functionals I and Ie over ΩT and Ωe by

I =

n∑i=1

(Ei , Ei)ΩT (12.168)

Ie =

n∑i=1

(Eei , Eei )Ωe (12.169)

Since ΩT = ∪eΩe we can write (using (12.168) and (12.169))

I =∑e

Ie =∑e

n∑i=1

(Eei , Eei )Ωe (12.170)

If φφφh is same as the theoretical solution φφφ then

Ei =∑j

Aij(φh)j − fi = 0, i = 1, 2, . . . , n˜ (12.171)

and

I = 0, Ie = 0 (12.172)

over ΩT and each Ωe. Minimally conforming space Vh ensures that integralsover ΩT are Riemann, hence proximity of I(φφφh) to zero (theoretical value offunctional I; that is, I(φφφ)) is a measure of error in the solution φφφh over ΩT .When I(φφφh) → 0, (EEE,EEE) → 0 ⇒ Ei → 0 ; i = 1, 2, . . . , n˜ ∀x ∈ ΩT , thus

Eei → 0 ∀x ∈ Ωe for each Ωe in ΩT , implying that differential equations(12.163) are satisfied in the pointwise sense. Thus, the main steps in aposteriori error computation can be summarized in the following.

(1) Choose minimally conforming space k ≥ 2m + 1 thereby ensuring inte-grals over ΩT in Riemann sense.

(2) Regardless of the method of approximation to construct integral formin the finite element process, the following steps are possible and helpin quantifying solution error. Calculate finite element solution φφφh andhence φφφeh.

(3) Calculate Eei , Ie =

n∑i=1

(Eei , Eei )Ωe for each element e with domain Ωe of

the discretization ΩT .

(4) Calculate I =∑eIe for ΩT .

(5) When I ' 0 (O(10−8) or lower), φφφh is reasonably converged to φφφ for theh, p, and k employed, hence no need for adaptive refinements.

12.8. ADAPTIVE PROCESSES IN FINITE ELEMENT COMPUTATIONS 751

(6) When I 6= 0, we examine Ie values for individual elements of ΩT todetermine which elements have Ie values larger than a certain thresholdvalue I˜e. These elements can be considered for adaptive refinement (hor p or both) depending on the strategy adopted. Some of these arepresented in the next section.

(7) In this approach, a posteriori error estimations derived and used presently(of little value in higher order spaces) are eliminated altogether.

(8) Errors in the computed solution are quantified without the knowledgeof theoretical solution and there is built-in adaptivity due to Ie for in-dividual elements. The elements with Ie values larger than a thresholdvalue I˜e are candidates for refinement.

(9) Adaptive processes based on Ie values for elements of descretization ΩT

are presented in the next section.

12.8 Adaptive processes in finite element computa-tions

As discussed earlier adaptivity refers to a mechanism or a process throughwhich the currently computed solution guides the most beneficial adjust-ments or changes primarily in h and p. Our view is that minimally conform-ing spaces are essential for local approximations so that the integrals over ΩT

are Riemann and the computed quantities for chosen h and p are true reflec-tion of how the physics is incorporated in the computations. With minimallyconforming approximation spaces the computed residual functional values Ie

are true measure of error in the computed solution for each element of the

discretization, hence I =∑eIe becomes a measure of the solution accuracy

over ΩT . A posteriori error estimation used currently is neither applicablein higher order spaces nor needed. The material presented in the follow-ing sections is based on residual functionals Ie for Ωe of the discretizationΩT . Secondly, the typical behavior of ||···||Hq norm shown in Figure 12.1 isessential to keep in mind at all times when considering adaptive processes.We make some remarks to refresh the concepts presented in earlier sections.Referring to figure 12.1, we note

(1) In the pre-asymptotic range adaptivity is not possible as φh does nothave sufficient accuracy to yield reliable values of Ie. Thus, in thisrange the only possible alternative is to rediscretize until we are in theonset of asymptotic range.

(2) In the onset of asymptotic range, computed φh show sufficient improve-ment (due to reducing ||···||Hq), hence adaptive process(es) can be initi-ated at any point in this range, keeping in mind that proximity to the


asymptotic range (CD), may have significant impact on the adaptiveprocess.

(3) In the asymptotic range adaptivity is rather straightforward due to op-timal convergence rate (shown in later sections).

(4) We consider 1D convection-diffusion equation and 1D Burgers equationdescribed by non-self adjoint and non-linear operators to present numer-ical studies. We consider non-self adjoint operator to develop adaptivestrategies and show their merit and then apply these to non-linear andself adjoint operators.

12.8.1 Adaptive processes for 1D convection-diffusionequation: non-self adjoint operator

We consider the following:

dφ

dx− 1

Pe

d2φ

dx2= 0 ∀x ∈ (0, 1) = Ω ⊂ R1

φ(0) = 1, φ(1) = 0

(12.173)

Theoretical solution of (12.173) is given by

φ(x) =ePe(x−1)

1− ePe; x ∈ [0, 1] (12.174)

As described in chapter 6, the theoretical solution is highly dependent onPeclet number Pe. Figure 12.25 shows graphs of φ(x) versus x for Pe = 10,100, and 1000. We observe progressively isolated high solution gradientsnear x = 1 with increasing Peclet number.

As illustrative example of adaptivity we choose Pe = 1000 and k = 3,p = 2k − 1 = 5 (minimum for k = 3) for which integrals over ΩT are alwaysRiemann. All computations are performed using LSP in which the integralform is VC. Equation (12.173) is used without auxiliary equation. In eachstudy we plot log(

√I) versus log(dofs).

12.8.1.1 Adaptivity in the pre-asymptotic range:uniform h-refinement

We start with a two element uniform mesh and continuously subdivideeach element in two, computing a solution for each discretization. Fig-ure 12.26 shows a plot of

√I versus dofs. In this approach we observe a

large pre-asymptotic range in which a large number of degrees of freedom arewasted through uniform mesh refinement in the portion of the domain wheresolution essentially is constant (value of one). For Pe = 1000 the solutiongradients are isolated near x = 1.0 over a length of O(10−3). A 128 element


0

0.5

1

0 0.2 0.4 0.6 0.8 1

φ

x

(a) Pe = 10

0

0.5

1

0 0.2 0.4 0.6 0.8 1

φ

x

(b) Pe = 100

0

0.5

1

0 0.2 0.4 0.6 0.8 1

φ

x

(c) Pe = 1000

Figure 12.25: Solution φ(x) versus x for (a) Pe = 10, (b) Pe = 100, and (c) Pe = 1000

uniform mesh (with he = 0.0078125) has appropriate he near x = 1.0. Thedata in Fig. 12.26 are also generated using uniform h-refinement beyond 128elements, for which we see reduction in

√I. We clearly observe this strategy

is totally ineffective.

12.8.1.2 Adaptivity in the pre-asymptotic range: adaptiveh-refinement

If we monitor Ie values for the elements of ΩT while performing uniformh-refinement, we observe that for a discretization all elements have roughlysame Ie value until we have a discretization with 128 elements with he =0.0078125 for each element. For this discretization the element located atx = 1.0 has the largest value of Ie. The other elements in its vicinity alsohave comparable Ie values. Thus, we need a threshold Ie value for adaptivity.For the sake of illustration we choose Iec = 0.3(max

eIe) as the threshold value

of Ie. All elements in the discretization (128 elements in this case)withIe values larger than Iec are subdivided into two elements of equal length.Figure 12.27 shows the graph of

√I versus dofs for this refinement strategy

and a comparison with uniform h-refiment. The dramatic improvement in


-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1 1.5 2 2.5 3 3.5

log(√I)

log(dofs)

Uniform h-refinement

Figure 12.26: 1D convection-diffusion equation, adaptivity using uniform h-refinement,LSP, k = 3, p = 5, Pe = 1000

this refinement strategy is obvious. Virtually no dofs are wasted beyond128 element mesh as obvious by almost vertical straight line behavior of

√I

versus dofs. In this approach adaptive process seems good, but arriving at128 element uniform mesh is obviously still quite wasteful due to additionof elements in the large part of the domain in which the solution is almostconstant.

12.8.1.3 Adaptivity in the pre-asymptotic range: gradedh-rediscretizations

Due to unreliable Ie values in the pre-asymptotic range, adaptivity isnot possible in this range, yet avoiding 128 element uniform refinement isalmost essential to eliminate wasteful dofs. This is accomplished by non-uniform rediscretizations. For this model problem he = O(10−3) is needednear x = 1 and he = O(100) will suffice near x = 0. Such decisions regardingelement lengths are based on physics as we have done here. It is shown in [7]that geometrically graded discretizations are effective in situations in whichisolated high gradients of the solution exist. We choose a geometric ratio (gr)of 1.5 (shown to be quite effective in most applications) and begin with two-element discretization and progressively increase the number of elements.The total length of 1 is the sum of n elements with geometric ratio of gr and


-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1 1.5 2 2.5 3 3.5

log(√I)

log(dofs)


Adaptive h-refinement

Figure 12.27: 1D convection-diffusion equation, adaptivity using adaptive h-refinement,LSP, k = 3, p = 5, Pe = 1000

the first element length h1 of the element located at x = 1.0. Thus,

1 = h1

(1− (gr)

n

1− gr

)(12.175)

For gr = 1.5 and n = 2, 3, 4, . . . , we determine h1 using (12.175). The lengthsof elements 2, 3, . . . (from x = 1.0 to x = 0.0) are obviously given by

hi+1 = grhi, i = 1, 2, . . . , n− 1 (12.176)

This rediscretization is generally continued using (12.175) and (12.176) untilwe reach the onset of asymptotic range which can be easily detected by ob-serving improvement in

√I as more dofs are added by newer discretizations.

In this study we continue rediscretizations without using adaptive refinementbased on Ie. Figure 12.28 shows a plot of

√I versus dof for this study and

comparisons with previous two. Improvement in√I and savings in dofs is

rather dramatic. Few comments regarding this approach or in order. First,if this rediscretization based on gr is the best strategy, then there is no needfor adaptivity. This is obviously not the case as we show in subsequentsections. This approach becomes ineffective beyond a certain number of el-ements because adding more elements hardly influences the element lengthnear x = 0.0; hence Ie in this region begin to dominate

√I for ΩT . However,

we clearly observe from Fig. 12.28 the effectiveness of geometrically gradedrediscretization. We note that the asymptotic range is achieved, straightline


portion of the graph in Fig. 12.28. It is perhaps instructive to examine ifthis rediscretization strategy would benefit due to increase in p-level or k,the order of space, once we enter the onset of asymptotic range.

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1 1.5 2 2.5 3 3.5

log(√I)

log(dofs)



Graded h-rediscretization

Figure 12.28: 1D convection-diffusion equation, adaptivity using graded h-rediscretization, LSP, k = 3, p = 5, Pe = 1000

(a) Choice of p-levelThe graded h-rediscretization process described above was carried outfor k = 3 and p = 5. It is perhaps meaningful to examine the graded h-rediscretization process at p-levels higher than 5. We repeat the gradedh-rediscretization study described in section 12.8.1.3 at p-level of 7, 9,and 11. Plots of

√I versus dofs are shown in Fig. 12.29. We note that

at p = 5 the asymptotic range is entered for lower degrees of freedomcompared to p = 7, 9, and 11. This is significant in the sense thatasymptotic range adaptive strategy for p = 5 could begin at far fewerdofs compared to higher p-levels. At lower p-levels the asymptotic rangeis rather short implying that lowest values of

√I achievable are limited.

Lowest value of√I continues to reduce with increasing p-level. Based

on this it is straightforward to conclude that minimum p-level (for min-imally conforming k) is the best choice in the pre-asymptotic range forgraded h-rediscretization designed primarily to enter into the onset ofasymptotic range for as few dofs as possible.

(b) Choice of kAnalogous to (a), we may also consider the same graded h-rediscretizationof section 12.8.1.3 but for k = 2, 3, 4 with p = 7 (minimum for k = 4)


-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1 1.5 2 2.5

log(√I)

log(dofs)

Graded h-rediscretization, C2 p=5




Figure 12.29: 1D convection-diffusion equation, influence of choice of p on graded h-rediscretization, LSP, k = 3, Pe = 1000

for all values of k. k = 2 has been chosen intentionally to observe theperformance of the process when the integrals over ΩT are Lebesgue.Plots of

√I versus dofs are shown in Fig. 12.30. For higher value of k,

the√I versus dof enters the onset of asymptotic range for fewer dofs.

With decreasing k the pre-asymptotic range increases, however highervalues of k require higher value of p that increases the pre-asymptoticrange. Thus, the best strategy at this stage (without any further investi-gation) is perhaps use of minimally conforming k with the correspondingminimum value of p (p = 2k − 1) in the pre-asymptotic range.

12.8.1.4 General Remarks

The onset of asymptotic range is rather important for the behavior of√I

versus dof due to the fact that in this range the solution shows improvementcompared to pre-asymptotic range. Rather than continuing with graded h-refinement in the onset of asymptotic and asymptotic ranges which has beenshown to be essential in the pre-asymptotic range, we consider alternatestrategies that are more effective in the onset of asymptotic and asymptoticranges. In the folowing we consider onset of asymptotic range and con-sider merits of a variety of adaptive strategies such as uniform p-refinement,adaptive p-refinement, adaptive h-refinement, etc.


-8

-7

-6

-5

-4

-3

-2

-1

0

1 1.5 2 2.5

log(√I)

log(dofs)




Figure 12.30: 1D convection-diffusion equation, influence of choice of k on graded h-rediscretization, LSP, p = 7, Pe = 1000

12.8.1.5 Adaptivity in the onset of asymptotic and asymptoticranges: uniform p-refinement

In this approach we choose the first point in the pre-asymptotic rangethat corresponds to 11 element graded discretization with gr = 1.5 at p = 5and initiate uniform p-refinement by increasing p-level by 2 up to p-level of15 while keeping fixed discretization and minimally conforming k (k = 3).Similar studies are repeated for 12, 13, and 14 element graded discretizationswith gr = 1.5. Plots of

√I versus dofs for these studies are presented in

Fig. 12.31. We observe that initiating uniform p-refinement progressivelycloser to the asymptotic range results in progressively lower values of

√I for

a given dofs. As the dofs increase this gain in√I for points progressively

closer to the asymptotic range decreases and ultimately all curves of√I

versus dof approach the same value. This study demonstrates the importanceof the onset of asymptotic range achieved using graded h-rediscretizationswith gr = 1.5 and significance of initiating uniform p-refinement as close tothe asymptotic range as possible.

12.8.1.6 Adaptivity in the onset of asymptotic and asymptoticranges: adaptive p-refinement

In this study we begin with the first point in the asymptotic range (11element mesh) arrived at using graded h-refinement in the pre-asymptoticrange with gr = 1.5 and p = 5. We choose Iec = 0.3(max

eIe) as threshold


-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1 1.5 2 2.5

log(√I)

log(dofs)







Figure 12.31: 1D convection-diffusion equation, uniform p-refinement, LSP, k = 3, Pe =1000

value of Ie. For the elements with Ie > Iec , p-level is increased in incrementsof one. Similar studies are conducted for 12, 13, 14, 15, 16, and 17 elementgraded rediscretizations (with gr = 1.5). We also conduct a p-adaptivitystudy for the 10 element graded discretization that marks the last pointin pre-asymptotic range before the beginning of onset of asymptotic range.Figure 12.32 shows graphs of

√I versus dofs for these studies. We observe

the following important features.

(a) Much higher convergence rates are achieved in this process compared touniform p-refinement (see Fig. 12.31) in which many dofs are wasted.

(b) All adaptive refinements starting with different locations in the onset ofasymptotic range yield good and similar convergence rates and accuracy.

(c) The first point in the onset of asymptotic range yields the best accuracyand fastest convergence rate.

(d) It is rather remarkable to note that adaptive p-refinement beginning with10 element graded rediscretization marking a location just before theonset of asymptotic range results in total failure (i.e. no improvement in√I as Ie remain inaccurate for p-adaptivity), hence the adaptive process

is misguided.


-10

-8

-6

-4

-2

0

1 1.5 2

log(√I)

log(dofs)


Adaptive p-refinement








Figure 12.32: 1D convection-diffusion equation, adaptive p-refinement, LSP, k = 3,Pe = 1000

12.8.1.7 Adaptivity in the onset of asymptotic and asymptoticranges: adaptive h-refinement

In this section we study the effectiveness of h-adaptivity in the onset ofasymptotic range. Using Iec = 0.3(max

eIe) as threshold value, the elements

with Ie greater than Iec are divided in two equal elements. We begin with11 element mesh arrived at using graded rediscretization with gr = 1.5 andp = 5, marking the first point in the onset of asymptotic range. The samestudy is repeated for 12, 13, etc. element graded rediscretizations markingprogressively closer points to the asymptotic range. Figure 12.33 showsplots of

√I versus dofs for these studies. We note that in this adaptive

process choice of particular point in the onset of asymptotic range is notvery critical as all

√I versus dofs graphs yield about the same convergence

rate and accuracy. Comparing results in Figs. 12.32 and 12.33 it is ratherobvious that p-adaptivity in the onset of asymptotic range is more effectivethan h-adaptivity.

12.8.1.8 Adaptivity using higher geometric ratios forh-rediscretization at Pe = 1000 and Pe = 106

In this section we study the effectiveness of the adaptive processes de-scribed in sections 12.8.1.1 – 12.8.1.7 for higher geometric ratios of gr = 1.75,2.0, and 3.0 as well as 1.5 used in earlier studies.


-10

-8

-6

-4

-2

0

1 1.5 2 2.5

log(√I)

log(dofs)










Figure 12.33: 1D convection-diffusion equation, adaptive h-refinement, LSP, k = 3,p = 5, Pe = 1000

(a) Adaptivity for Pe = 1000Figure 12.34 shows plots of

√I versus dofs for various adaptive stud-

ies. For each gr the best convergence rate and lowest values of√I are

achieved using adaptive p-refinement beginning with the first point inonset of asymptotic range. Higher gr values are more effective in general.In this study

√I of O(10−3) is possible with only 27 dofs when gr = 3

as opposed to√I of the same order for gr = 1.5 requiring 51 dofs.

(b) Adaptivity for Pe = 106

Figure 12.35 shows graphs of√I versus dofs for gr = 1.5, 1.75, and

2.0 at Pe = 106. The details follow the studies presented earlier andin (a). In this case we also observe that for all values of gr adaptivep-refinement is most effective when we begin with the first point in theonset of asymptotic range arrived at using graded h-rediscretization.

12.8.2 Adaptive processes for 1D Burgers equation:non-linear operator

Consider the following 1D Burgers equation with its boundary conditions:

φdφ

dx− 1

Re

d2φ

dx2, φ(0) = 1, φ(1) = 0 (12.177)

We consider finite element formulation based on LSP (chapter 7) in whichthe integral form is VC. We choose k = 3 with minimum p-level of 5 (2k−1).


-10-9-8-7-6-5-4-3-2-1 0

1 1

.5 2

2.5

3 3

.5

log(√I)

log

(do

fs)

Un

ifo

rm h

-refi

nem

en

t

Ad

ap

tiv

e h

-refi

nem

en

t

Gra

ded

h-r

ed

iscre

tizati

on

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Ad

ap

tiv

e h

-refi

nem

en

t

Ad

ap

tiv

e p

-refi

nem

en

t

(a)Pe

=1000,g r

=1.5

-10-9-8-7-6-5-4-3-2-1 0

1 1

.5 2

2.5

3 3

.5

log(√I)

log

(do

fs)

Un

ifo

rm h

-refi

nem

en

t

Ad

ap

tiv

e h

-refi

nem

en

t

Gra

ded

h-r

ed

iscre

tizati

on

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Ad

ap

tiv

e h

-refi

nem

en

t

Ad

ap

tiv

e p

-refi

nem

en

t

(b)Pe

=1000,g r

=1.7

5

-10-9-8-7-6-5-4-3-2-1 0

1 1

.5 2

2.5

3 3

.5

log(√I)

log

(do

fs)

Un

ifo

rm h

-refi

nem

en

t

Ad

ap

tiv

e h

-refi

nem

en

t

Gra

ded

h-r

ed

iscre

tizati

on

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Ad

ap

tiv

e h

-refi

nem

en

t

Ad

ap

tiv

e p

-refi

nem

en

t

(c)Pe

=1000,g r

=2.0

-10-9-8-7-6-5-4-3-2-1 0

1 1

.5 2

2.5

3 3

.5log(√I)

log

(do

fs)

Un

ifo

rm h

-refi

nem

en

t

Ad

ap

tiv

e h

-refi

nem

en

t

Gra

ded

h-r

ed

iscre

tizati

on

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Ad

ap

tiv

e h

-refi

nem

en

t

Ad

ap

tiv

e p

-refi

nem

en

t

(d)Pe

=1000,g r

=3.0

Figure

12.34:

1D

convec

tion-d

iffusi

on

equati

on,

com

ple

teadapti

ve

pro

cess

es,

LSP

,k

=3,Pe

=1000

usi

ngGR

=1.5,1.7

5,2.0,3.0


-10-9-8-7-6-5-4-3-2-1 0

1 1

.5 2

2.5

3 3

.5

log(√I)

log

(do

fs)

Un

ifo

rm h

-refi

nem

en

t

Ad

ap

tiv

e h

-refi

nem

en

t

Gra

ded

h-r

ed

iscre

tizati

on

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Ad

ap

tiv

e h

-refi

nem

en

t

Ad

ap

tiv

e p

-refi

nem

en

t

(a)Pe

=1000,g r

=1.5

-10-9-8-7-6-5-4-3-2-1 0

1 1

.5 2

2.5

3 3

.5

log(√I)

log

(do

fs)

Un

ifo

rm h

-refi

nem

en

t

Ad

ap

tiv

e h

-refi

nem

en

t

Gra

ded

h-r

ed

iscre

tizati

on

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Ad

ap

tiv

e h

-refi

nem

en

t

Ad

ap

tiv

e p

-refi

nem

en

t

(b)Pe

=1000,g r

=1.7

5

-10-9-8-7-6-5-4-3-2-1 0

1 1

.5 2

2.5

3 3

.5

log(√I)

log

(do

fs)

Un

ifo

rm h

-refi

nem

en

t

Ad

ap

tiv

e h

-refi

nem

en

t

Gra

ded

h-r

ed

iscre

tizati

on

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Ad

ap

tiv

e h

-refi

nem

en

t

Ad

ap

tiv

e p

-refi

nem

en

t

(c)Pe

=1000,g r

=2.0

Figure

12.35:

1D

convec

tion-d

iffusi

on

equati

on,

com

ple

teadapti

ve

pro

cess

es,

LSP

,k

=3,Pe

=106

usi

ngGR

=1.5,1.7

5,2.0


-10-9-8-7-6-5-4-3-2-1 0

1 1

.5 2

2.5

3 3

.5

log(√I)

log

(do

fs)

Un

ifo

rm h

-refi

nem

en

t

Ad

ap

tiv

e h

-refi

nem

en

t

Gra

ded

h-r

ed

iscre

tizati

on

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Ad

ap

tiv

e h

-refi

nem

en

t

Ad

ap

tiv

e p

-refi

nem

en

t

(a)Re

=1000,g r

=1.5

-10-9-8-7-6-5-4-3-2-1 0

1 1

.5 2

2.5

3 3

.5

log(√I)

log

(do

fs)

Un

ifo

rm h

-refi

nem

en

t

Ad

ap

tiv

e h

-refi

nem

en

t

Gra

ded

h-r

ed

iscre

tizati

on

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Ad

ap

tiv

e h

-refi

nem

en

t

Ad

ap

tiv

e p

-refi

nem

en

t

(b)Re

=1000,g r

=1.7

5

-10-9-8-7-6-5-4-3-2-1 0

1 1

.5 2

2.5

3 3

.5

log(√I)

log

(do

fs)

Un

ifo

rm h

-refi

nem

en

t

Ad

ap

tiv

e h

-refi

nem

en

t

Gra

ded

h-r

ed

iscre

tizati

on

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Ad

ap

tiv

e h

-refi

nem

en

t

Ad

ap

tiv

e p

-refi

nem

en

t

(c)Re

=1000,g r

=2.0

-10-9-8-7-6-5-4-3-2-1 0

1 1

.5 2

2.5

3 3

.5log(√I)

log

(do

fs)

Un

ifo

rm h

-refi

nem

en

t

Ad

ap

tiv

e h

-refi

nem

en

t

Gra

ded

h-r

ed

iscre

tizati

on

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Ad

ap

tiv

e h

-refi

nem

en

t

Ad

ap

tiv

e p

-refi

nem

en

t

(d)Re

=1000,g r

=3.0

Figure

12.36:

1D

Burg

ers

equati

on,

com

ple

teadapti

ve

pro

cess

es,

LSP

,k

=3,Re

=1000

usi

ngGR

=1.5,1.7

5,2.0,3.0


For this choice of k the integrals are Riemann over ΩT . We also chooseIec = 0.3(max

eIe) as threshold value of Ie for adaptive refinements. Reynolds

number Re = 1000 is considered so that the solution gradients are significantto show effectiveness of adaptive processes. Adaptive strategies described insections 12.8.1.1 – 12.8.1.7 for 1D convection-diffusion equation are consid-ered here for adaptive studies. Geometric ratios gr of 1.5, 1.75, 2.0 and 3.0are considered for graded h-rediscretizations. Figure 12.36 shows graphs of√I versus dofs for various adaptive studies. Conclusions are exactly similar

to those presented in section 12.8.1.8. All geometric ratios are effective buthigher value of gr are meritorious. p-adaptivity at the onset of asymptoticrange is most effective.

12.8.3 Adaptive processes for 1D diffusion equation:self-adjoint operator

Consider the following BVP in R1 in which the differential operator isself adjoint:

−d2φ

dx2= f(x) ∀x ∈ (0, 1) = Ω ⊂ R1

φ(0) = 0,

(dφ

dx

)x=1

= 0

f = (x+ σ)θ − (1 + σ)θ

σ = 0.005, θ = −1.5

(12.178)

For this BVP both GM/WF and LSP based on residual functional will yieldVC integral form. We consider finite element formulation of BVP (12.178)using LSP based on residual functional without using auxiliary variables.For this problem the behavior of φ(x) is localized near x = 0.0. Severityof gradients are dependent on σ and θ. The values of σ and θ chosen areadequate to create significant gradients of φ near x = 0.0. We choose k = 3with minimum p-level of 5. Adaptive studies parallel to those described inearlier sections are conducted. Plost of

√I versus dofs for various adap-

tive strategies are shown in Fig. 12.37. Observations and conclusions fromthese studies are similar to those drawn from previous studies for non-selfadjoint and non-linear operators. However, we note that in this BVP the so-lution gradients isolated near x = 0.0 are due to non-homogeneous functionf(x) whereas in case of convection-diffusion and Burgers equations the highgradients are caused by Pe, Re, and boundary conditions. Once again wesee the influence of the starting point in the onset of asymptotic range onthe adaptivity processes. Graded p-refinement is the most effective adaptivestrategy when initiated from the first location in the onset of asymptoticrange arrived at by using graded h-rediscretization for any of the geometricratios.


-10-9-8-7-6-5-4-3-2-1 0

1 1

.5 2

2.5

3 3

.5

log(√I)

log

(do

fs)

Un

ifo

rm h

-refi

nem

en

t

Ad

ap

tiv

e h

-refi

nem

en

t

Gra

ded

h-r

ed

iscre

tizati

on

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Ad

ap

tiv

e h

-refi

nem

en

t

Ad

ap

tiv

e p

-refi

nem

en

t

(a)θ

=−

1.5

,σ

=0.0

05,g r

=1.5

-10-9-8-7-6-5-4-3-2-1 0

1 1

.5 2

2.5

3 3

.5

log(√I)

log

(do

fs)

Un

ifo

rm h

-refi

nem

en

t

Ad

ap

tiv

e h

-refi

nem

en

t

Gra

ded

h-r

ed

iscre

tizati

on

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Ad

ap

tiv

e h

-refi

nem

en

t

Ad

ap

tiv

e p

-refi

nem

en

t

(b)θ

=−

1.5

,σ

=0.0

05,g r

=1.7

5

-10-9-8-7-6-5-4-3-2-1 0

1 1

.5 2

2.5

3 3

.5

log(√I)

log

(do

fs)

Un

ifo

rm h

-refi

nem

en

t

Ad

ap

tiv

e h

-refi

nem

en

t

Gra

ded

h-r

ed

iscre

tizati

on

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Ad

ap

tiv

e h

-refi

nem

en

t

Ad

ap

tiv

e p

-refi

nem

en

t

(c)θ

=−

1.5

,σ

=0.0

05,g r

=2.0

-10-9-8-7-6-5-4-3-2-1 0

1 1

.5 2

2.5

3 3

.5log(√I)

log

(do

fs)

Un

ifo

rm h

-refi

nem

en

t

Ad

ap

tiv

e h

-refi

nem

en

t

Gra

ded

h-r

ed

iscre

tizati

on

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Un

ifo

rm p

-refi

nem

en

t

Ad

ap

tiv

e h

-refi

nem

en

t

Ad

ap

tiv

e p

-refi

nem

en

t

(d)θ

=−

1.5

,σ

=0.0

05,g r

=3.0

Figure

12.37:

1D

diff

usi

on

equati

on,

com

ple

teadapti

ve

pro

cess

es,

LSP

,k

=3,θ

=−

1.5

,σ

=0.0

05

usi

ngGR

=1.5,1.7

5,2.0,3.0

12.9. SUMMARY 767

12.8.4 General Remarks

(a) The adaptive strategies and the numerical studies are only presentedfor 1D model problems in R1 to illustrate merits and shortcomings ofvarious approaches.

(b) Extensions of these concepts in R2 and R3 are quite simple but requiresufficient care at the interelement boundaries when there is a p-levelmismatch and the graded discretizations require transition strategies,but are well established.

(c) The most significant aspect of this work on adaptivity is the recognitionof the behavior of

√I versus dofs and various ranges that are involved

and their significance to adaptivity when it is initiated from a locationthat lies within these ranges.

(d) Graded h-rediscretizations and their effectiveness based on geometricallygraded discretization is of utmost significance in overcoming the pre-asymptotic range in which adaptive processes are not possible due tounreliable values of computed Ie for the elements of ΩT

12.9 Summary

In this chapter we have considered a posteriori and a priori error es-timation, a posteriori error computation, convergence rates, and adaptiveprocesses in finite element computations for BVPs described by self ad-joint, non-self adjoint, and non-linear differential operators. Concepts ofh-, p-, and k-versions and h-, p-, and k-convergences in finite element pro-cesses are presented and discussed. It is shown that a desired measure oferror norm or residual functional versus degrees of freedom behavior hasdistinct features that can be classified as pre-asymptotic range, onset ofasymptotic range, asymptotic range, onset of post-asymptotic range, andpost-asymptotic range. The significance and importance of these rangesin adaptive processes has been discussed and demonstrated through threemodel problems described by self adjoint, non-self adjoint, and non-lineardifferential operators.

The a priori estimates only hold in asymptotic range and their derivationin the currently published literature are only valid for self adjoint operatorsin GM/WF when functional B(···, ···) is symmetric, thus GM/WF has bestapproximation property in B-norm. New work presented in this chapter es-tablishes correspondence between best approximation property of an integralform in some norm and the variational consistency of the integral form anddemonstrates that when one exists the other is ensured. Thus, for establish-ing error estimates, variational consistency becomes an essential property ofthe integral form. Of course best approximation property in some norm if it


exists is equally good as best approximation property and variational con-sistency of integral form can not exist without each other, i.e. these co-exist.In case of GM/WF, VC integral form is possible for self adjoint operator andin case of LSP VC integral form is possible for all three classes of differentialoperators, hence a priori estimates for GM/WF for self adjoint operators anda priori estimates for LSP for all three classes of operators can be derived.The derivation of a priori error estimates presented in proposition 12.2 ap-plies to GM/WF for self adjoint operators and in case of LSP for all threeclasses of operators as well as any other integral form resulting from a chosenmethod of approximation as long as the integral form is VC. Numerical stud-ies for the model problems containing the three classes of operators confirmthat when the integral form is VC, same a priori estimates and convergencerates hold. Thus, we have a priori error estimates for non-self adjoint andnon-linear differential operators. Extensive numerical studies are presentedfor various p and k values for uniform h-refinements demonstrating that thetheoretically derived convergence rates from a priori estimates are alwaysin agreement with calculated values when the integral forms are VC. The apriori error estimates derived here also hold for 2D and 3D BVPs as long asthe integral forms in these BVPs are variationally consistent. This can beconfirmed numerically and is in agreement with published literature for selfadjoint operators.

Extensive adaptive studies based on various h-, p-, and k-adaptive pro-cesses are presented in various regions of an appropriate norm (||···||) versusdof behavior to demonstrate: (a) ineffectiveness of pre-asymptotic range inadaptivity and significance of graded h-rediscretizations to overcome thisrange, (b) effectiveness of the onset of asymptotic range in adaptivity, and(c) merits of selective p-adaptivity by commencing the adaptive process inthe onset of asymptotic range.

A posteriori error estimations based on the work presented here areviewed unnecessary when the approximation spaces are minimally conform-ing or of orders higher than minimally conforming due to the fact that whenusing such spaces a posteriori error computations of any desired quantity(for example Ie and I) are accurate measures of error to guide adaptivity.Ie residual values for elements of ΩT are shown to be a perfect choice foradaptivity.

In short, VC integral form permits derivation of a priori error estimatesand determination of convergence rates for all three classes of differentialoperators and use of minimally conforming spaces make a posteriori errorestimation unnecessary and permit calculations of desired a posteriori mea-sures (such as Ie and I) to quantify errors in the currently computed solutionand to design adaptive processes based on these measures as shown in thischapter.


[1–4,7–35]


for self-adjoint operators in BVP. Int. J. Comp. Eng. Sci., 3(2):155–218, 2002.



[4] K. S. Surana, S. Allu, and J. N. Reddy. The k-version of finite element method forinitial value problems: Mathematical and computational framework. Int. J. Comp.Eng. Sci., 8(3):123–136, 2007.

[5] J. T. Oden and G. F. Carey. Finite Elements: Mathematical Aspects. Prentice Hall,1983.


[7] I. Babuska and W.C. Rheinboldt. Adaptive approaches and reliability estimations infinite element analysis. Comp. Methods Appl. Mech. Eng., 17:519–540, 1979.

[8] I. Babuska and W.C. Rheinboldt. A posteriori error estimates for the finite elementmethod. Int. J. Num. Meth. Eng., 12:1597–1615, 1978.

[9] I. Babuska and W.C. Rheinboldt. Error estimates for adaptive finite element compu-tations. SIAM J. Num. Anal., 18:736–754, 1978.

[10] I. Babuska and W.C. Rheinboldt. A posteriori error analysis of finite element solutionsfor one dimensional problems. SIAM J. Num. Anal., 18:435–463, 1981.

[11] Mark Ainsworth and J. Tinsley Oden. A Posteriori Error Estimation in Finite Ele-ment Analysis. Wiley-Interscience, 2000.

[12] B. A. Szabo and I. Babuska. Finite Element Analysis. Wiley-Interscience, 1991.

[13] Ch. Schwab. p and hp Finite Element Methods. Clarendon Press, Oxford, 1998.

[14] G. Guo and I. Babuska. The hp version of the finite element method. part1: Thebasic approximation results. part 2: General results and applications. Comput. Mech,1:21–41,203–220, 1986.

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Appendix ANumerical Integration using

Gauss Quadrature

A.1 Gauss quadrature in R1, R2 and R3

The element maps of line elements, quadrilateral elements and the hex-ahedron elements in the natural coordinate systems ξ, ξη and ξηζ are

Ωm = [−1, 1], Ωm = [−1, 1]× [−1, 1] and Ωm = [−1, 1]× [−1, 1]× [−1, 1]

domains in R1, R2 and R3.

We note that the element interpolation functions in the natural coordi-nate system ξ, ξη and ξηζ are algebraic polynomials of degree pξ, pη andpζ and that the element maps in the natural coordinate system are alwaysin two-unit length, two-unit square or two-unit cube with the origins of thenatural coordinate systems at the center of the element and hence the lim-its of integrations in the element coefficient matrices and vectors are alwaysbetween -1 to +1 for ξ, η and ζ. Since the approximation functions arealgebraic polynomials in ξ, η and ζ, the integrands of the coefficients of theelement matrices and vectors are algebraic polynomials in ξ, η and ζ also.Hence, gauss quadrature can be used to integrate numerically without intro-ducing any error or approximation in the value of the integral due to the factthat the locations of sampling points and the values of the weight factorsare derived based on the idea that a certain order Gauss quadrature inte-grates algebraic polynomials of certain degree exactly. In the following weconsider three specific cases in which the integrals are line, area and volumeintegrals in R1, R2 and R3 over Ωm = [−1, 1], Ωm = [−1, 1] × [−1, 1] andΩm = [−1, 1]× [−1, 1]× [−1, 1].

A.1.1 Line integrals over Ωm = Ωξ = [−1, 1]

Consider

I =

1∫−1

f(ξ) dξ (A.1)

771

772 APPENDIX A: NUMERICAL INTEGRATION USING GAUSS QUADRATURE

in which f(ξ) is a polynomial of highest degree pξ. Let nξ be the number ofsampling points then we have

2nξ − 1 = pξ

or

nξ =pξ + 1

2(round to next higher integer) (A.2)

Since pξ is known, obtain nξ using (A.2). Then from table A.1 we obtain

the locations of the sampling points ξi ; i = 1, . . . , nξ and weight factors W ξi

; i = 1, . . . , nξ. The integral of f(ξi) in (A.1), then can be calculated using

I =

1∫−1

f(ξ) dξ =

nξ∑i=1

W ξi f(ξi) (A.3)

Remarks.

(1) nξ integration points integrate an algebraic polynomial of highest degreepξ exactly provided nξ is obtained using (A.2).

(2) If we use higher number of sampling points than nξ, then the compu-tational effort increases but the accuracy of the integral remains unaf-fected.

(3) Sampling points are always symmetrically located about ξ = 0.

A.1.2 Area integrals over Ωm = Ωξη = [−1, 1]× [−1, 1]

Consider the following

I =

1∫−1

1∫−1

f(ξ, η) dξ dη =

1∫−1

( 1∫−1

f(ξ, η) dξ)dη (A.4)

Let f(ξ, η) be an algebraic polynomial in ξ and η and let pξ = highestdegree of the polynomial f(ξ, η) in ξ and pη = highest degree of the polyno-mial in f(ξ, η) in η. Let nξ and nη be the number of sampling points in ξand η. Then nξ and nη can be determined using

nξ =pξ + 1

2

nη =pη + 1

2

(A.5)

From table A.1, obtain locations of the sampling points and correspond-ing weight factors in ξ and η. Let

ξi,Wξi , i = 1, . . . , nξ

ηj ,Wξj , j = 1, . . . , nη

(A.6)

A.1. GAUSS QUADRATURE IN R1, R2 AND R3 773

be the sampling points and the weight factors in ξ and η. Now we canintegrate f(ξ, η) in (A.4) numerically using gauss quadrature. First, let usintegrate with respect to ξ, holding η constant.

I =

1∫−1

( 1∫−1

f(ξ, η) dξ)dη =

1∫−1

( nξ∑i=1

W ξi f(ξi, η)

)dη (A.7)

Next integrate with respect to η.

I =

nη∑j=1

W ηj

( nξ∑i=1

W ξi f(ξi, ηj)

)(A.8)

and we have the desired result i.e. numerical value of the integral I.

A.1.3 Volume Integrals over Ωm = Ωξηζ = [−1, 1]× [−1, 1]× [−1, 1]

Consider the following integral,

I =

1∫−1

1∫−1

1∫−1

f(ξ, η, ζ) dξ dη dζ =

1∫−1

[ 1∫−1

( 1∫−1

f(ξ, η, ζ) dξ

)dη

]dζ (A.9)

Let f(ξ, η, ζ) be an algebraic polynomial in ξ, η and ζ and let pξ, pη andpζ be the highest degree of the polynomial f(ξ, η, ζ) in ξ, η and ζ. Let nξ,nη and nζ be the number of sampling points in ξ, η and ζ determined using

nξ =pξ + 1

2

nη =pη + 1

2

nζ =pζ + 1

2

(round to next higher integer) (A.10)

From table A.1, obtain locations of the sampling points and correspond-ing weight factors in ξ, η and ζ. Let

ξi,Wξi , i = 1, . . . , nξ

ηj ,Wηj , j = 1, . . . , nη

ζk,Wζk , k = 1, . . . , nζ

(A.11)

be the sampling points and the weight factors in ξ, η and ζ. We now canintegrate f(ξ, η, ζ) in (A.9) using gauss quadrature. First, we integrate withrespect to ξ, holding η and ζ constant.

I =

1∫−1

[ 1∫−1

( nξ∑i=1

W ξi f(ξi, η, ζ)

)dη

]dζ (A.12)


Table A.1: Sampling points and weight factors for gauss quadrature for integration limits[−1, 1]

I =+1∫−1

f(x) dx =n∑i=1

Wi f(xi)

±xi Wi

n = 10 2.00000 00000 00000

n = 20.57735 02691 89626 1.00000 00000 00000

n = 30.77459 66692 41483 0.55555 55555 555560.00000 00000 00000 0.88888 88888 88889

n = 40.86113 63115 94053 0.34785 48451 374540.33998 10435 84856 0.65214 51548 62546

n = 50.90617 98459 38664 0.23692 68850 561890.53846 93101 05683 0.47862 86704 993660.00000 00000 00000 0.56888 88888 88889

n = 60.93246 95142 03152 0.17132 44923 791700.66120 93864 66265 0.36076 15730 481390.23861 91860 83197 0.46791 39345 72691

n = 70.94910 79123 42759 0.12948 49661 688700.74153 11855 99394 0.27970 53914 892770.40584 51513 77397 0.38183 00505 051190.00000 00000 00000 0.41795 91836 73469

n = 80.96028 98564 97536 0.10122 85362 903760.79666 64774 13627 0.22238 10344 533740.52553 24099 16329 0.31370 66458 778870.18343 46424 95650 0.36268 37833 78362

n = 90.96816 02395 07626 0.08127 43883 615740.83603 11073 26636 0.18064 81606 948570.61337 14327 00590 0.26061 06964 029350.32425 34234 03809 0.31234 70770 400030.00000 00000 00000 0.33023 93550 01260

n = 100.97390 65285 17172 0.06667 13443 086880.86506 33666 88985 0.14945 13491 505810.67940 95682 99024 0.21908 63625 159820.43339 53941 29247 0.26926 67193 099960.14887 43389 81631 0.29552 42247 14753

A.2. GAUSS QUADRATURE OVER TRIANGULAR DOMAINS 775

Integration with respect to η yields

I =

1∫−1

[ nη∑j=1

W ηj

( nξ∑i=1

W ξi f(ξi, ηj , ζ)

)]dζ (A.13)

and, finally, the integration with respect to ζ, we have

I =

nζ∑k=1

W ζk

[ nη∑j=1

W ηj

( nξ∑i=1

W ξi f(ξi, ηj , ζk)

)](A.14)

And, we have the desired result i.e. the numerical value of the integral I.

A.2 Gauss quadrature over triangular domains

The distorted triangular domain in the physical coordinate space xy(Fig. A.1 (a)) is mapped into ξη natural coordinate space. The mastertriangular domain Ωm in the ξη coordinate space is a two-unit equilateraltriangle (Fig. A.1 (b)). We choose three vertex nodes, three mid-side nodesand an internal node. The mid-side nodes and the internal node are hier-archical nodes when the local approximation is p-version hierarchical. Theorigin of the ξη coordinate system is located at node 2 of the equilateraltriangle.

Using area coordinates L1, L2 and L3 we can define the geometry of thetriangular domain

xy

=

n∑i=1

Ni(L1, L2, L3)

xiyi

(A.15)

(xi, yi) are the Cartesian coordinates of the element nodes (figure A.1 (a))and Ni(L1, L2, L3) are shape functions. The shape functions for triangularelements are in terms of standard barycentric or area coordinates L1, L2, L3.We could use six-node configuration with parabolic shape functions for thispurpose (see chapter 8). The area coordinates L1, L2, L3 can be related tothe orthogonal natural coordinates ξη through the relations introduced bySzabo and Babuska [1]:


23

(a)

1 31

46

5

7

2

467

5

2

1 2 3

567

8 94

(c)

(b)

y

x

ξ

η

ηr

ξr

Map of triangular domain of(b) into a two-unit square

2D distorted triangular do-main in xy space

Map of triangular domain of(a) into natural coordinatespace ξη

√3

Figure A.1: Distorted triangular element and its map in ξ, η and ξr, ηr spaces.

L1 =1

2

(1− ξ − η√

3

)L2 =

1

2

(1 + ξ − η√

3

)L3 =

η√3

(A.16)

The relations (A.16) can be used to convert the interpolation functions in(A.17) from L1, L2, L3 to natural coordinates ξ, η; that is, instead of (A.17)we can write

xy

=

n∑i=1

Ni(ξ, η)

xiyi

(A.17)

A.2. GAUSS QUADRATURE OVER TRIANGULAR DOMAINS 777

From (A.17) we can write

dxdy

= [J ]

dξdη

, [J ] =

dx

dξ

dx

dη

dy

dξ

dy

dη

(A.18)

anddx dx = det[J ] dξ dη (A.19)

Consider a two-unit square domain with the origin of the coordinatesystem ξr, ηr located at the center of the two-unit square (A.1 (a)). Then,the relationship between (ξr, ηr) and (ξ, η) coordinates can be written asin (A.20). We note that in mapping (A.20), nodes 1,2,3,4,6 and 7 of figurescorrespond to nodes 1,2,3,4,6 and 9 of figure A.1 (c). Whereas nodes 5,6 and7 of figure A.1 (c) correspond to node 5 of figure A.1 (b).

ξ =1

2ξr(1− ηr), η =

√3

2(1 + ηr) (A.20)

dξdη

=

[dxdξr

dxdηr

dydξr

dydηr

]dξrdηr

=

[1−ηr

2−ξr

2

0√

32

]dξrdηr

= [Jr]

dξrdηr

(A.21)

anddξ dη = det[Jr] dξr dηr (A.22)

In going from ξη to ξrηr space, the motivation being that in ξrηr space,the map of the triangular element of figure A.1 (a) (and hence Fig. A.1 (b)) isa two unit square. Now we can consider the integration over Ωe encounteredin the finite element processes. Consider

I =

∫Ωe

f(x, y) dx dy (A.23)

in which Ωe is the distorted triangular domain of Fig. A.1 (a). Using (A.19),integral in (A.23) can be transformed to ξη domain of Fig. A.1 (b).

I =

∫Ωξη

f(ξ, η) det[J ] dξr dηr (A.24)

Using (A.22) we can further transform (A.24) to ξrηr domain, a two unitsquare.

I =

∫Ωξrηr

(f(ξr, ηr)det[J ] det[Jr]

)dξr dηr =

1∫−1

1∫−1

g(ξr, ηr) dξr dηr (A.25)


in whichg(ξr, ηr) = f(ξr, ηr)det[J ] det[Jr] (A.26)

Numerical values of I in (A.25) can be obtained using standard Gaussquadrature over a two unit square.

References for additional reading[1] B. Szabo and Ivo Babuska. Finite Element Analysis. Wiley-Interscience, 1st edition,

1991.

INDEX

A

A posteriori error estimation, 747A priori error estimation, 694–719Accuracy of finite element method, 693,

749–751Adaptive processes, 751–767

1D Burgers equation, 61–7651D convection-diffusion equation, 752–

767adaptive h-refinement, 753–757,

760adaptive p-refinement, 758–760graded h-rediscretizations, 754–757higher geometric ratios, 760–761uniform h-refinement, 752–753uniform p-refinement, 758

1D diffusion equation, 765–766Admissible

change (δu), 42spaces, 95–96

Advection, 401, 408–409, 411Alternate

approach to error, 154approach to FE processes, 609derivation (Euler’s equation), 36derivation of C11 approximation, 576formulation, LSP, 743methods of approximation

boundary element method, 2collocation method, 120–121finite difference method, 2finite volume method, 2meshless methods, 2

Approximate solution, 3, 177, 182Approximation error, 327Approximation functions, 7, 70, 203–205,

233, 283, 493–604Approximations

classical methods, 69–94local 1DC0 p-version hierarchical, 501–506CJ p-version hierarchical, 507Lagrange interpolation, 497polynomial, 495

local 2Dquadrilaterals C00, 522–531rectangular family CJJ , 532–535

localC0, 495–507C1, 507–511C2, 511–515CJ , 515–516

one, two, ... parameters, 153spaces, 95–96, 231, 369spaces Vh, V , 231, 275

Area coordinates, 558–577Assembled equations, 9, 212–214Assembly of element equations, 9, 195,

212–213Asymptotic range, 752–760Axial spar (rod) elements, 651–668

in R1, 651–655in R2, 656–666in R3, 666–668

Axisymmetric elements, 644–648, 681Axisymmetric heat conduction, 113, 300Axisymmetric problems, 113, 300, 644–

648Axisymmetric solids, 644–648, 681

B

Banach space, 18Banach theorem, 186Bars (see axial rods)Basic concepts, FE method, 3–13

discretization, 3–5

779

780 INDEX

element equation assembly, 8–9hpk framework, 11–13integral forms, algebraic equations

over element, 7–8k -version, 11–13local approximation, 5–7post-processing, 9solution computation, 9

Basis (functions), 13, 16, 17, 71, 260,496, 501–503, 520, 521, 559

Beams, 105, 305–310Euler-Bernoulli, 668–674Timoshenko, 677–681

Bending of Beams (see beams)Bilinear functionals, 26, 96, 719Boundary conditions

essential, 81–82, 109, 122, 178, 243,670

homogeneousimposition of, 242–243specification, 242–243natural, 82, 83, 101, 104, 122, 128,

145, 150, 178, 256Boundary value problem, 1–2Brick element, 348–358, 584–642Burgers equation, 422–442, 741–747, 761

1D non-linear operator, 422–441Galerkin method with weak form,

423–428LSP based on residual functional,

428–441

C

C0 interpolation, 495–507, 567C00 interpolation, 522–532, 556–566C00 interpolations, serendipity family of,

578–584deriving serendipity interpolation func-

tions, 579–584C1 interpolation, 507–511C11 interpolation, 538, 539, 549, 550C2 interpolation, 511–515C22 interpolation, 540, 541, 550, 551C33 interpolation, 551, 552Cj interpolation, 515, 516Cij interpolation, 532–537, 542–550, 553,

554, 571,575Calculus of variations

Euler’s equation, 34–41fundamental lemma, 31fundamental lemma, integral forms

using, 72–94

Galerkin method, 73–78Galerkin method with weak form,

81–86Petrov-Galerkin method, 78–80weighted-residual method, 78–80

variation of functional, 33–34, 41–42

variational consistency, 37–38variational inconsistency, 38

Cauchy-Schwarz inequality, 20–22, 704–706, 708

Chebyshev polynomials, interpolations basedon, 577–578

1D C0 p-version hierarchical inter-polations, 577–578

2D Cij p-version interpolation func-tions for quadrilateral elements,578

Classical methods of approximation, 69–192

approximation methods, approxima-tion spaces for, 95–97

calculus of variations, fundamentallemma of, integral forms using,72–94

Galerkin method, 73–78Galerkin method with weak form,

81–86Petrov-Galerkin method, 78–80weighted-residual methods, 78–80

Galerkin methodnon-linear differential operator, 77–

78non-self-adjoint linear differential

operator, 74–77self-adjoint linear differential op-

erator, 74–77Galerkin method with weak form

linear differential operator, 85–86non-linear differential operator, 86

integral forms, steps based on, 70–72

integral formulation of BVPs using,97–153

non-linear differential operator, 141–153

non-self-adjoint differential oper-ator, 128–141

self-adjoint differential operator,98–128

numerical examples, 153–185Petrov-Galerkin method

INDEX 781


non-self-adjoint linear differentialoperator, 78–80

self-adjoint linear differential op-erator, 78–80

weighted-residual methodsnon-linear differential operator, 80–

81non-self-adjoint linear differential

operator, 78–80self-adjoint linear differential op-

erator, 78–80Convergence, h-, p-, k -versions of FEM,

684–689h-version of FEM, h-convergence, 685hp-version of FEM, hp-convergence,

686k -version of FEM, k -convergence, 687–

689p-version of FEM, p-convergence, 686

Convergence rates, 689–693computations, convergence behavior

of, 690–692Coordinates

Cartesian, 233, 545, 559, 600, 602cylindrical, 113, 300, 644global, 658–661local, 652–676natural, 203–204, 227–228, 493–605

D

Dirichlet boundary condition, 166Discontinuity

inter-element, 249, 253, 327, 380,693

subdomain, 501Direction cosines, 28–29, 310–312, 658Discretization, 3–11Displacement

field, local approximation, linear elas-ticity, 611

nodal, 652, 659–660, 665–666strain, 330, 353transverse, 105, 669

E

Eigenvalueseigenvalue problem, 188–191positive-definite coefficient matrices,

38–39

Elasticity, linear, principle of minimumtotal potential energy, 609–624

approach, 611element equations, 611–617

displacement field, local approxi-mation, 611

strain energy Πe1, potential energy

of loads Πe2, 612–613

strains, 612total potential energy Πe, elemente, 614–617

finite element formulation for 2D lin-ear elasticity, 617–624

body forces, 620–621element stiffness matrix [Ke], 619–

620initial strains, 621–622

local approximation of u, v over Ωe

or Ωn 618nodal degrees of freedom, strains

to, [B] matrix relating to, 619pressure acting normal to element

faces, 622–624strains, constitutive equations, 618–

619stresses, 618–619transformations from (ξ, η) to (x, y)

space, 620new notation, 610

Energy, kinetic, 443, 451, 455, 626Energy product, 27Equilibrium

condition 747considerations, 243equations, 655

Error computation, 693–694a posteriori, 747–751a priori, 694–719

GM/WF, 703–719GM/WF for non-self adjoint, non-

linear operator, 697–698integral forms, approximation meth-

ods, 702least-squares method based on resid-

ual functional for non-linear op-erator, 699–702

LSP, 703–719non-self adjoint, non-linear oper-

ator, GM/WF, 697–698non-self-adjoint operator, residual

functional, lease-squares methodbased on, 698–699

782 INDEX

proposition, proof, 706–714, 716–719

residual functional, lease-squaresmethod based on, 698–699

self-adjoint, non-self-adjoint oper-ator, 698–699

self-adjoint operator, 694–697a priori GM/WF, 703–704

convergence rates, 714–716LSP model problem, 704–706model problem, 703–704proof, 706–714, 716–719proposition, 706–714, 716–719

Essential boundary condition, 81–82, 122,243

Euler-Bernoulli beam element, (see Beams)Euler’s equation, (see Calculus of Varia-

tions)

F

Field displacement, local approximation,linear elasticity, 611

Finite element formulation2D linear elasticity

body forces, principle of minimumtotal potential energy, 620–621

element stiffness matrix [Ke], prin-ciple of minimum total poten-tial energy, 619–620

initial strains, principle of mini-mum total potential energy, 621–622

local approximation of u, v overΩe or Ωn, principle of minimumtotal potential energy, 618

nodal degrees of freedom, strainsto, [B] matrix relating to, 619

pressure acting normal to elementfaces, equivalent nodal loads F ep ,622–624

principle of minimum total poten-tial energy, 617–624

strains, constitutive equations, 618–619

stresses, principle of minimum to-tal potential energy, 618–619

thermal loads, 621–622transformations from (ξ, η) to (x, y)

space, principle of minimum to-tal potential energy, 620

linear structural mechanics, R2,R3, 681

Function spaces, 22–23Banach spaces, 18Hilbert spaces, 18–22, 72metric spaces, 17–18scalar product spaces, 340, 716

Functional, 26bilinear, 26, 122linear, 26,quadratic, 27, 40–41, 103, 124–126residual, 87–89, 97

in error computations, 748–751symmetric, 26

Functional analysis, 15–67energy product, 27function spaces, 15–30Hilbert spaces Hk(Ω), 18integration by parts, 27–30operator, 15–30

properties of, 44–64operator properties


non-self-adjoint operator, 58–62self-adjoint differential operator,

44–58operator types, 24–27scalar product, Hk(Ω) space, defined,

19scalar product properties, 19sets, 15–30spaces, 15–30u in Hilbert space Hk(Ω)

norm of, 20seminorm of, 20–22

Fundamental lemma (see Calculus of Vari-ations)

G

Galerkin method, classical method, 73–78

non-linear differential operator, 77–78, 142–143, 148

non-self-adjoint linear differential op-erator, 74–77, 129, 135–136

self-adjoint linear differential opera-tor, 74–77, 98–99, 106–107, 119–120

Galerkin method, FEM, 199, 212–214Galerkin method with weak form, classi-

cal method, 81–86non-linear differential operator, 86,

144–146, 150–152

INDEX 783

non-self-adjoint differential operator,85, 130–133, 137–139

self-adjoint differential operator, 85,100–104, 108–117, 121–126

Galerkin method with weak form, FEM,200, 206–210, 215


non-self-adjoint differential operator,367

self-adjoint differential operator, 224,227–243, 301–303, 320–337

Gauss elimination, 216Gauss quadrature, numerical integration,

771–778

H

h-adaptivity, 752–757hpk framework, 11-13h-refinement, 752–757h-version of FEM, 11Hamilton’s principle, 625Hermite interpolation functions, 673Hexahedron elements, 3D elements, in-

terpolation functions forderivatives of φeh(ξ, η, ζ) with respect

to x, y, z, 587–588mapping of lengths, 586mapping of points, 584–585mapping of volumes, 586–587

Hilbert spaces Hk(Ω), functional analy-sis, 18

I

IBP. See Integration by partsIntegral forms, steps based on, 70–72Integral formulation of BVPs, classical

approximation, 97–153Integrals

Lebesgue, 42–44Riemann, 42–44

Integration, numerical (see Gauss quadra-ture)

Integration by parts, 27–30Interpolation functions, 3D elements, 584–

604Interpolation theory

2D triangular elements, 556–565Pascal triangle, Lagrange familyC00 basis functions based on,556–558

2D triangular elements area coordi-nates

higher degree C00 basis functions,559–565

Lagrange family C00 basis func-tions based on, 558–559

mapping, 493–607

C00 interpolations, serendipity fam-ily of, 578–584

2D Cij(Ωe) p-version, local ap-proximation, 532–542

1D interpolations based on, 577–578

2D interpolations based on, 577–578

2D triangular elements, interpo-lation theory for, 556–565

interpolation theory over Ω, ele-ments of, 495–516

local approximation over Ωm, 520–532

one dimension, mapping, 493–495

three-dimensional elements, inter-polation functions for, 584–604

two dimensions, mapping in, quadri-lateral elements, 516–520

mapping Legendre polynomials

1D approximation based on, 566–577

two-dimensional approximation basedon, 566–577

quadrilateral element mapping

approximations for, two-dimensionalCij(Ωe), 542–555


Interpolation theory over Ω, elements, 495–516

higher order global differentiabilityapproximations, one dimension,p-version, 507–516

class Ci(Ωe), 515–516

class C1(Ωe), 508–511

interpolations or local approxima-tions of class C2(Ωe), 511–515

Lagrange interpolating polynomials,one dimension, 497–501

p-version hierarchical functions, onedimension, 501–507

polynomial approximation, one di-mension, 495–497

784 INDEX

J

Jacobian of Mapping, 494Jacobian of transformation, 349, 518, 544,

586

K

k-version of FEM, 11–13Kinetic energy, 443, 451, 455, 626

L

L2-norm, 20, 689–749Laplace operator, 29Lax-Milgram theorem, 186Least squares finite element formulation,

non-self-adjoint differential op-erator

1D convection-diffusion equation, 378–379


residual functional, two-dimensionalconvection-diffusion equation, 395–397

Least squares formulation, non-self-adjointdifferential operator, first ordersystem, 1D convection-diffusionequation, 380–390

Least-squares method, classical method,87–94

non-linear differential operator, 93–94, 146–147, 152–153

non-self adjoint linear differential op-erator, 92–93, 133-134, 140–141

self-adjoint linear differential oper-ator, 92–93, 104–105, 111-112,117–119, 126–128

Least-squares method, FEM, 202, 210–212, 214

non-linear differential operator, 428–429, 452–455, 471–473

non-self adjoint linear differential op-erator, 378–390, 395–397

self-adjoint linear differential opera-tor, 225, 265–285, 303–305, 337–344

Lebesque integral, calculus of variations,functional analysis, 42–44

Legendre polynomials1D, 2D approximations based on,

566–577

2D C00 p-version interpolationsfunctions for triangular elements,568

2D Cij interpolation functions fortriangular elements, 571–577

2D Cij p-version interpolations func-tions for quadrilateral elements,568

1D p-version C0 hierarchical ap-proximation functions, 567

2D p-version C00 hierarchical in-terpolation functions for quadri-lateral elements, 567

Legendre polynomials, 5661D approximation based on, 566–

577Lengths, mapping of, mapping in one di-

mension, 494Linear, non-linear solid mechanics, prin-

ciple of virtual displacements,625–649

axisymmetric solid finite elements,644–648

2D solid continua, finite element for-mulation, 637–641


solution method, 635–637solution procedure, 636–637virtual displacements principle, 626virtual work statements, 627–635stiffness matrix, 633–635

Linear elasticity, principle of minimumtotal potential energy, 609–624





stresses, 612total potential energy Πe elemente, 614–617

finite element formulation for 2D lin-ear elasticity, 617–624


620initial strains, 621–622local approximation of u, v over

Ωe or Ωn, 618nodal degrees of freedom, strains

INDEX 785


faces, equivalent nodal loads F ep ,622–624


stresses, 618–619transformations from (ξ, η) to (x, y)

space, 620new notation, 610

Linear structural mechanics, 651–6811D axial spar

rod element in R1, 651–655rod element in R2, 656–666rod element in R3, 666–668

1D axial spar rod element in R1 stresses/strains,653

total potential energy, Πe, 653–655

1D axial spar rod element in R2

coordinate transformation, 656–659

two member truss, 659–666Euler-Bernoulli beam element, 668–

674derivation of element equations,

670–671local approximation, 671–674

Euler-Bernoulli frame elements in R2,675–677

finite element formulation, R2, R3,681

Timoshenko beam elements, 677–681element equations, 678–681

Local approximation, finite element method,5–7

Local approximation over Ωm, quadrilat-eral elements, 520–532

C00 p-version hierarchical local ap-proximations based on Lagrangepolynomials, 529–532

polynomial approach, C00 local ap-proximations over Ωn, 522–525

tensor product, C00 Lagrange typelocal approximation, 525–529

M

Mapping in one dimension, 493–495dependent variable φ over Ωe , be-

havior of, 495lengths, mapping of, 494

points, mapping of, 494Mapping in two dimensions, quadrilat-

eral elements, 516–520Mapping/interpolation theory, 493–607

C00 interpolations, serendipity fam-ily of, 578–584

2D Cij(Ωe) p-version, local approx-imation, 532–542

1D interpolations based on, 577–5782D interpolations based on, 577–5782D triangular elements, interpola-

tion theory for, 556–565interpolation theory over Ω, elements

of, 495–516Legendre polynomials




one dimension, mapping, 493–495quadrilateral elements

approximations for, two-dimensionalCij(Ωe), 542–555


three-dimensional elements, interpo-lation functions for, 584–604

two dimensions, mapping in, quadri-lateral elements, 516–520

Mapping of lengths, 3D elements, inter-polation functions for, hexahe-dron elements, 586

Mapping of points, hexahedron elements,3D elements, interpolation func-tions for, 584–585

Mapping of volumes, hexahedron elements,3D elements, interpolation func-tions for, 586–587

Master element, 518, 522, 542, 565Mesh (see Discretization)Minimally conforming spaces, 96–97

N

Natural boundary condition, 81–83Navier-Stokes equations, two-dimensional

steady-state, non-linear differ-ential operator, 450–471

asymmetric backward facing step, 461–467

786 INDEX

flow past circular cylinder, 467–471LSP based on residual functional

first order system of PDEs, 452–453

higher order systems of PDEs, 454–455

slider bearing, flow of viscous lubri-cant, 455–457

square lid-driven cavity, 457–461Neumann boundary condition, 289–290Newton-Raphson method, 89, 419

Newton’s method with line search,91

Newtonian fluid flow, two-dimensional com-pressible, non-linear differentialoperator, 471–486

Carter’s plate, 474–483higher Mach number flows, 478–

479Mach 1 flow, 476–478Mach 2 flow, 479–480Mach 3 flow, 480–481Mach 5 flow, 481–483

Mach 1 flow past circular cylinder,483–486

Nodal displacement, 652, 659–660, 665–666

Non-linear, linear solid mechanics, prin-ciple of virtual displacements,625–649

axisymmetric solid finite elements,644–648



solution method, 635–637solution procedure, 636–637

virtual displacements principle, 626virtual work statements, 627–635

stiffness matrix, 633–635Non-linear differential operator, 419–492

1D Burgers equation, 422–441Galerkin method with weak form,

423–428LSP based on residual functional,

428–441Navier-Stokes equations, 2D steady-

state, 450–471asymmetric backward facing step,

461–467first order system of PDEs, 452–

453flow past circular cylinder, 467–

471higher order systems of PDEs, 454–

455slider bearing, flow of viscous lu-

bricant, 455–457square lid-driven cavity, 457–461

Newtonian fluid flow, 2D compress-ible, 471–486

Carter’s plate, 474–483Carter’s plate higher Mach num-

ber flows, 478–479Carter’s plate Mach 1 flow, 476–

478Carter’s plate Mach 2 flow, 479–



483Mach 1 flow past circular cylin-

der, 483–486operator, properties of, functional

analysis, 63–64parallel plates (polymer flow), fully

developed flow of Giesekus fluidbetween, 442–450

Non-self-adjoint differential operator, 363–417


analytical solution, 365–367first order system, least squares

formulation, 380–390Galerkin method with weak form

(GM/WF), 367–378least squares finite element for-

mulation, 378–3792D convection-diffusion equation, 390–

413advection of cosine hill in rotat-

ing flow field, 408–411advection skewed to square do-

main, 401–408convection dominated thermal flow,

401–408least squares finite element for-

mulation, 397–400residual functional, least squares

finite element formulation basedon, 395–397

INDEX 787

thermal boundary layer, 411–413Non-self-adjoint operator

1D convection-diffusion equation, modelproblems, 730–741

GM/WF, 737–738LSP first order system, 731–736LSP higher order system, 738–741

operator, properties, functional anal-ysis, 58–62

NormsL2-norms, 20, 689–749Hk-norms, 20, 689–749Hk-seminorms, 20, 689–749

Numerical integration using Gauss quadra-ture, 771–778

O

One-dimensional, two-dimensional inter-polations based on, Chebyshevpolynomials, 577–578

Chebyshev polynomials, 5771D C0 p-version hierarchical inter-

polations, 577–5782D Cij p-version interpolation func-

tions for quadrilateral elements,578

1D p-version C0 hierarchical approx-imation functions, 578

One-dimensional axial spar, linear struc-tural mechanics

rod element in R1, 651–655stresses/strains, 653total potential energy, Πe, 653–

655rod element in R2, 656–666

coordinate transformation, 656–659

two member truss, 659–666two member truss computations,

660–666two member truss post-processing,

665–666rod element in R3, 666–668

One-dimensional Burgers equation, non-linear operator, model problem,741–747

GM/WF, 747LSP, higher-order system, 743–746

One-dimensional convection-diffusion equa-tion, non-self-adjoint differen-tial operator, 365–390

analytical solution, 365–367first order system, least squares for-

mulation, 380–390Galerkin method with weak form (GM/WF),

367–378least squares finite element formu-

lation, 378–379One-dimensional p-version C0 hierarchi-

cal approximation functions, Leg-endre polynomials, 1D, 2D ap-proximations based on, 567

One-dimensional steady-state diffusion equa-tion, self-adjoint differential op-erator, single dependent vari-able, 1D BVPs in, 226–251

analytical solution, comparison withfinite element solutions, 246–251

approximation space Vh, 231–232assembly of element equations, so-

lution computation, 235–236discretization, 227EBCs, imposition of, 243element equations, 234–235integral form using GM/WF (weak

form) of BVP for element e withdomain Ωe , 227–230

inter-element continuity conditionson PVs, dependent variables,237

local approximation φeh , mappingΩe to Ω, 233

numerical study, special case, 244–245

post-processing, solution, 246rules for assembling element matri-

ces, vectors, 237–242sum of secondary variables, inter-

element continuity conditions,242–243

unknown degrees of freedom, solv-ing for, 243–244

Onset of post-asymptotic range, 691Onset of pre-asymptotic range, 691Operator

functional analysis, 15–30properties of, functional analysis, 44–

64non-linear differential operator, 63–

64non-self-adjoint operator, 58–62self-adjoint differential operator, 44–

788 INDEX

58types of, 24–27

Orthogonality, 23Orthotropic material, 354Overview, finite element method, 3–13

discretization, 3–5element equation assembly, 8–9hpk framework, 11–13integral forms, algebraic equations

over element, 7–8k -version, 11–13local approximation, 5–7post-processing, 9solution computation, 9

P

p-adaptivity, 758–760p-version of FEM, 11–12Pascal’s prism, 591Pascal’s pyramid, 601Pascal’s rectangle, 524Pascal’s triangle, 557Parallel plates (polymer flow), fully de-

veloped flow of Giesekus fluidbetween, non-linear differentialoperator, 442–450

Petrov-Galerkin method, classical method,78–80

non-linear differential operator, 80–81, 143–144, 149

non-self-adjoint linear differential op-erator, 78–80, 129–130, 136–137

self-adjoint linear differential oper-ator, 78–80, 99–100, 107–108,120–121

Petrov-Galerkin method, FEM, 199, 205,212, 214

Philosophy, finite element method, 1–3Plate

Carter’s plate, 474–483clamped plate, 342–344simply supported plate, 341–344

Points, mapping of, mapping in one di-mension, 494

Poisson’s equation, 44, 119, 320Post-processing of FEM solution, 9, 220–

221, 246Primary variables, 81–82Principle of minimum total potential en-

ergy, linear elasticity using, 609–624





stresses, 612total potential energy Πe, elemente, 614–617

finite element formulation, 2D lin-ear elasticity, 617–624


620initial strains, 621–622local approximation of u, v over

Ωe or Ωn, 618nodal degrees of freedom, strains


faces, equivalent nodal loads F ep ,622–624


transformations from (xi, η) to (x, y)space, 620

new notation, 610

Q

Quadratic functional, 27, 40–41, 103, 124–126

Quadrilateral elements, local approxima-tion over Ωm, 520–532

C00 p-version hierarchical local ap-proximations based on Lagrangepolynomials, 529–532

polynomial approach, C00 local ap-proximations over Ωn, 522–525

tensor product, C00 Lagrange typelocal approximation, 525–529

R

Residualfunctional, LSP based on, self-adjoint

differential operator, 225–226functions, 87–89, 687, 716

Reynolds number, 63–65, 427, 463, 465Riemann integral, calculus of variations,

functional analysis, 42–44Ritz method, 84Rodriguez formula, 566

INDEX 789

Rolle theorem, 707

S

Scalar product, Hk(Ω) space, defined, func-tional analysis, 19

Scalar product properties, functional anal-ysis, 19

Secondary variables, 81–82Self-adjoint differential operator, 223–362

2D boundary value problems, 310–344

2D plane elasticity, 328–3442D Poisson’s equation, numerical

studies, 320–328Galerkin method with weak form,

332–337multi-variables, 328–344residual functional, least-squared

method using, 337–344single dependent variable, general

2D BVP in, 310–320GM/WF, 224–225operator, properties of, functional

analysis, 44–58residual functional, LSP based on,

225–226single dependent variable, 1D BVPs

in, 226–309convective boundary, 1D heat con-

duction, 289–3001D axisymmetric heat conduction,

300–3051D steady-state diffusion equation,

226–251finite element processes based on

GM/WF, 226–251fourth-order differential operator,

A 1D BVP governed by, 305–309

least-squares finite element formu-lation, 265–278

LSFEP using auxiliary equations,numerical studies, 285–289

LSFEP using auxiliary variables,auxiliary equations, numericalstudies, 285–289

three-dimensional boundary value prob-lems, 344–358

multivariables, 353–358single dependent variable, 344–353

Self-adjoint differential operator 2D bound-ary value problems, single de-

pendent variable, general 2D BVPin

approximation space Vh, 316–317definition of Ωe , element geometry,

314–315element matrix [Ke] computation,

317secondary variable vector P e, 317–

320vector F e, computation, 317

Self-adjoint differential operator single de-pendent variable, 1D BVPs in

convective boundary, 1D heat con-duction

approximation space Vh, 296–297numerical study, 297–300

1D axisymmetric heat conductionGalerkin method with weak form,

301–303residual functional, LSM based on,

303–3051D steady-state diffusion equationanalytical solution, comparison with

finite element solutions, 246–251

approximation space Vh, 231–232assembly of element equations, so-

lution computation, 235–236discretization, 227EBCs, imposition of, 243element equations, 234–235integral form using GM/WF (weak



local approximation φeh , 231–232local approximation φeh mapping

Ωe to Ω, 233numerical study, special case, 244–

245post-processing, solution, 246rules for assembling element ma-

trices, vectors, 237–242sum of secondary variables, inter-


test function space V , 231–232unknown degrees of freedom, solv-

ing for, 243–244fourth-order differential operator, A

790 INDEX

1D BVP governed by, approx-imation space Vh, 309

least-squares finite element formu-lation

approximation space Vh, 275numerical studies, 275–278

LSFEPauxiliary equations, 278–289auxiliary equations approximation

spaces for φeh and τeh , 285auxiliary variables, 278–289auxiliary variables approximation

spaces for φeh and τeh, 285Self-adjoint differential operator three-dimensional

boundary value problemsmultivariables

approximation spaces, 357Galerkin method with weak form,


single dependent variableapproximation space, 347definition of Ωe , 348–350element matrix [K],vector F e,

computations of, 350Galerkin method with weak form,

345–346local approximation T eh , 347–348secondary variable vector P e,

350–353Serendipity family of C00 interpolations,

578–584deriving serendipity interpolation func-

tions, 579–584Sets, functional analysis, 15–30Single dependent variable, 226–309

convective boundary, 1D heat con-duction, 289–300

approximation space Vh, 296–297numerical study, 297–300

1D axisymmetric heat conduction,300–305

Galerkin method with weak form,301–303

residual functional, LSM based on,303–305

1D steady-state diffusion equation,226–251

analytical solution, comparison withfinite element solutions, 246–251

approximation space Vh, 231–232

assembly of element equations, so-lution computation, 235–236

discretization, 227EBCs, imposition of, 243element equations, 234–235integral form using GM/WF (weak



local approximation φeh , 231–232numerical study, special case, 244–

245post-processing, solution, 246rules for assembling element ma-

trices, vectors, 237–242sum of secondary variables, inter-


test function space V , 231–232unknown degrees of freedom, solv-

ing for, 243–244finite element processes based on GM/WF,

226–251fourth-order differential operator, A

1D BVP governed by, 305–309approximation space Vh, 309

least-squares finite element formu-lation, 265–278

approximation space Vh, 275numerical studies, 275–278

LSFEPauxiliary equations, 278–289auxiliary variables, 278–289using auxiliary equations, numer-

ical studies, 285–289using auxiliary variables, auxiliary

equations, numerical studies, 285–289

Single dependent variable 1D steady-statediffusion equation, local approx-imation φeh , mapping Ωe to Ω,233

Single dependent variable LSFEPauxiliary equations, approximation

spaces for φeh and τeh , 285auxiliary variables, approximation spaces

for φeh and τeh, 285Singular matrices, 377Spaces, functional analysis, 15–30Stability

INDEX 791

unconditionally stable computations,97, 186, 198

Stiffness matrix, virtual work statements,linear, non-linear solid mechan-ics, 633–635

Strain displacement, 330, 353Strain energy Πe

1, potential energy of loadsΠe

2, linear elasticity, 612–613

T

Tensor productR2, 525–529R3, 592–599

Tetrahedral family of elements, 600–604Three-dimensional boundary value prob-

lems, self-adjoint differential op-erator, 344–358

multivariables, 353–358approximation spaces, 357Galerkin method with weak form,


single dependent variable, 344–353approximation space, 347definition of Ωe, 348–350element matrix [Ke], vector F e,

computations of, 350Galerkin method with weak form,

345–346local approximation T eh 347–348secondary variable vector P e,350–

353Three-dimensional elements

interpolation functions for, 584–604hexahedron elements, 584–588

interpolation functions for dependentvariable φ over Ωm, local ap-proximation for, 588–604

four-node linear tetrahedron ele-ment (p-level of one), 604

hexahedron elements, 588–590higher degree approximations ofφ over Ωm, 590–592

Lagrange interpolations, basis func-tions of class C000(Ωe), inter-polation theory for 3D tetrahe-dron elements, 600–601

node element, 3D Cijk(Ωe) p-versioninterpolations for distorted hex-ahedron elements, 600

ten-node tetrahedron element (p-level of 2), 604

Three-dimensional elements interpolationfunctions for, hexahedron ele-ments

derivatives of φeh(ξ, η, ζ) with respectto x, y, z, 587–588

mapping of lengths, 586mapping of points, 584–585mapping of volumes, 586–587

Three-dimensional elements interpolationfunctions for dependent variableφ over Ωm, local approxima-tion for

hexahedron elements, 3D Cijk(ωe)p-version local approximations,599–600

tensor productC000 Lagrange type local approx-

imations using, 592–597C000 p-version 3D hierarchical lo-

cal approximations, 597–599volume coordinates

higher degree C000 basis functions,603–604

Lagrange family C000 interpola-tions based on, 601–603

Three-dimensional solid continua, linear,non-linear solid mechanics, prin-ciple of virtual displacements,641–644

Timoshenko beam elements, linear struc-tural mechanics, 677–681 ele-ment equations, 678–681

Total potential energy Πe, element e, lin-ear elasticity, 614–617

Transverse displacement, 105, 669Triangular family of elements, 556–559Two-dimensional boundary value prob-

lems, self-adjoint differential op-erator, 310–344

2D plane elasticity, 328–3442D Poisson’s equation, numerical stud-

ies, 320–328multi-variables, 328–344

Galerkin method with weak form,332–337

residual functional, least-squaredmethod using, 337–344

single dependent variable, general 2DBVP in, 310–320

approximation space Vh, 316–317definition of Ωe, element geome-

try, 314–315

792 INDEX

element matrix [Ke] computation,317

secondary variable vector P e,317–320

vector F e computation, 317

Two-dimensional C00 p-version interpo-lations functions for triangularelements, Legendre polynomi-als, 1D, 2D approximations basedon, 568

Two-dimensional Cij(Ωe)

approximations, quadrilateral elements,542–555

C11 HGDA, 2D distorted quadri-lateral elements in xy space, 549–550



p-version local approximations, 532–542

p-levels of pξ, pη, 2D Cij(Ωe) in-terpolations, 542

type C11(Ωe) with p-levels of pξ, pη,2D interpolations of, 538–539

type C22(Ωe) with p-levels of pξ, pη,2D interpolations of, 540–541

Two-dimensional Cij(Ωe) approxi-mations, quadrilateral elements,distorted quadrilateral elementsderivation of Cij approximationsfor, 553–554

limitations of 2D C11 global differ-entiability local approximations,554–555

Two-dimensional Cij interpolation func-tions for triangular elements,Legendre polynomials, 1D, 2Dapproximations based on, 571–577

Two-dimensional Cij p-version interpo-lations functions for quadrilat-eral elements, Legendre poly-nomials, 1D, 2D approximationsbased on, 568

Two-dimensional convection-diffusion equa-tion, non-self-adjoint differen-tial operator, 390–413

advection of cosine hill in rotating

flow field, 408–411advection skewed to square domain,

401–408convection dominated thermal flow,

401–408least squares finite element formu-

lation, 397–400residual functional, least squares fi-

nite element formulation basedon, 395–397

thermal boundary layer, 411–413Two-dimensional p-version C00 hierarchi-

cal interpolation functions forquadrilateral elements, Legen-dre polynomials, 1D, 2D ap-proximations based on, 567

Two-dimensional solid continua, linear,non-linear solid mechanics, prin-ciple of virtual displacements,637–641

Two-dimensional triangular elements, in-terpolation theory, 556–565

area coordinateshigher degree C00 basis functions,

559–565Lagrange family C00 basis func-

tions based on, 558–559Pascal triangle, Lagrange family C00

basis functions based on, 556–558

U

U in Hilbert space Hk(Ω)norm of, functional analysis, 20seminorm of, functional analysis, 20–

22

V

Variation of functional, calculus of vari-ations, functional analysis, 33–34, 41–42

Variational consistency, 37–38Variational inconsistency, 38–39Virtual work statements, linear, non-linear

solid mechanics, principle of vir-tual displacements, 627–635

W

Weighted-residual methods, classical method,78–80

INDEX 793


non-self-adjoint linear differential op-erator, 78–80

self-adjoint linear differential oper-ator, 78–80

Weighted-residual methods, FEM, 199,212–214

Y

Young’s modulus, 360

Z

Zones of convergence behavior, 690

The Finite Element

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