A POSTERIORI ERROR ESTIMATION METHODS FOR PDE's...This method can be easily extended to other linear...

A POSTERIORI ERROR ESTIMATION

METHODS FOR PDE’s

S. Repin

Saint Petersburg Department of V.A. Steklov Institute of Mathematicsand University of Jyvaskyla

With warm greetings from EIMI !

S. Repin Zurich Summer School, 2012 2

Error estimation theory for PDE’s contains three main parts:

A priori estimates of approximation errors

A posteriori indicators of approximation errors

Guaranteed (fully reliable) estimates of deviation from the exact solution


Part I. Historical overview.


A priori and a posteriori conceptions for PDE’s

In the 20th century, the theory of differential equations was mainlydeveloped in the context of the a priori conception:

Existence ⇒ Regularity ⇒ Approximation

Classical approximation theory gives an asymptotic answer to the errorcontrol problem:General scheme:Au = f , where A : V → V ∗ is projected to Vh ⊂ V ,dimVh = N ∼ 1/h <∞,uh: Ahuh = fh. It is proved that uh → u. Ultimate goal is to provequalified estimates of the type

‖u − uh‖V ≤ Chk , C > 0, k > 0.


Features:

estimate has a purely asymptotic sense.

crucially based on additional regularity of u and regularity of themeshes used.

valid only for Galerkin solutions uh.

gives no real information on the error encompassed in a particularapproximation uh.

A priori estimates show theoretical behavior of errors, which arise ifcomputations are performed on an ideal computer with infinite power andunlimited memory.


Evolutionof a posteriori approaches

for differential equations


Heuristic rule of C. Runge:

If the difference between two approximate solutions computed on a coarsemesh Th and a refined mesh Thref

with mesh size href (e.g., href = h/2) issmall, then both uhref

and uh are probably close to the exact solution.

In other words, this rule can be formulated as follows:

If N (uh − uhref) is small then uhref

is close to u

where N (·) is a certain functional or mesh-dependent norm.S. Repin Zurich Summer School, 2012 8

It terms of modern terminology Runge’s rule can be viewed asthe very first a posteriori Error Indicator :

N (uh − uhref) = Error Indicator(uh) = EIRunge(uh)

Runge’s heuristic rule is simple!

It looks very natural and was easily accepted by numerical analysts.


However, this simple rule cannot provide reliable error analysis.The simplest contr-example:

xk+1 =√δ + x2

k , k = 1, 2, 3, ... xk → +∞,|xk+1 − xk | → 0.

Another example: if the spaces Vh are refined ”improperly”, e.g., new trialfunctions do not really extend the space, then approximations may bequite close to each other and far from the exact one.

Also, in practice, we need to now precisely what the word ”close” means,i.e. we need to have a more concrete presentation on the error.For example, it would be useful to know thatIf uk − uk+1 ≤ ǫ then ‖uk − u‖ ≤ δ(ǫ), where the function δ(ǫ) isknown and computable.


50-60’: creation of numerical methods for PDEs.

”A posteriori ideas” are mentioned only in a few publications.

W. Prager and J. L. Synge, 1947.R. D. Richtmyer, 1950.M. Slobodetskii, 1952.S. Mikhlin, 1962-64.


One of the earliest a posteriori methods has the origin in the paper byPrager and Synge, 1947 (and later publications of Synge).

In modern terminology, the main tool they have used is orthogonal Helmholz

decomposition of the ”energy” space.

Figure: W. Prager and J. L. Synge


.


Simplest case: Orthogonal decomposition of L2.

L2(Ω,Rd) = H10 (Ω)

⊕S(Ω).


∆u + f = 0, in Ω, (1)

u = 0, on ∂Ω. (2)

∫

Ωq0 · ∇wdx = 0 ∀w ∈ H1

0 (Ω), q0 ∈ S(Ω).

Let

q ∈ Qf :=

η :

∫

Ωη · ∇wdx =

∫

Ωfw dx ∀w ∈ H1

0 (Ω)

.

The set Qf contains vector–valued functions that satisfy (in a generalizedsense) the relation divq + f = 0.


Since ∇u − q ∈ Q0, we have the orthogonality relation

∫

Ω

∇(u − v) · (∇u − q) dx = 0,

which implies the estimate

‖∇(u − v)‖2 + ‖∇u − q‖2 = ‖∇v − q‖2

and the error bound

‖∇(u − v)‖ = infq∈Qf

‖∇v − q‖.


This method can be easily extended to other linear elliptic PDE’s(elliptic systems) of the divergent form.

Since 47’, the Prager–Synge estimate has been several times rediscoveredand represented in different notation for elliptic BVPs

and elliptic systems (such as, e.g., linear elasticity).


Mikhlin’s identity for quadratic functionals (1962).Let

J(u) := infv∈

H1(Ω)

J(v), J(v) :=1

2‖∇v‖2 −

∫

Ωfvdx .

Then

1

2‖∇(u − v)‖2 = J(v)− J(u),

Proof.

J(v)− J(u) =1

2(‖∇v‖2 − ‖∇u‖2)−

∫

Ωf (u − v) =

=1

2‖∇v − u‖2 +

∫

Ω(∇u · ∇(u − v)− f (u − v))dx .


In 60-70’, mechanical ”complementary energy” principles generated anelegant mathematical conception ”duality theory”.

J(u) = I ∗(p) := supq∈Qf

−1

2‖q‖2

,

We transform the Mikhlin estimate as follows

J(v)− J(u) ≤∫

Ω

(1

2‖∇v‖2 +

1

2‖q‖2 − fv

)dx .

Since q ∈ Qf , we arrive at the same estimate:

‖∇(u − v)‖ = infq∈Qf

‖∇v − q‖.

These observations have generated the orthogonal projection method (H.Weil, S. Zaremba, M. Vishik (40-47’)) and similar methods.

In all these methods getting a guaranteed upper bound amounts solvingthe dual problem on the space Qf , which involves differential equation(s).


70’- early 80’. Big progress in approximation theory and computer simulationmethods.

First attempts to solve numerically highly nonlinear problems.

Necessity of a posteriori methods is understood theoretically, but in computations

they are not used (except some rare cases).

Main ”a posteriori” publications:A. Ostrowski, 1972 (iteration schemes).I. Babuska and W.C. Rheinboldt, 1978 (residual method for FEM).H. Gajewski, K. Groger, K. Zacharias, 1974 (variational methods).P. Mosolov and V. Myasnikov, 1976-80 (variational methods).K. Babenko and M. Vasiliev, 1983 (computational proofs).


General a posteriori estimates for iteration methods

Consider a Banach space (X , d) and a continuous operator

T : X → X , x⊙ = Tx⊙ .

Approximations of a fixed point are usually constructed by the iterationsequence

xi = Txi−1 i = 1, 2, ... . (3)

Let T : X → X be q-contractive on S ⊂ X , i.e., there exists a positivereal number q such that the inequality

d(Tx , Ty) ≤ q d(x , y)

holds for any elements x and y of the set S .


.

Figure: A. Ostrowski

A. Ostrowski. Les estimations des erreurs a posteriori dans les procedesiteratifs, C.R. Acad.Sci. Paris Ser. A–B, 275(1972), A275-A278.


Theorem

Let xj∞j=0 be a sequence obtained by the iteration process contractivewith a mapping T . Then, for any xj , j > 1, the following estimate holds:

M j⊖ :=

1

1+qd(xj+1, xj) ≤ ej ≤ M j

⊕ :=q

1−qd(xj , xj−1).

Modern interpretations: H. Zeidler Nonlinear functional analysis and itsapplications. I. Fixed-point theorems. Springer, 1986,


Explicit residual method

It is commonly accepted that the origin of this approach is in works of I.Babuska and W.C. Rheinboldt, 1978-80.

This method is based on special type interpolation estimates derived byPh. Clement (1975).

In the last decades thousands of papers using this approach has beenpublished.

Detailed description can be found in: R. Verfurth, Teubner, 1996.


Starting from mid 80’, numerical analysis of PDEs is based on the Adaptive

Modeling conception, which cannot be realized without indication of a

posteriori errors.

Indicators of approximation errors have different origins and forms:

hierarchical methods;

weighted sums of local residuals and interelement jumps;

comparison with post processed solutions;

dual-weighted residual method for goal-oriented error estimates.

I. Babuska, P. Clement, R. Bank,T. Oden, R. Nochetto,O. C. Zienkiewicz and J. Z. Zhu,R. Rannacher, C. Johnson, E. Suli, C. Carstensen, R. Hoppe, W. Dorfler,W. Wendland, C. Schwab, R. Verfurth, M. Ainsworth, T. Stroboulis,.........................


Hierarchically based error indicators

Use the same idea as the Runge’s rule, but construct Vh and Vhrefin a

special form.

Vhref= Vh

⊕Wh and EIRunge =||| uh − uhref

|||.O.C. Zienkiewicz, D. W. Kelly, J. Gago, I Babuska (1982),A. Weiser (1981), R. Bank and K. Smith (1993).Since

a(u − uhref, uhref

− uh) = (f , uhref− uh)− (f , uhref

) + (f , uh) = 0,

we have

||| u − uh |||2=||| u − uhref|||2 + ||| uh − uhref

|||2 . (4)

Further analysis is based on the saturation assumption (if Wh isconstructed by higher order approximations, then usually λ ∼ hq)

||| u − uhref|||≤ λ ||| u − uh |||, λ ≤ 1

We obtain:

(1− λ2) ||| u − uh |||2=||| EIRunge(uh) |||2≤||| u − uh |||2 .S. Repin Zurich Summer School, 2012 27

A posteriori indicators based on post–processing

Post–processing of approximate solutions is a numerical procedureintended to modify already computed solution in such a way that thepost–processed function would fit some a priori known properties muchbetter than the original one.

Let u ∈ V + ⊂ V and Π(v) : Vh → V + be a mapping generated by thepost–processing operator Π.

v

u V +

V.

..(v)ΠΠ


General form of these indicators:

EIΠ(vh) := Φ(vh − Π(vh)),

where Φ is a certain measure (e.g., some suitable norm).Π is a ”projector” onto V + (e.g., averaging ‖∇vh − Gh∇uh‖).

Superconvergence:J. Bramble and A. Schatz. Math. Comp. 1977.M. Zlamal. Math. Comp. 1978.L. Oganesjan and L. Ruchovets. USSR Comput. Math. Math. Phys.1969.Efficient error indicators based on averaging:O. C. Zienkiewicz and J. Z. Zhu, I. Babuska and R. Rodriguez,C. Carstensen, S. A. Funken, S. Bartels, J. Wang,R. Lazarov, G. Carey, R. Ewing...........................................


Error indicators using adjoint problems

Usually these indicators are used to evaluate ℓ(u − uh), where ℓ is a given(goal–oriented) functional. Consider

Au = f ,

where A is a positive operator and f is a given vector. Let v be an approximatesolution. Let uℓ by the solution of adjoint problem

A⋆uℓ = ℓ.

Then,

ℓ · (u − v) = A⋆uℓ · u − ℓ · v = f · uℓ − ℓ · v = (f − Av) · uℓ

We have

< ℓ, u − v >=< f − Av , uℓ > . (5)

and find the error with respect to a linear functional by the product of the

residual and the exact solution of the adjoint problem.S. Repin Zurich Summer School, 2012 30

Very often these type methods are called Dual Weighted Residual (DWR)(initiated in the works of R. Becker, C. Johnson, R. Rannacher and others).A more detailed exposition of these works can be found inW. Bangerth and R. Rannacher. Adaptive finite element methods fordifferential equations. Birkhauser, Berlin, 2003.R. Becker and R. Rannacher. A feed–back approach to error control infinite element methods: Basic approach and examples, East–West J.Numer. Math., 4(1996), 237-264.Concerning error estimation in goal–oriented quantities we refer, e.g., toJ. T. Oden, S. Prudhomme. Goal-oriented error estimation and adaptivityfor the finite element method, Comput. Math. Appl., 41, 735-756, 2001.


Further development of the theory was focused on getting estimates,which satisfy the following (practical) conditions:

A For a concrete solution the estimate must give a guaranteed andrealistic estimate of the error.

B It must be fully computable, CPU time required for the computationshould be taken into account as a substantial parameter.

C It must be ”universal”, i.e., applicable for ANY approximation;

D It must not attract extra regularity or other properties of the exactsolution.

In mid 90’ these studies have formed a new line in numerical analysis,which can be named Fully Reliable Mathematical Modeling


Estimates of Deviations from Exact Solutions.A different look at the problem.

Quantitative analysis of mathematical models requires solving the followinggeneral mathematical problem: Let u be an exact (generalized) solution ofa problem and Ok be a sequence of neighborhoods. Let v be a functionfrom the energy space. We need a method able to detect arbitrary closeneighborhoods such that O1 ⊂ O2 and v ∈ O2 but v 6∈ O1.

u

v

O1

O2


In this conception, finding computable estimates of the distance betweenexact solution of a PDE and a function in the corresponding energy spaceis a necessary part of analysis, which must be done together with existenceand energy (stability) estimates before introducing approximationspaces (Vh, Qh, etc.) and particular numerical methods (e.g., FEM,FD, FV,...).

Such estimates must be derived by purely functional methods and mustnot use such properties as, e.g., ”superconvergence”, ”Galerkinorthogonality”, which arise in a pair V ↔ Vh associated with a particularnumerical method.


Sources

Mathematical techniques used to derive such type estimates rely onmethods of functional analysis used in PDE theory. They are

A. Fixed point theorems.

B. Maximum principle and theory of super- and sub-solutions.

C. Duality theory of convex variational problems.

D. Transformations of integral identities that define generalized(weak) solutions.


B. Maximum principle and sub/supersolutions

Pointwise estimates of approximation errors can be derived with the helpof known estimates for partial differential equations, which follow from themaximum principle (e.g., see D. Gilbarg and N. S. Trudinger). In thesimplest case, it reads as follows:

Theorem

Let A be a uniformly elliptic operator of the second order, which is definedin a bounded domain Ω with Lipschitz boundary Γ. Assume thatu+ ∈ C 2(Ω)

⋂C 0(Ω) and

Au+ ≥ 0. (6)

Then, the function u+ attains its maximum on Γ, i.e.,

supΩ

u+ = supΓ

u+. (7)


This principle holds for many elliptic operators. In particular, it holds forthe operator

Av := divA∇v + b · ∇v + cv

provided that c ≤ 0, the coefficients are bounded, and |b| is small withrespect to the ellipticity constant c1 (so that the ellipticity condition issatisfied).The function u+ in Theorem 2 is called a sub-solution associated with A.A function u− ∈ C 2(Ω)

⋂C 0(Ω) that satisfies the condition Au− ≤ 0 is

called a super-solution.If A is the operator ∆, then sub- and super-solutions are presented by sub-and super-harmonic functions, respectively.


Consider the problem Au = f in Ω with homogeneous Dirichlet boundaryconditions. Assume that we have an approximate solutionu ∈ C (Ω) ∩ C 2(Ω) that satisfies the condition

Au ≥ f , in Ω.

Then A(u − u) ≥ 0 and by the maximum principle we obtain a pointwiseestimate (earliest versions in L. Collatz, Funktionanalysis und numerischemathematik, Springer,1964) :

supΩ

(u − u) ≤ supΓ

(u − u) = supΓ

u.

If u satisfies the relation Au ≤ f , then we find that infΩ

(u − u) ≥ infΓ

u.

Main difficulty: sub/super subsolutions are related to strong/classical typesolutions of differential equations, which are not easy to construct.


Part II.

Guaranteed and computable bounds of energy errornorms.

Linear PDE’s.


Why we need computable estimates of deviations from exact solutions?

Mathematical Model:Hookes law?Small deformations?"Dead" load F?

Model ParametersYoung and Poisson’s constants?

Exact geometry?

F

Exact F and U?

Purely isotropic media?

U


Approximation

We have u_h with #100000. How the errors are balanced?

?

Model

Data uncertainty


Estimates we discuss are valid for analysis of all these errors.They yield

A posteriori estimates of approximation errors for conforming andnonconforming approximations of all types if they are applied toapproximations;

Modeling errors if they are applied to exact solutions of simplifiedmodels;

Errors caused by incomplete information on the data, if we studysensitivity of majorants with respect to the data.


Linear

Nonvariationalproblems

Variational

method

Transformationof

integral identities

Orthogonaldecomposition

Linear growth energy

Derivation methods

96’−−97’

01’−−03’

09’−−10’


Transformation of integral identities

Integral Identity

(AΛu,Λw) = (f,w) w ∈ V e.g., Λ = ∇

is the source for:

Energy estimates of u (set w = u) ;

Local estimates of u (set w = φu);

Regularity estimates of u (set w = u,i or w = φu,i);

estimates of deviations from u (set w = u− v).


Nonvariational method in the simplest case

Consider ∆u + f = 0 with u = 0 on Γ. We have∫

Ω∇(u− v)∇wdx =

∫

Ω(fw −∇v · ∇w)dx

Main idea is to split the right hand side by means of the integration by partsidentity associated with the differential operator of a BVP.

In the case considered it is∫

Ω

(divyw +∇w · y)dx = 0 ∀w ∈ V0, y ∈ H(Ω,div).


We have

∫

Ω(∇v · ∇w − fw)dx =

∫

Ω(∇v · ∇w − fw − (divyw +∇w · y))dx

∫

Ω((∇v − y) · ∇w − (f + divy)w)dx ≤

‖∇v − y‖‖∇w‖+ ‖f + divy‖‖w‖ ≤≤ (‖∇v − y‖+ CΩ‖f + divy‖)‖∇w‖.

Set w = u− v.∫

Ω|∇(u− v)|2dx ≤ (‖∇v − y‖+ CΩ‖f + divy‖)‖∇(u− v)‖.

Thus, we find that

‖∇(u− v)‖ ≤ ‖∇v − y‖+ CΩ‖f + divy‖.


Meaning and properties

For the problem

∆u + f = 0, u = 0 on ∂Ω

we have

‖∇(u− v)‖ ≤ ‖∇v − y‖+ CFΩ‖divy + f‖, v ∈ V0, y ∈ H(Ω,div)

Two terms in the right–hand side have a clear sense:they s present measures of the errors in two physical relations

p = ∇u, divp + f = 0 inΩ

that jointly form the equation.


If set v = 0 and y = 0, we obtain the energy estimate for the generalizedsolution

‖∇u‖ ≤ CΩ‖f‖.

Therefore, no constant less than CFΩ can be stated in the second term.The estimate has no gap. If set y = ∇u, than the inequality holds as theequality.

Thus, the estimate is exact in the sense that the multipliers of both terms cannot

be taken smaller and in the set of admissible y there exists a function that

makes the inequality hold as equality.


BVP ∆u + f = 0 has another variational formulation

infv∈V0,

β>0,

y∈H(Ω,div),

M⊕( v, y,β,CFΩ, f)

Minimum of this functional is zero;

it is attained if and only if v = u and y = A∇u !;

M⊕ contains only one global constant CFΩ, which is problemindependent;


In principle, one can select certain sequences of subspaces Vhk ∈ V0 andYhk ∈ H(Ω,div) and minimize the Error Majorant with respect to thesesubspaces

infv∈Vhk ,

β>0,

y∈Yhk,

M⊕( v, y,β,CFΩ, f)

If the subspaces are limit dense, then we obtain a sequence of approximatesolutions (vk, yk) and the sequence of numbers

γk := infβ>0

M⊕( vk, yk,β,CFΩ, f)→ 0


Generalizations and modifications

divA∇u + f = 0,

c1|ξ|2 ≤ Aξ · ξ ≤ c2|ξ|2.

||| y ||| 2 =

∫

ΩAy · y dx , ||| y ||| 2∗ =

∫

ΩA−1y · y dx

With the paradigm of this problem we present a unified scale of functionaltype error majorants.


Majorant M0⊕: General Error Majorant

We impose minimal requirements on y , namely, y ∈ Hdiv(Ω).

||| ∇(u − v) |||A≤ 1 ||| A∇v − y |||A∗ + C ‖divy + f‖L2(Ω)

Here v ∈ V0 + u0,C : ‖w‖Ω ≤ C |||∇w |||Ω, C ≤ CFΩ

c1

On y ∈ H(Ω,div) NO SMALLER WEIGHTS CAN BE USED!

Majorant vanishes if and only if

y = A∇v ,

divy + f = 0.


Majorant M1⊕ : Error Majorant with the global Poincare constant

We reduce the space for y by appending the condition

y ∈ H(Ω,div) +

∫

Γ

y · n ds =

∫

Ωfdx

.

Then |divy + f |Ω = 0 and

||| ∇(u − v) |||A≤||| A∇v − y |||A∗ +C‖divy + f ‖L2(Ω)

Here C : ‖w‖Ω ≤ C |||∇w |||Ω, ∀w ∈ H1(Ω), |w | = 0.It is easy to see that C ≤ CPΩ

c1.

Advantage: if Ω is convex or star-shaped, we can evaluate the constant.Disadvantage: extra requirement on y .


Majorant MN⊕ . Error Majorant with N local Poincare constants

Decompose Ω into a collection of Ωi , i = 1, 2, ...,N and reduce the spacefor y

∫

∂Ωi

y · nds =

∫

Ωi

fdx

We will have an upper bound

||| ∇(u − v) |||A≤||| A∇v − y |||A∗ +

(N∑

i=1

CPΩ2i‖divy + f ‖2L2(Ωi )

)1/2

.

Advantage: we can use local Poincare estimates.Disadvantage: N additional requirements on y .


Majorant M∞⊕ . Error Majorant with total equilibration

Reduce the space for y by assuming that the balance equation holds not ina ”mean” sense, but almost everywhere, i.e.,

divy + f = 0, in Ω.

This is the MAXIMAL restriction which leads to PS estimate.

Note that the general Majorant

‖∇(u − v)‖ ≤ ‖∇v − y‖+ C‖divy + f ‖,

holds for any C ≥ CFΩ.Thus, the PS estimate is a very special (limit) form of it: C = +∞.Then we have

‖∇(u− v)‖ ≤ ‖∇v − q‖, ∀q ∈ Qf := divq + f = 0


From M0⊕ to M∞

⊕

Hdiv Hdiv

Hdiv

+dvg y+f

+dvg y+f+dvg y+f

Ω

ΩΩ

1

2

ΩΩ

Ω Ω1 2

=0

=0=0

Error Majorant

dvg y+f=0

Ω

PS estimate

M M

M M

0 1

2

MΝ


Equilibrate or Not to Equilibrate ?

In ”academic” type problems, as, e.g., ∆u + f = 0 fully equilibrated fieldscan be indeed constructed. However in more complicated cases (e.g.,Oseen, convection diffusion, nonlinear terms) it is hardly possible.

Moreover, in many cases full (or almost full) equilibration leads tocomputational expenditures, which are not required:indeed if the first part of the Majorant dominates then further equilibrationcannot essentially improve the estimate.


CONCLUSION 1: Guaranteed and computable error bounds for ellipticproblems form a sequence, in whichthe PS estimateandthe functional Error Majorant are the two boundary points.For PS the space of admissible fluxes is minimal, for EM it is maximal.


CONCLUSION 2: Getting sharp and computable estimates for constantsin the estimates likeFriedrichs-Poincare,LBB,trace inequalitiesis an important problem in modern analysis.


Some known results:Scalar valued functions: L. Payne and H. Weinberger.d = diamΩ, convex domain

infc∈R

‖w − c‖L2 ≤d

π‖∇w‖L2

G. Acosta and R. Duran (2003). Convex domain Ω.

infc∈R

‖w − c‖L1 ≤d

2‖∇w‖L1

For vector fields we need

infη∈RD

‖w − η‖Lp≤ c‖ε(w)‖Lp

.

Case p = 1: M. Fuchs and S. Repin, Numer. Func. Anal. Optim. 2011.


How to use the Majorants in practice?

Consider first CLASSICAL CONFORMING FEM APPROXIMATIONS.

We have 3 basic ways to use the deviation estimate:(a) Direct (via flux averaging on the mesh Th);(b) One step delay (via flux averaging on the mesh href);(c) Minimization (minimization via y).


(a) Use recovered gradients

Let uh ∈ Vh, then

ph := ∇uh ∈ L2(Ω,Rd), ph 6∈ H(Ω,div).

Use an averaging operator Gh : L2(Ω,Rd)→ H(Ω,div) and have a

directly computable estimate

‖∇(u− uh)‖ ≤ ‖∇uh − Ghph‖+ CFΩ ‖divGhph + f‖


(b) Use recovered gradients from Thref

Let u1, u2, ..., uk , ... be a sequence of approximations on meshes Thk.

Compute pk := ∇uk , average it by Gk and for uk−1 use the estimate

‖u− uk−1‖ ≤ ‖∇uk−1−Gkpk‖+ CFΩ ‖divGkpk+f‖

This estimate gives a correct form of the Runge’s rule!


(c) Minimize M⊕ with respect to y .

Select a certain subspace Yτ in H(Ω,div).In general, Yτ may be constructed on another mesh Tτ and with help ofdifferent trial functions.Then

‖∇(u− uh)‖ ≤ infyh∈Yh

‖∇uh−yh‖+ CFΩ ‖divyh +f‖

The wider Yh ⊂ H(Ω,div) the sharper is the upper bound.

This method results in a reconstruction of the flux, which is much better than

any other reconstruction on Th.


From the technical point of view it is better to use a quadratic form of themajorant, namely

‖∇(u − uh)‖2 ≤ infyh∈Yh

(1+β)‖∇uh−yh‖+ CFΩ

1 + β

β‖divyh+f ‖2

Here, the positive parameter β can be also used to minimize theright–hand side.

If computations are performed by classical FEM, then flux is usuallyinaccurate, so that we start from Gh∇uh and minimize the majorant (e.g.,by direct method) and improve the flux unless the second (reliability)becomes smaller than the first one.


Complicated/Oscillating f

Ω

Ω

Ω

1

m

Ωk

Figure: Decomposition of Ω into subdomains.

e(f ) :=1

c1

(∑

i

C 2P(Ωi )‖f − fi‖2Ωi

)1/2

e(f ) represents the error generated by simplification of f (fi = |f |ΩI).


Indeed, let u be the exact solution of the problem with f . Then,

∫

ΩA∇(u − u) · ∇w dx =

∫

Ω(f − f )w dx

Hence,

||| u − u |||2=N∑

i=1

∫

Ωi

(f − fi )(u − u) dx .

Since∫Ωi

(f − fi ) dx = 0 ∀ i = 1, 2, ...,N we find that

N∑

i=1

∫

Ωi

(f − fi )(u − u) dx =∑

Ωi

∫

Ωi

(f − fi )(u − u − ci )dx ,

where ci are arbitrary constants.


Set ci = |u − u|Ωi

‖u − u − ci‖Ωi≤ CP(Ωi )‖∇(u − u)‖Ωi

≤ CP(Ωi )

c1‖∇(u − u)‖A, Ωi

.

Then,

||| u − u |||2≤ 1

c1

N∑

i=1

CP(Ωi )‖f − fi‖Ωi‖∇(u − v)‖A, Ωi

≤

1

c1

(N∑

i=1

C 2P(Ωi )‖f − fi‖2Ωi

)1/2

‖∇(u − v)‖A

and, therefore,||| u − u ||| ≤ e(f ). (8)

e(f ) as a simple modeling error generated by simplification of f .Depending on the desired accuracy ǫ, we may have two different situations.


A. If ǫ≫ e(f ), then the boundary value problem with f can be efficientlyused instead of the original one. In the context of finite elementapproximations the value of e(f ) on a particular mesh is easy to compute.Since this quantity is proportional to h, we can always detect when themesh is so fine that we can ignore local oscillations of f within theaccepted tolerance level.

B. if e(f ) is of the order ǫ or larger, then first of all, we need to find asuitable mesh, which makes the simplification error sufficiently small(e.g. 0.1 ∗ ǫ). This is a cheap procedure, WHICH DOES NOT REQUIRESOLVING PDE.

Similar combined modeling/discretization adaptive may be applied in morecomplicated situations, e.g., if we simplify A.


Two–sided error bounds in combined norms

Which error measure is physically more correct?

If the pair (u, p) is viewed as the solution, then errors should be computedin terms of a different (primal–dual) norm which is the norm of theproduct space W := V × Q∗:

‖(v , y)‖W := ‖∇v‖+ ‖y‖+ ‖divy‖.

InS. R., S. Sauter, A. Smolianski. SIAM J. Num. Anal. 5, 2007error bounds has been derived for combined/mixed approximations.


Mixed method on H1 × L2

∫

Ω

(A−1p −∇u

)· q dx = 0 ∀q ∈ Q = L2(Ω) ,

∫

Ω

p · ∇w dx − ℓ(w) = 0 ∀w ∈ V0 = H10 (Ω) .

In PMM=Classical FEM, we need to find a pair of functions( uh, ph) ∈ (V0h + u0)× Qh where Qh ⊂ L2(Ω) such that

∫

Ω

(A−1 ph −∇uh

)· qh dx = 0∀ qh ∈ Qh , (9)

∫

Ω

ph · ∇wh dx − ℓ(wh) = 0∀wh ∈ V0h . (10)

In the simplest case, uh is constructed by means of piecewise affine (C 0)elements and ph by piecewise constant functions.Equation of balance is satisfied in H−1 only!


Dual mixed method on L2 × H(Ω,div):

∫Ω

(A−1p · q + (divq)u

)dx = 0 ∀q ∈ H(Ω,div) ,

∫Ω

(divp + f )v dx = 0 ∀v ∈ V = L2 .

A discrete analog: find (uh, ph) ∈ Vh × Qh such that

∫

Ω

(A−1ph · qh+uhdivqh

)dx = 0 ∀qh ∈ Qh,

∫

Ω

(divph + f )vhdx = 0 ∀vh ∈ Vh.

Approximations of the primal and dual variables must satisfy a discreteanalog of the ”infsup” condition and are produced with the help ofRaviart–Thomas (RT) elements.Mean values |divph + f |T = 0!Similar properties in FV approximations.


From the viewpoint of physics/engineering fluxes/stresses are mostinteresting. It is important to work with a method, which generates asequence of fluxes, which are elementwise balanced and converge inH(Ω,div).


Theorem

Majorant is equivalent to the error norm

‖(u − v , p − y)‖W := ‖∇(u − v)‖︸︷︷︸+ ‖p − y‖+ ‖div(p − y)‖︸︷︷︸ .

E (v) E (y)

Namely,

c⊖M∆(v, y) ≤ ‖(u − v, p − y)‖W ≤ c⊕M∆(v, y),

where c⊖ = 1max1,CFΩ and c⊕ = max

3, 2 + 1

CFΩ

.


Hence, the efficiency index of the majorant is estimated from above asfollows:

Ieff ≤c⊕

c⊖

= max1,CFΩ max3, 2 + 1

CFΩ

.

Thus, M∆ is an efficient and reliable measure of the error in thecombined norm ‖(u − v , p − y)‖W .

Since the Majorant is equivalent to the total error it is natural to use it as a basisfor the corresponding mesh adaptation, which takes into account BOTH

components of the solution pair.

Two parts ‖∇uh − qh‖ and CFΩ‖divqh + f ‖ can be used to balance two

components of the error.


Remark

If the error is measured in a different (but equivalent) norm

‖(v , y)‖(1)W := ‖∇v‖+ ‖y‖+ CFΩ‖divy‖,

then

M∆(v, y) ≤ ‖(u − v, p − y)‖(1)W ≤ 3M∆(v, y).

Thus, the majorant is equivalent to such a norm and the respectiveefficiency index does not depend on CFΩ.


Reaction–diffusion problem

−divp + 2u = f in Ω,

p = A∇u in Ω,

u = u0 on Γ.

Several different majorants can be derived. The simplest one:

|[u − v ]|2 ≤||| A∇v − y |||2∗ +‖1(f − 2v + divy)‖2, (11)

|[w ]|2 :=||| ∇w |||2 +

∫

Ω|w |2dx , ||| y |||2= (Ay , y).

However, this estimate has an essential drawback: if is small then thesecond term has a large multiplier that makes the whole estimate sensitivewith respect to the residual

r(v , y) := f − 2v + divy .


||| ∇(u − v) ||| ≤||| A∇v − y |||2∗ +CΩ,A‖(f − 2v + divy)‖. (12)

‖w‖ ≤ CΩ,A ||| ∇w ||| ∀w ∈ V0

This estimate has a different drawback: we cannot prove that thisestimate has no gap between left and right hand sides.


In S. R. and S. Sauter, Comptes Rendus, 2006, it was derived an upperbound (hybrid of (11) and (12), which is free of both drawbacks:

|[u − v |]2 ≤

≤(CΩ,A‖(1− α)(f − 2v + divy)‖+ ||| A∇v − y |||∗

)2+

+ ‖α

(f − 2v + divy)‖2. (13)

It is easy to see that above simple majorants are particular cases of (13).Here α(x) ∈ L∞(Ω) with values in [0, 1].


Minimum of the right–hand side with respect α is attained at

α =C 2

Ω,A2(1 + β)

C 2Ω,A

2(1 + β) + 1∈ [0, 1)

which leads to the estimate

|[u − v ]|2 ≤∫

Ω

C 2Ω,A(1 + β)

C 2Ω,A

2(1 + β) + 1r2(v , y)dx+

+1 + β

β||| A∇v − y |||2∗:= MRD(v , y , β).


Since

MRD(v , p, β) =

∫

Ω

( C 2Ω,A(1 + β)

C 2Ω,A

2(1 + β) + 14(v − u)2+

+1 + β

β|∇(v − u)|2

)dx

we observe that

MRD(v , p, β)→||| u − v |||2 as β → +∞.

Therefore, the estimate has no ”gap”. At the same time the structure ofthe first term is such that it is not sensitive with respect to small values of.


Convection–diffusion problem

−divA∇u + a · ∇u = f in Ω,

u = 0 on Γ.

Here a is a given vector–valued function satisfying the conditions

a ∈ L∞(Ω,Rd), diva ∈ L∞(Ω), diva ≤ 0.


In this case, the estimates are derived for the norm

|[u − v ]|2 :=||| ∇(u − v) |||2 +δ2‖u − v‖2,

where δ2 = −12diva ≥ 0.

First estimate:

|[u − v ]| ≤||| y − A∇v |||∗ +CΩ‖f − a · ∇v + divy‖.

If δ(x) > 0, then we have another an upper bound:

|[u − v ]|2 ≤ ‖1δ (f − a · ∇v + divy)‖2+ ||| y − A∇v |||2∗ .


Elliptic systems.

Linear elasticity.

σ = Lε(u) in Ω,

Divσ + f = 0 in Ω,

u = u0 on Γ1, measΓ1 > 0,

σn = F on Γ2.

Here f and F are given forces and L = Lijkm is the tensor of elasticityconstants, which is subject to the conditions

c21 |ε|2 ≤ Lε : ε ≤ c2

2 |ε|2, ∀ε ∈Md×ds ,

Lijkm = Ljikm = Lkmij , i , j , k ,m = 1, . . . , d ,

Lijkm ∈ L∞(Ω).


Theorem (96’)

For any v ∈ V0 + u0, β > 0,τ ∈ H(Ω,Div) :=

τ ∈ H(Ω,Div); τn ∈ L2(Γ2,R

d)

we have

1

2

∫

ΩL ε(v − u) : ε(v − u) dx ≤

≤ 1+β

2

∫

Ω(ε(v)−L

−1τ) : (L ε(v)−τ) dx+

+1+β

2βc1C 2

Ω(‖Div τ−f ‖2Ω+‖F +τn‖2Γ2),

where ‖w‖2Ω + ‖w‖2Γ2≤ CΩ‖ε(w)‖2Ω

Estimate has a clear meaning: distance to the exact solution is controlledby error in the Hooke’s law τ = Lε(v), equilibrium equation andNeumann boundary condition.


Maxwell type equation

curlµ−1curl u + κ2u = j in Ω, (14)

j is a given current density,µ is the permeability of a media.

ν × u = 0 on ∂Ω. (15)

Here basic spaces are H(Ω, curl)

‖w‖curl :=(‖w‖2Ω + ‖curlw‖2Ω

)1/2.

and

V0(Ω) := w ∈ H(Ω, curl) |w × ν = 0 on ∂Ω


Approximation methods for the Maxwell’s equation were investigated bymany authors (e.g., (Monk, Haase, Kuhn, Langer, Hiptmair...)).A posteriori estimates were obtained e.g. in the works R. Beck, R.Hiptmair, R. Hoppe, B. Wohlmuth. and by Braess and Schoberl with thehelp of equilibrated approach.


Estimate for Maxwell problem

For any v ∈ V0(Ω).

||| u − v |||2≤

≤ M2Max(v , y) := ‖1

κ(j − κ2v − curl y)‖2 + ‖µ1/2(y − µ−1curl v)‖2,

where y ∈ H(Ω, curl).


It is easy to see that

infv∈V0,

y∈H(Ω, curl)

MMax(v , y) = 0

and the exact lower bound is attained if and only if

curl y + κ2v = j a.e. in Ω (16)

y = µ−1curl v a.e. in Ω. (17)

By assumption v × ν = 0 on ∂Ω. Therefore, (16) and (17) mean that vcoincides with the exact solution u and y with µ−1curlu.


Estimate has no gap

infy∈V0

M2(v , y) ≤ M2(v , µ−1curlu) =

= ‖1κ

(j − κ2v − curlµ−1curlu)‖2 + ‖µ−1/2curl (u − v)‖2 =

= ‖κ(u − v)‖2 + ‖µ−1/2curl (u − v)‖2 =||| u − v |||2 .

Therefore, the upper bound is sharp.


Viscous fluids

Models considered

Stokes problem;

Generalized Stokes problem;

Oseen problem;

Stokes and Oseen problems with polymerization;

Non-Newtonian fluids;

Navier Stokes problem.


For the above models numerical methods were developed in numerouspublications, see, e.g.,

F. Brezzi and M. Fortin. Springer, 1991.V. Girault and P. A. Raviart. Springer, 1986M. Feistauer. Longman, Harlow, 1993.M. Gunzburger. Academic press. 1989.R. Glowinski. North–Holland, 2003.R. Rannacher. Birkhauser, 2000.R. Temam. North-Holland, 1979.A posteriori methods for FEM in CFD can be found e.g. in publications of:M. Ainsworth, C. Bernardi, R. E. Bank, M. Boman,L. Demkowicz, P. Devloo, C. Johnson, J. T. Oden,R. Rannacher, T. Strouboulis, R. Verfurth, B. D. Welfert, W. Wu.........................................


Indicators

Usually automatic mesh adaptation is guided by some cell/elementindicators computed with help of approximate solution (uh, ph).Vorticity indicator:‖∇ × uh‖T.

Pressure–gradient indicator‖∇ph‖T.Energy–norm error indicator

‖div uh‖T + ‖ − ν∆uh + uh · ∇uy +∇ph + f‖T + h1/2‖j(uh, ph)‖∂T,

where j(uh, ph) are certain jumps on the edges.


Stokes Problem

A linearization of NS system known as Stokes model is:

ut − ν∆u+ = f −∇ p in Ω,

u(x, 0) = U(x),

u = u0 on ΓD

∇u · ν + pν = gN onΓN

Consider the stationary Stokes problem with Dirichlet boundary conditions

− ν∆u = f −∇p in Ω,

u = u0 on ∂Ω,

ν

∫

Ω∇u : ∇wdx =

∫

Ωf ·wdx ∀w ∈

J

12(Ω).


Get a computable upper bound of the velocity error in 4 steps:

Step 1. Let v ∈J 1

2(Ω) + u0 be an approximation of u

ν

∫

Ω∇(u− v) : ∇wdx =

∫

Ω(f ·w −ν∇ v : w)dx ∀w ∈ V0.

Step 2. Let τ be any in H(Div,Ω). Then

ν

∫

Ω∇(u− v) : ∇wdx =

∫

Ω(f ·w −ν∇ v : ∇w)dx+

+

∫

Ω(Divτ · w+τ : ∇w)dx =

=

∫

Ω((f + Divτ ) · w + (τ −ν∇ v) : ∇w)dx


Step 3. Set w = u − v and use Helmholtz theorem on orthogonaldecomposition

ν

∫

Ω∇(u− v) : ∇wdx = ν‖∇(u− v)‖2 ≤

=

∫

Ω((f + Divτ +∇q) · ( u − v) + (τ −ν∇ v) : ∇( u − v))dx ,

where q ∈L.

Step 4. By Holder and Friederichs inequalities we obtain

ν‖∇(u− v)‖ ≤ ‖τ −ν∇ v‖+ CF‖f + Divτ +∇q‖.

S. R. (2002), J. Math. Sci, (New York)


How to use it?

First way is to substitute v , q and τ , compute

‖τ −ν∇ v‖ and ‖f + Divτ +∇q‖

and define a bound.Another way: use post–processed τ as the ”initial guess” for the stressand minimize the majorant (e.g., by a direct minimization method) withrespect to τ . In this case, it is better to transform the estimate in aquadratic form:

ν2‖∇(u− v)‖2≤(1+β)‖ν∇(v)−τ‖2+1 + β

βC2

Ω‖divτ+f−∇q‖2

Here β is any positive real number.


A form valid for q ∈ L2

We may also rewrite this estimate in a different form, which is valid for awider set of pressure functions q.

Set τ = η + qI, where I is the unit tensor, q ∈L, and η ∈ H(Div,Ω).

ν ‖∇(u−v)‖≤‖ν∇v−η−qI‖+CFΩ‖divη+f‖.


Estimates applicable also for non–solenoidal approximations has beenderived with the help of

Lemma. For any g ∈L there exists a function u ∈ V0 satisfying the

relation div u = g and the condition

‖∇u‖ ≤ κΩ‖g‖, (18)

where κΩ is a positive constant that depends on Ω.Proof in 2D I. Babuska and A.K. Aziz,general case O. Ladyzhenskaya and V. Solonnikov,also can be viewed as a form of Closed Range Lemma in the functionalanalysis (see, e.g., F. Brezzi and M. Fortin).


Lemma implies a projection type inequality

infv∈

J12(Ω)

‖∇(v − v)‖ ≤ κΩ‖divv‖. (19)

and the estimate

ν‖∇( u−v)‖ ≤ ‖ν∇v−τ‖+CFΩ‖divτ+f −∇q‖+2νκΩ‖divv‖,

Three terms in the right–hand side of the estimate present three naturalparts of the error, namelyerrors in the constitutive (Newton) lawerror in the differential equationerror due to non-solenoidality of approximations.


Estimates for ‖ p − q‖ are also derived with the help of Lemma

Key idea: Since ( p − q) ∈L, we know that

∃w ∈ V0, divw = p − q, and ‖∇(w)‖ ≤ κ‖ p − q‖.

Hence, ‖ p − q‖2 =∫Ω divw( p − q) dx

1

2κ‖p− q‖ ≤ ‖ν∇v − τ − qI‖+ CFΩ‖Divτ + f‖+ νκ‖div v‖.

Estimate has the same principal terms as before and vanishes if and only if,

v = u, τ = σ, and p = q.

However, in this case, the dependence of the penalty multipliers on theconstant κ is stronger.


Oseen Problem: another linearization of the NS system

ut − ν∆u + Div(a⊗ u) = f −∇ p in Ω,

u(x, 0) = U(x),

diva = 0,

u = u0 on ΓD


Here a is a given vector–valued function.


Oseen equation. Guaranteed upper bound of the deviation from u

−ν∆u + div(a⊗ u) = f −∇p inΩ,

div u = 0 inΩ,

u = u0 on Γ.

Let v ∈J 1

2(Ω) and τ ∈ H(Ω,Div) and q ∈L ∩H1. Then,

ν‖∇(u− v)‖ ≤ ‖τ − ν∇v‖+ CFΩ‖f −∇q− div(a⊗ v) + divτ‖,


Generalized Stokes and Oseen models

In particular, such type problems arise from semidiscrete approximations ofNS equations, e.g.,:

uk − uk−1

∆t− ν∆uk + Div(uk−1 ⊗ uk−1) = f −∇pk in Ω,

divuk = 0

and

uk − uk−1

∆t− ν∆uk + Div(uk−1 ⊗ uk) = f −∇pk in Ω,

divuk = 0.


Semidiscrete NS approximations lead to generalized Stokes andOseen Problems

α u − ν∆ u+ = f −∇ p in Ω,

u = u0 on ΓD


α u − ν∆ u + Div(a⊗ u) = f −∇ p in Ω,

diva = 0,

u = u0 on ΓD



Another version of the generalized Stokes and Oseen problems is related tomodels of fluids with polymerization (J. Bonvin, M. Picasso, R. Stenberg),

Divσ + f = 0, in Ω,

div u = 0 in Ω,

σ = − pI + κ+ ν∇ u, in Ω,

u = 0 on ∂Ω,

where κ is a given tensor–valued function such that trκ = 0.A posteriori estimates for such a problem with mixed Dirichlet–Neumannconditions has been recently derived:S. R. and R. Stenberg, 2007.


Navier–Stokes equation

Certainly the most known model is the Navier–Stokes problem: find

u(x, t) ∈J1

2(Ω) and p(x , t) ∈L such that

ut − ν∆u + Div(u⊗ u) = f −∇p in Ω,

u(x, 0) = U(x),

u = u0 on ΓD

∇u · ν + pν = gN onΓN.

From the mathematical point of view NS is not completely understood.Existence of a unique solution in 3D is not yet proved even for (0,T ]×R

n

(this is one of the Millenium Prize Problems).It is known that for sufficiently regular solenoidal U(x) there exists a weakLeray-Hopf solution, i.e., a function

u ∈ L∞(0,T;L2(Rn)) ∩ L2(0,T;H1(Rn))


Stationary NS problem in Ω ⊂ R2

− ν∆u + Div(u⊗ u) = f −∇p in Ω,

u = u0 on Γ

ν‖∇(u− v)‖ ≤ ‖τ − ν∇v‖+

+CFΩ‖f −∇q−Divτ + Div(v ⊗ v)−∇q‖

if ν := ν − C4‖∇v‖ > 0, C4 is the constant of the embedding inequalityV0 → L4(Ω,Rd).


Flow problems in a rotating frame

ut − ν∆u + (u · ∇u) + ω × u = f −∇p in Ω,

where ω = |ω|iz is the rotation parameter.

A posteriori estimates for such models has been derived and studied byE. Gorshkova, A. Mahalov, P. Neittaanmaki, and S. R., 2006


GENERALIZATIONS

HB←− V0

Λ−→ U (Y,Y∗)m

HB∗−→ V∗

0Λ∗

←− U

Find p ∈ H and u ∈ V0 that satisfy the relation

(AΛ u, Λw) + 〈f − B∗p,w〉 = 0 ∀w ∈ V0,

where

V0 = KerB := v ∈ V0 | Bv = 0 .


Let the operator B possesses the following property: there exists aconstant κ such that for any g ∈ ImB := z ∈ H | ∃v ∈ V0 : Bv = zone can find ug ∈ V0 such that

Bug = g and ‖ug‖V ≤ κ‖g‖.

||| Λ(u− v) ||| 2√ν2κ‖Bv‖+ ||| AΛv − y |||∗ +1√ν1||| f + Λ

∗y − B∗q ||| .

||| y |||:= (Ay , y)1/2, ||| y |||∗:= (A−1y , y)1/2

〈Λ∗σ + f − B∗p,w〉 = 0 ∀w ∈ V0,σ = AΛu,Bv = 0.


Conjecture

Deviation estimates (error majorants) consist of the terms, which are aspenalties for the unconformity in each of the basic relations forming aBVP. Multipliers (factors) of these terms are defined by constants in theembedding inequalities for the spaces arising in the mathematicalformulation of a BVP.


Optimal control problems

Jmin := minξ∈K

J(ξ, uξ)

A(ξ)uξ = f(ξ), uξ ∈ V

Major difficulty is that the control ξ and state uξ functions are subject toa differential relation.In general, uξ is never known, so that the value of J(ξ,uξ) is impossible tocompute exactly.Assume that J satisfies the estimate

J(ξ, u + v) ≤ J(ξ, u) + Φ(‖v‖),

where Φ is a nonnegative function and M is the majorant for thedifferential problem.


We have (now v is viewed as an approximation of uξ)

Jmin ≤ J(ξ, uξ) ≤ J(ξ, v) + Φ(‖v − uξ‖) ≤ J(ξ, v) + Φ(M(v,D)),

This estimate holds on K×V and allows to transform the problem into an”unconstrained” form. Moreover, it is not difficult to show that

Jmin = minξ∈K,v∈V

J(ξ, v) + Φ(M(v,D)),

This idea was applied to the problem (A. Gaevskaya, R. Hoppe, and S. R.,2007).Given ψ ∈ L∞(Ω), yd ∈ L2(Ω), ud ∈ L2(Ω), f ∈ L2(Ω), and a ∈ R+,consider the distributed control problem

min J(y(u), u) :=1

2‖y − yd‖2 +

a

2‖u − ud‖2

over (y , u) ∈ H10 (Ω)× L2(Ω),

subject to −∆y = u + f a.e.inΩ,

over u ∈ K := v ∈ L2(Ω)|v ≤ ψa.e. inΩ


Publications

By the variational method majorants has been derived inS. R. A posteriori error estimation for nonlinear variational problems byduality theory, Zapiski Nauchn, Semin, Steklov Mathematical Institute inSt.-Petersburg, 243(1997), pp. 201-214.

See also S. R. A posteriori error estimates for variational problems withuniformly convex functionals, Mathematics of Computation, 69(230), pp.2000, 481-500.


By means of the orthogonal decomposition method error majorants hasbeen derived in

S. R., S. Sauter, A. Smolianskii.A posteriori error estimation for the Dirichlet problem with account of theerror in the approximation of boundary conditions,Computing, 70(2003), pp. 205-233. In this paper first estimates fornonconforming approximations (fictitious domain method) has beenderived and tested.Mixed methods:S. R. and A. Smolianskii. Functional-type a posteriori error estimates formixed finite element methods, Russian J. Numer. Anal. Math. Modelling20 (2005), pp. 365–382.S. R., S. Sauter, and A. Smolianskii. Two-Sided A Posteriori ErrorEstimates for Mixed Formulations of Elliptic Problems, SIAM J. Numer.Anal., 45(2007), pp. 928–945.


DG method: R. Lazarov, S. R., and S. Tomar, Numer. Meth. PDE, 2008.S. R. and S. Tomar. IMA J. Numer. Anal., 2010.

FV method: S. Coshes-Dohdt, S. Nicaise, and S. R. Mat. Anal. Nat.Phenom., 2009.

Analysis of indeterminacy errors: O. Mali and S. R., 2009-2011

Coupled elasto-porosity. J. Nordbotten, S. R., and J. Valdman. 2010.

Maxwell type problems: I.Anjam, O. Mali, A. Muzalevski, P.Neittaanmaki, and S. R. Rus. J. Numer Anal., 2009 D. Pauly and S. R.Zapiski Nauchn. Semin. POMI, 2010

Exterior problems: D. Pauly and S. R., 2010.


A systematic exposition of the variational (duality) method for derivingFunctional a Posteriori Estimates can be found in the book (Elsevier,2004):


Nonvariational method of derivation was introduced inS. R. Two-sided estimates of deviation from exact solutions of uniformlyelliptic equations, Proc. St. Petersburg Math. Society, IX(2001), pp.143–171.

Consequent exposition in deGruyter 2008


Part 3. Modeling errors.


Estimates of deviations are valid for ANY admissible function.Therefore, we can use them as a tool of deriving guaranteed boundsof modeling errors. We substitute the exact solution of a simplifiedmodel u into the deviation estimate derived for the original one.Then M(u,D) yields a guaranteed bound of the modeling error.


ACCURACY OF DIMENSION REDUCTION MODELS

(d)u u(d−k) u(d−k)

R

P(d)P

(d−k)P(d−k)

τ

τ

Figure: Dimension reduction and reconstruction


Let u and u are the solutions of the basic and simplified models,respectively.

In many cases, these two functions belong to different spaces, so tomeasure the difference between them we need a certainRECONSTRUCTION operator ℜ.

Now the problem is to estimate the difference

‖u−ℜu‖in terms of a suitable (e.g., energy norm).


Dimension reduction for diffusion problems

We consider the problem

Div(A∇u) + f = 0,

u = 0 on Γ0,

A∇u · ν⊕,⊖ = F⊕,⊖ on Γ⊕,⊖

in a plate type domain

Ω = x ∈ R3 | (x1, x2) ∈ Ω, d⊖ ≤ x3 ≤ d⊕.

Γ⊖ and Γ⊕ are lower and upper faces of a plate type body.Γ0 is a lateral surface.


x

x

x

1

2

3

2

3

x

x

-

+

00 ΓΓ Γ

Γ

x

x

1

2

Ω Γ


Problem (P): Find u ∈ V0 such that

∫

ΩA∇u · ∇w dx =

∫

Ωf w dx +

∫

Γ⊖

F⊖ w ds +

∫

Γ⊕

F⊕ w ds ∀w ∈ V0 .

(20)

Div τ =∂τ 1

∂x1+∂τ 2

∂x2+∂τ 3

∂x3, div τ =

∂τ 1

∂x1+∂τ 2

∂x2.

F⊖(x) := F⊖(x , d⊖(x)), F⊕(x) := F⊕(x , d⊕(x)) for any x ∈ Ω.

|||v ||| :=(∫

ΩA(x)∇v · ∇v dx

)1/2

∀v ∈ V0 . (21)


We assume thatd << diamΩ

and accept the hypothesis that

The exact solution is almost constant with respect to thex3-coordinate.

This gives rise to the so-called zero-order reduced model for the originalproblem (30). Discussion of hierarchy of reduced models of different orderscan be found inM. Vogelius and I. Babuska, Math. Comp. 1981 andI. Babuska and C. Schwab, SIAM J. N. Anal., 1996.For the case of a plate with plane parallel faces Γ⊖ and Γ⊕ (i.e. whend⊖ = −d0

2 , d⊕ = d02 , d0 = const > 0 is the plate thickness) and f = 0, it

was proved that

|||e||| ≤ C d1/20 (‖F⊖‖L2(Ω)

+ ‖F⊕‖L2(Ω)) as d0 → 0 .


We take the subspace

V0 := v ∈ V0 | ∃ v ∈ H10 (Ω) such that v(x) = v(x) x = (x , x3) ∈ Ω .

(22)

In this case, the reconstruction operator ℜ is very simple

Now, the energy-norm projection of u onto the subspace V0 yields thereduced problemProblem (P):Find u ∈ V0 such that

∫

ΩA∇u · ∇w dx =

∫

Ωf w dx +

∫

Γ⊖

F⊖ w ds +

∫

Γ⊕

F⊕ w ds ∀w ∈ V0. (23)


Computable estimates for the difference between u and u are derived andinvestigated inS. Repin, S. Sauter, A. Smolianski, SIAM J. Numer. Anal. 2004.They are based on the estimate of deviation from the exact solution oflinear elliptic problem

1

2

∫

ΩA∇(v − u) · ∇(v − u) dx ≤

1 + β

2

∫

Ω(∇v − A−1y) · (A∇v − y) dx +

1 + β

2β

c2Ω

c21

‖Div y − f ‖2

in which we set v = u and y = Ap∇u + τ ∗, τ ∗ = 0, 0, ψ is a correctionvector and Ap is the ”plane” part of A.

B := A−1 (B(x) = (bij(x))i ,j=1,3 , B = BT ) ,

Bp := (bij)i ,j=1,2 ,

b3 := b31 , b32T .


M21 =

∫

Ω

d (x)(BpAp − I)∇u · Ap∇u d x+

∫

Ω

(b33ψ(x)2 + 2(b3 · Ap∇u)ψ(x))dx

M22 =

∥∥∥∥∥f − f − F⊖

√1 + |∇d⊖|2 + F⊕

√1 + |∇d⊕|2

d− ∇d

d· Ap∇u +

∂ψ

∂x3

∥∥∥∥∥

2

L2(Ω)

M23 = ‖F⊖ − Ap∇u · ν⊖ − ψν⊖3‖2L2(Γ⊖) + ‖F⊕ − Ap∇u · ν⊕ − ψν⊕3‖2L2(Γ⊕)

General Estimate

|||u − u||| ≤ M1 + CΩ M2 + CΓ

√1 + C 2

Ω M3. (24)

C−2Ω = inf

w∈V0\0

|||w |||2‖w‖2

L2(Ω)

, ,

C 2Γ = sup

w∈V0\0

‖w‖2L2(Γ⊕) + ‖w‖2

L2(Γ⊖)

|||w |||2 + ‖w‖2L2(Ω)

,


The function ψ can be selected in different forms. In particular, we canuse it to exactly satisfy 3D Neumann boundary conditions. Then M3 = 0,and the general estimate comes in a simplified from

|||u − u||| ≤ M := M1 + CΩ M2 (25)


Particular case. Plate with plane parallel facesWe assume that d⊕ = d0

2 , d⊖ = −d02 (d0 = const > 0) and the function

ψ1 takes the simple form

ψ1(x) =F⊕(x) + F⊖(x)

d0x3 +

F⊕(x)− F⊖(x)

2

Then

|||u − u||| ≤√

d0

3

(∫

Ωa−133 (F 2

⊕ + F 2⊖ − F⊕F⊖) dx

)1/2

+ CΩ ‖f − f ‖L2(Ω) .

(26)If we set here f = 0, a33 = 1 and F⊕ = F⊖ = F , we obtain

|||u − u||| ≤√

d0

3‖F‖

L2(Ω)(27)

that is exactly the Babuska-Schwab estimator. for the zero-order reducedmodel, which can be obtained as a particular case of the error majorant.


A two-dimensional test problem in the “sine-shape’ domain

d⊕,⊖(x) = sin(kπx)± d0

2, k = 1, 2, . . . ,

Ω = (x , y) ∈ R2 | x ∈ Ω , d⊖(x) < y < d⊕(x). The considered problem

is

−∆u = f in Ω ,

u = 0 at x = 0 and x = 1 ,

∇u · ν⊕,⊖ = F⊕,⊖ at y = d⊕,⊖ ,

Y

X

u(x , y) = sin(πx) · ym (m = 1, 2, . . .)


100

101

102

103

104

10−2

10−1

100

101

102

1/d0

|||e|

||, M

1

0.5

Figure: (left) The domain geometry; (right) Convergence rate of the exactmodelling-error and of the error majorant, k = 2, m = 4 (solid lines) and m = 5(dash-dot lines), the majorant is indicated by “”.


0 0.2 0.4 0.6 0.8 10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

x0 0.2 0.4 0.6 0.8 1

0

0.005

0.01

0.015

0.02

0.025

0.03

x

Figure: Local error distribution provided by the exact modelling-error (solid line)and by the M1-term of the majorant (dash-dot line), k = 4, m = 4: (left)d0 = 0.1, (right) d0 = 0.05.


Note that the presented error estimator provides a reliable upper bound forthe exact error at any positive values of the domain thickness d0, i.e. alsoin the cases when the domain is not so “thin”.

m = 4 m = 5

d−10 |||e||| M M

|||e||| |||e||| M M|||e|||

100 3.2108 9.5598 2.9774 5.0842 16.8434 3.3129101 0.5058 0.5690 1.1250 0.6399 1.3481 2.1066102 0.1581 0.1598 1.0106 0.1991 0.3937 1.9770103 0.0500 0.0501 1.0010 0.0630 0.1237 1.9650104 0.0158 0.0158 1.0000 0.0199 0.0391 1.9638

Table: Convergence of the exact modelling-error in the energy norm (|||e|||) andof the error majorant (M) as d0 → 0 (k = 2); the results are rounded up to 10−4.

Development of the method for higher order models: S. R. and T.Samrovski, J. Math. Sci., 2011.


ACCURACY OF DIMENSION REDUCTION MODELS IN LINEARELASTICITY


Plane stress model.

Ω

Ω

^

Ωγ

γ

Γ

Γ−

+

1

2

x

xx

x

x

1

3

1

23


Let Ω = Ω × (−d, +d) , Ω ∈ R2 with boundary γ. Assume that

d ≪ diam(Ω) := sup(x1,x2)∈ Ω

|x1 − x2| .

u and σ are the exact solutions of 3D elasticity problem:

σ(x) = Lε(u)(x), in Ω, (28)

ε(u)(x) = 12(∇u(x) + (∇u(x))T) in Ω, (29)

divσ(x) + f(x) = 0 in Ω, (30)

u(x) = u0(x) on Γ1, σ(x)ν = F(x) on Γ2 . (31)


Here c1 |ε|2 ≤ L ε : ε ≤ c2 |ε|2 ∀ε ∈M3×3s .

Γ⊕ :=

x ∈ R3 || x ∈ Ω, x3 = +d

, Top face

Γ⊖ :=

x ∈ R3 || x ∈ Ω, x3 = −d

, Bottom face

Γi := γi × (−d , +d) i = 1, 2 ,

F = (F1(x),F2(x), 0), f = (f1(x), f2(x), 0).

Boundary conditions:

σν = F on Γ2, (32)

σν = 0 on Γ⊕ ∪ Γ⊖, (33)

u = u0 on Γ1 u0 = (u01(x), u02(x), u03(x)) . (34)

This is a classical problem in the theory of thin elastic bodies.


To compare 2D and 3D solutions we use the estimate

||| ε(u − v) |||≤ 1+β

∫

Ω(ε(v)−L

−1y) : (L ε(v)−y) dx+

+1+β

βc1C2

Ω(‖Div y−f‖2Ω+‖F+yν‖2

∂2Ω). (35)

Here u is a 3D solution and v is a 2D one taken from the plane stressmodel and ∂2Ω = Γ2 + Γ⊕ + Γ⊖.


Exact 3D solutions are approximated by u and σ that are solutions of asimplified 2D problem constructed with help of”a priori plane stress assumptions”:

σ13 = σ23 = σ33 = 0,

σαβ = σαβ(x) α, β = 1, 2, (36)

uα = uα(x) ,

Thus, it is required to find

u = (u1(x), u2(x)) and σ = σαβ(x) α, β = 1, 2

that satisfy a simplified system


Plane stress problem

σ = L ε in Ω, (37)

ε = 12(∇u + (∇u)T ) in Ω, (38)

∇ · σ + f = 0 in Ω, (39)

u = u0 on γ1, (40)

σν = F on γ2, (41)

∇ = ( ∂∂x1, ∂

∂x2), f = (f1(x), f2(x)),

F = (F1(x),F2(x)), u0 = (u01(x), u02(x))


Reconstruction operators ℜu and ℜσ Since u = (u1, u2) and σ is a 2× 2tensor, we define an approximate 3D solution as follows:

u = (u1, u2, φ(x1, x2, x3)); σαβ = σαβ, σ3α = 0,

where φ ∈ H1(Ω) and satisfies the same boundary conditions as u03 on theDirichlet part of ∂Ω.

φ is in our disposal!


Isotropic media

If the media is isotropic, then

L ε = K0 tr(ε) I + 2µεD , (42)

L−1 τ =

1

9K0tr(τ)I +

1

2µτD . (43)

K0 and µ are positive (elasticity) constants, tr is the first invariant of atensor, εD is the deviator of ε, and I is the unit tensor.

σ = L ε = K0 tr(ε) I + 2µεD , (44)

ε = L σ =1

K0

tr(σ) I +1

2µσD , . (45)

K0 = 9K0µ3K0+4µ , and tr denotes the first invariant of a plane tensor.


In S. R. J. Math. Sci. New York, 2001 it was shown that the errorgenerated by plane stress model is given by the relation

Cε ‖ ε(u − u)‖2Ω + Cτ ‖ σ − σ‖2

Ω ≤

≤(

K0

2+

2µ

3

)∫

Ω(ρ(u1,1+ u2,2)+ φ,3)

2 dx + µ

∫

Ω

(φ2

,1+ φ2,2

)dx

Here µ and K0 are the elasticity coefficients

ρ =3K0 − 2µ

3K0 + 4µ=

λ

λ+ 2µ=

ν

1− ν ,

Cε = min2µ, 3K0, Cτ = (max2µ, 3K0)−1.


Another form of the estimate

Cε ‖ ε(u − u)‖2Ω + Cτ ‖ σ − σ‖2

Ω ≤

≤ (λ+ 2µ)

∫

Ω(ρ(u1,1+ u2,2)+ φ,3)

2 dx + µ

∫

Ω

(φ2

,1+ φ2,2

)dx

In general, the right–hand side of the above estimate is positive. Really, ifthe second integral is equal to zero then φ = φ(x3). Then the first integralis positive. The only one exception is the case, in which u03 = 0 and ρ = 0or divu = 0. The condition ρ = 0 means that the Poisson coefficient νis equal to zero, i.e., we have a very special material (similar to cork).


KL plate model.Let Ω = Ω× (−h/2, +h/2) , Ω ∈ R

2 with boundary γ.

x

x

x1

2

3

Ω

ω0

h

n

Γ

S −

S+

n−

n +S0

^

F^

f

Let h ≪ diam(Ω) := sup(x1,x2)∈ Ω

|x1 − x2| , and u and σ denote the exact

solutions of 3D elasticity problem


Here

Γ⊕ :=

x ∈ R3 || x ∈ Ω, x3 = +h/2

, Top face

Γ⊖ :=

x ∈ R3 || x ∈ Ω, x3 = −h/2

, Bottom face

Γi := γi × (−h/2, +h/2) i = 1, 2 ,

F = (F1(x),F2(x), 0), f = (f1(x), f2(x), 0).

Boundary conditions:

σν = F on Γ2, (46)

σν = 0 on Γ⊕ ∪ Γ⊖, (47)

u = u0 on Γ1 u0 = (u01(x), u02(x), u03(x)) . (48)

f = h2f0, F = h3F0.


One of the most known is Kirchhoff-Love (KL) plate model.

u1 ( x) = −x3w,1; u2 ( x) = −x3w,2; u3 (x) = w (x) , x = (x1, x2),

σi3 = 0, i = 1, 2, 3

w,1111 + 2w,1122 + w,2222 =g

D,

where g ( x) = hf + F , D := Eh3

12(1−ν2)= µh3

6(1−ν) .

Asymptotic analysis:D. Morgenstern, B. A. Shoikhet, S. Nazarov, R. Monneau, P. G. Ciarletand P. Destuynder,...

Our main goal is different: we wish to explicitly estimate the modelingerror related to a concrete value of h.


Certainly, we can use the general deviation estimate for the linear elasticity:

||| ε(u − v) |||≤ 1+β

∫

Ω(ε(v)−L

−1τ ) : (L ε(v)−τ ) dx+

+1+β

βc1C2

FΩ(‖Div τ−f‖2Ω+‖F+τν‖2

∂2Ω), (49)

where 3D vector function v and 3D tensor τ are reconstructed from 2DKL solution.However, CFΩ comes from 3D inequality and in general is not known.

We construct another deviation majorant, which is well adapted to thespecific geometry of the problem.


Denote ri (τ ) := div τij3j=1 = τi1,1 + τi2,2 + τi3,3, i = 1, 2, 3.

Theorem

Let v ∈ V0, and τ ∈ H(Ω,Div) satisfies the conditionsr3(τ ) + h2 f0 = 0 a.e. in Ω and for i = 1, 2, 3

τ33

(x ,

h

2

)= h3F , τ13

(x ,

h

2

)= τ23

(x ,

h

2

)= 0, τi3

(x ,−h

2

)= 0.

Then

||| ε(u − v) |||2≤ (1 + β)

∫

Ω(Lε(v)− τ ) : (ε(v)− L

−1τ )dx+

+1 + β

c21β

2C 2ω

(‖r1(τ )‖2Ω + ‖r2(τ )‖2Ω

),

where β > 0 and Cω is the 2D constant: Cω ≤ CΠ = 1π

ab√a2+b2

.


How to reconstruct the solution using w? Simple version: (110)-reconstruction:

v110 := R110v (w) = (−x3w,1,−x3w,2, w)⊺ . (50)

D. Morgenstern (1959) have shown that we should use an ”advanced” or(112)-reconstruction:

v112 := R112v (w) =

(−x3w,1,−x3w,2, w + x2

3W (x))

⊺

, (51)

where W ( x) ∈ H10 (ω) is a special correction function.


Necessary properties of W has been thorough investigated byA. L. Alessandrini, D. N. Arnold, R. S. Falk, and A. L. Madureira (1996),A. Rossle, M. Bischoff, W. Wendland, and E. Ramm (1999),D. Braess, S. Sauter, and C. Schwab (2010).It was shown that the reconstruction is ”good” (in the asymptotic sense),

if W (x) solves a specially constructed singularly perturbed problem.


Improved reconstruction of the stress includes W :

τ imkl :=

−2µx3

(w,11 + ν

1−ν ∆w)

−2µx3w,12 θ(x3)q1

−2µx3w,12 −2µx3

(w,22+ ν

1−ν ∆w)

θ(x3)q2

θ(x3)q1 θ(x3)q2 ψ(x3)h2 f0

.

Here the correction functions θ, q, and ψ are specially selected.

q1 =2µ

1− ν ∆w,1, q2 =2µ

1− ν ∆w,2

We choose θ(x3), q1, q2, and ψ(x3) in such a way that conditions ofTheorem are satisfied and arrive at the estimate


Deviation estimate implies:

||| ε(u − v112kl ) |||2 ≤ m11h

5‖∇W ‖2ω + m21h3‖2W − ν

1− ν ∆w‖2ω +

h5R(w , f0),

where γ, and λ are arbitrary positive numbers and

R(w , f0) = m124µ2

120(1− ν)2 ‖∇∆w‖2ω + m22h2

210‖f0‖2ω.

m11 =µ(1 + γ)

80, m12 =

1 + γ

µγ,

m21 =µ(1− ν)(1 + λ)

6(1− 2ν),m22 =

1

µ

(1

2(1 + ν)+

(1− 2ν)

λ(1− ν)

).

How to select W ?


Lemma (AAFM)

Let φ ∈ H1(ω) and W∗ be a minimizer of the functional

infW∈H1(ω)

h2‖∇W ‖2ω + ‖W − φ‖2ω

, (52)

where h is a small positive number. Then

h2‖∇W∗‖2ω + ‖W∗ − φ‖2ω ≤ C(h‖φ‖∂ω + h2‖φ‖1,ω

), (53)

We use this Lemma for φ→ c∆w , recall the trace theorem and have

h‖∇W∗‖ω + ‖ρ(w , W∗)‖ω ≤ Ch1/2‖∆w‖ω.

whereρ(w , W ) := 2W − ν

1− ν ∆w .


Therefore,

M6(w , W∗, q0) ≤ Ch4 (m11 + m21) ‖∆w‖2ω + h5R(w , f0) (54)

Since

||| ε(u) |||2 and ||| ε(v112) |||20 ∼ h3

we get necessary convergence

M6(w , W∗, q0)

||| ε(v112) |||2 ≤ ch andM6(w , W∗, q0)

||| ε(u) |||2 ≤ ch. (55)


Details are exposed in:S. R and S. Sauter.Estimates of the modeling error for the Kirchhoff-Love plate model.C. R. Math. Acad. Sci. Paris, 348(2010), no. 17-18, 1039-1043.

S. R and S. Sauter.Computable estimates of the modeling error related to Kirchhoff-Loveplate model.Anal. Appl. 8 (2010), no. 4, 409428.


Indeterminant data

No one real-life problem has ”exact” data!

Examples:Viscosity ν depends on many factors (e.g., on T );Diffusion matrix A depends on media properties which expose certainoscillations.Source term fDomain Ω.

What is the total level of indeterminacy (in terms of the energy spacemetric) generated by these factors???

Regrettably this question is often ignored in applied analysis.


Converging sequence and a ”cloud” of equally possible exact solutions

.

v

ee

−

+

ee−

+=?

.

improving is senseless

Indeterminacy set "cloud"

. v


Estimates of deviations include problem data EXPLICITLY:Simple example:

||| u − v |||2≤ (1 + β)

∫

Ω

(A∇v · ∇v + A−1y · y − 2y · ∇v)dx

+ CFΩ1 + β

β‖f + divy‖2

This fact allows to compute e− and e+ and adequately judge about thesituation.Moreover, we can estimate the diameter of the indeterminacy set. Thisvalue states the limit of any quantitative analysis of a problem.


In 2008-2010 two sided bounds of indeterminacy errors for problems withthe operator divA∇u has been derived inO. Mali and S. R., 2010, Comput. Meth. Appl. Sci.O. Mali and S. R., 2009, Adv. Math. Sci. Appl.This techniques can be extended to operators

Λ∗AΛu

with incompletely known operator A.Linear elasticity (indeterminacy effect with respect to the Poisson’s ratio):O. Mali and S. R., 2011, Russ. J. Numer. Anal. Math. Mod.


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