Finite Element Analysis for Mechanical and Aerospace Designibilion/... · N. Zabaras (05/01/2014)...

CCOORRNNEELLLL U N I V E R S I T Y 1

MAE 4700 – FE Analysis for Mechanical & Aerospace Design N. Zabaras (05./01/2014)

Finite Element Analysis for Mechanical and Aerospace

Design

Prof. Nicholas Zabaras Materials Process Design and Control Laboratory

Sibley School of Mechanical and Aerospace Engineering 101 Rhodes Hall

Cornell University Ithaca, NY 14853-3801 [email protected]

http://mpdc.mae.cornell.edu

mailto:[email protected]

http://mpdc.mae.cornell.edu/


MAE 4700 – FE Analysis for Mechanical & Aerospace Design N. Zabaras (05/01/2014)

References • FEM for the incompressible NS equations, A.Segal • Incompressible Computational Fluid Dynamics: Trends and

Advances, M. D. Gunzburger and R. A. Nicolaides • Finite Elements and Fast Iterative Solvers: with Applications

in Incompressible Fluid Dynamics, by Howard C. Elman, David J. Silvester and Andrew J. Wathen.

• Numerical Methods for Incompressible Flow, M. Burgen

http://ta.twi.tudelft.nl/users/vuik/burgers/fem_notes.pdf

http://www.cfd-online.com/Books/show_book.php?book_id=17

http://www.cfd-online.com/Books/show_book.php?book_id=17

http://www.cfd-online.com/Books/list_books.php?author=Gunzburger,+M.+D.

http://www.amazon.com/gp/reader/019852868X/ref=sib_dp_pt#reader-link

http://www.amazon.com/gp/reader/019852868X/ref=sib_dp_pt#reader-link

http://www.wepapers.com/Papers/54603/Lecture_Notes_Numerical_Methods_for_Incompressible_Flow



Advection-diffusion problems • Consider the following advection-diffusion problem:

• Note that the BCs above are similar to those for a diffusion (heat conduction) problem.

• But what if the problem is convection dominated?

1

2

3

1 2 3

( ) ( ) ,

( ,0) ,

( , ) ( , ),

,,

cx A c u c c f xt

c x givenc x t c x t x

A c n q xA c n g c x

ρ β

σ

∂−∇ ∇ + ∇ + = ∈Ω

∂=

= ∈Γ

∇ = ∈Γ∇ = − ∈Γ

Γ ∪Γ ∪Γ = Γ



Advection-diffusion problems • When the convection term strongly dominates the

diffusive term, the equation resembles more that the pure convection equation than the diffusion equation.

• For a pure convection equation, boundary conditions should only be given at inflow not at outflow.

• Since for the convection-diffusion equation, boundary conditions must be given at outflow, one should use BCs that influence the solution as little as possible.

• This means that at outflow one usually applies natural BCs. Essential BCs may result in unwanted wiggles.



Standard Galerkin for Advection-Diffusion • Multiplication of the PDE with a time-independent

weight function w, integration over the domain & using Gauss theorem gives:

• Substitution of the BCs finally results in:

• Finite element implementation:

( ) 10cwd A c w cw u cw d wA c nd f wd admissible w with w ont

ρ βΩ Ω Γ Ω

∂Ω + ∇ ∇ + + ∇ Ω− ∇ Γ = Ω∀ = Γ

∂∫ ∫ ∫ ∫

( )3

2 3

10

cwd A c w cw u cw d cwdt

f wd wqd gwd admissible w with w on

ρ β σΩ Ω Γ

Ω Γ Γ

∂Ω + ∇ ∇ + + ∇ Ω+ Γ

∂

= Ω + Γ + Γ ∀ = Γ

∫ ∫ ∫

∫ ∫ ∫

1( , ) ( ) ( )

n

h j jj global FE

basis functions

c x t c t N x=

=∑



Standard Galerkin for Advection-Diffusion • Find such that:

• This system is of the form:

( )3

2 3

1 1 1

, 1,2,...,

n n nj

i j j i i j j i i jj j j

i i i

cN N d A N N N N u N N d N N d

t

f N d N qd gN d i n

ρ β σ= = =Ω Ω Γ

Ω Γ Γ

∂Ω + ∇ ∇ + + ∇ Ω+ Γ

∂

= Ω + Γ + Γ =

∑ ∑ ∑∫ ∫ ∫

∫ ∫ ∫

( )jc t

( )3

2 3

1

( )

( )

( )

ij i j

n

ij j i i j j i i jj

i i i i

M c Kc F

M N N d mass matrix

K A N N N N u N N d N N d stiffness matrix

F f N d N qd gN d load vector

ρ

β σ

Ω

= Ω Γ

Ω Γ Γ

+ =

= Ω

= ∇ ∇ + + ∇ Ω+ Γ

= Ω + Γ + Γ

∫

∑∫ ∫

∫ ∫ ∫



Standard Galerkin: Time integration

• Among the many available time-integration methods we review once mote the θ-method ( ):

M c Kc F+ =

11 1

1

11 1

11 1

(1 ) (1 )

0 :

1 :

1 11/ 2 : ( ) ( )2 2

k kk k k k

k kk k

k kk k

k kk k k k

c cM Kc Kc F Ft

c cexplicit method M Kc Ft

c cimplicit method M Kc Ft

c cCrank Nicolson method M Kc Kc F Ft

θ θ θ θ

θ

θ

θ

++ +

+

++ +

++ +

−+ + − = + −

∆−

= + =∆−

= + =∆

−= − + + = +

∆

0 1θ≤ ≤



Explicit time integration

• In the explicit method we still have to solve a system of equations unless the mass matrix is lumped

• The time-step must be restricted in order to get a stable solution. E.g. in the case of a pure time-dependent diffusion problem , C-constant, h=local diameter of the elements.

1

0 :k k

k kc cexplicit method M Kc Ft

θ+ −

= + = ⇒∆

2t Ch∆ ≤

( )1k k k kMc Mc t F Kc+ = + ∆ −



Implicit time integration

• The implicit method is unconditionally stable for the convection

equation. The time step is restricted only for accuracy reasons.

• The accuracy of both the implicit and the explicit Euler time-discretization is Ο(∆t).

• In this implicit Euler method the errors in time are always damped.

11 11 :

k kk kc cimplicit method M Kc F

tθ

++ +−

= + =∆

( )1 1 1k k k kMc tK c Mc tF+ + ++ ∆ = + ∆



Crank Nicolson integration

• This scheme is unconditionally stable and the accuracy is one order higher, i.e. Ο(∆t2). • This scheme does not have the damping property of Euler implicit and

as a consequence once produced errors in time will always be visible.

• This scheme starts with one Euler implicit step.

M c Kc F+ =

11 11 11/ 2 : ( ) ( )

2 2

k kk k k kc cCrank Nicolson method M Kc Kc F F

tθ

++ +−

= − + + = +∆



Accuracy of the SGA for advection-diffusion • Consider the following problem:

• The analytical solution of this problem is:

• We use 3-node linear elements and 6-node quadratic triangulars with various values of ε.

, [0,1] [0,1]( , ) sin sin ,

, 2 sin sin cos sin sin cos

c u c f x xc x y x y x

xu f x y x x y y x y

y

ε

ε

− ∆ + ∇ = ∈Ω == ∈Γ

= = + +

sin sinc x y=



Accuracy of the SGA for advection-diffusion

• The error shown above is in the max-norm. Note from this table the following: – for relatively small convection the accuracy of the linear

elements is O(h2), and for the quadratic elements at least O(h3),

– for convection-dominant flow, the numerical solution is very inaccurate especially for coarse grids,

– the use of quadratic elements makes only sense for problems with small convection.

SGA is not a good method for convection dominant flows




Rotating cone problem



Streamline upwind Petrov Galerkin • We have seen that the SGA method may lead to wiggles and

inaccurate results for convection-dominant flows. In finite differences one has tried to solve it by so-called upwind methods.

• This has motivated researchers in FEM to construct schemes, which are comparable to classical FDM upwind schemes. Among the various upwind techniques for the FEM, the streamline upwind Petrov-Galerkin method (SUPG) is the most popular one (Brooks and Hughes (1982)).

• Instead of choosing the test function in the same space as the solution, a test function is introduced according to

where w is the classical test function and p denotes a correction in

order to take care of the upwinding part.

ww w p= +



Streamline upwind Petrov Galerkin

• The function w is chosen in the same space as the solution, which means that the 1st derivative is square integrable.

• The function p may be discontinuous over the elements. Thus the Gauss’ divergence theorem can only be applied to the w part.

• The 2nd derivative of c does not have to exist over the element boundaries and is certainly not integrable for the FE considered.

• Thus we cannot compute the integral containing the p term. In order to solve this problem, this integral is split into a sum of integrals over the elements, and the inter-element contributions are neglected.

( )( ) ( ) ( ) ( )c w p d A c u c c w p d f w p dt

ρ βΩ Ω Ω

∂+ Ω+ −∇ ∇ + ∇ + + Ω = + Ω

∂∫ ∫ ∫

( )3

2 3

1

( )

0


c A c u c c f pdt

f wd wqd gwd admissible w with w on

ρ β σ

ρ β

Ω Ω Γ

Ω

Ω Γ Γ

∂Ω + ∇ ∇ + + ∇ Ω+ Γ

∂

∂ + −∇ ∇ + ∇ + − Ω ∂

= Ω + Γ + Γ ∀ = Γ

∫ ∫ ∫

∫

∫ ∫ ∫



Streamline upwind Petrov Galerkin

• This approximation is consistent since it consists of a standard Galerkin part, which itself is consistent, and a summation of residuals of the differential equation per element multiplied by a function. Consistency implies that at least the constant and first term of the Taylor series expansion of the solution are represented exactly.

• The choice of p defines the type of SUPG method used. A complete

class of different SUPG methods may be defined by different choices of p.

( )3

2 3

1

11

( )

0

e

e

e

e

N

e

N

e


c A c u c c pdt

f wd wqd gwd fpd admissible w with w on

ρ β σ

ρ β

Ω Ω Γ

= Ω

=Ω Γ Γ Ω

∂Ω + ∇ ∇ + + ∇ Ω+ Γ

∂

∂ + −∇ ∇ + ∇ + Ω ∂

= Ω + Γ + Γ + Ω∀ = Γ

∫ ∫ ∫

∑ ∫

∑∫ ∫ ∫ ∫



Choosing p

Solution for ε = 0.01 and u = 1: — exact solution, − − numerical solution for ∆x = 0.1 and central differences.

The solution shows wiggles as long as ∆ x > 2/Pe ,where the Peclet number Pe is defined as

Consider the following advection-diffusion problem modeled with finite differences.

Note that with SGA and linear finite elements we obtain exactly the same discretized FEM equations



Choosing p • For a pure convection problem all information is transported from left to

right and hence the discretization of the convective term should also be based upon information from the left. Let us return to the central differences approximation:

• The above equation can be written as:

( ) ( )1 1 1 12 02i i i i i

Pe xc c c c c+ − + −∆

− − + + − = ⇒

1 11 1(1 ) (1 )2 2 2 2i i i

Pe x Pe xc c c+ −∆ ∆

= − + + ⇒ 12

Pe x∆<

for information to propagate from left to right



Choosing p • In the classical FD upwind scheme one tries to get rid of these wiggles

by replacing the first derivative by a backward difference scheme provided the velocity u is positive.

• How do we make the FEM equations to give us this short of approximation? Note that the above backward difference approximation

can be written as: • This is our initial approximation but with diffusion

1 1 1 12

2 02 2

i i i i ic c c c cu x ux x

ε + − + −− + −∆ − + + = ∆ ∆

in t

2artificial diffusion

he direction offlow

u xε ∆+



Choosing p • The figure shows the result of the upwinding; the wiggles have been

disappeared and the numerical solution has been smoothed. • Although backward differences suppress the wiggles, they also make

the solution inaccurate. • In the literature many upwinding schemes for FD methods have been

derived which are much more accurate than the backward difference scheme. But the idea is similar.

Solution for ε = 0.01 and u = 1; —exact solution, -+- numerical solution for ∆x = 0.1 and backward differences



Choosing p • In principle one may consider the backward difference approximation as

the discretization of a convection-diffusion equation with a diffusion of

• As we showed earlier it can be derived by taking the central difference scheme of the differential equation



Artificial diffusion • The is usually called artificial diffusion. Many of the

upwind schemes in finite differences may be considered as a central difference scheme with artificial diffusion.

• For example the exact solution of of is constructed by the

central difference scheme with artificial diffusion equal to • Other choices for the parameter have been developed. ξ



Petrov-Galerkin formulations • This observation motivates us to choose the

function p such that an artificial diffusion of the forms above is constructed. If we confine ourselves

to linear elements, the second derivative of the approximate solution is zero per element and hence SUPG applied to our problem reduces to:

• The accuracy of the scheme is O(∆x2) + εO(∆x), which may be considered to be of O(∆x2) for small values of ε.



SUPG in 2D • If we apply the SUPG method in 2D in each of the

directions, a typical cross-wind diffusion arises. That means that the solution perpendicular to the flow direction is smoothed and becomes inaccurate.

• The SUPG method must be extended in such a way that the upwinding is applied in the direction of the flow only.

• Brooks and Hughes (1982) have solved this problem by giving the perturbation parameter p a tensor character – p is in the direction of flow (as in FD artificial diffusion is introduced in the direction of flow)

• In this formula h is the local element width, which may depend on the element type.



SUPG performance

• The improvement for small values of ε and coarse grids is immediately clear (compare with an earlier Table).

• The use of upwind makes the matrices to be solved more diagonally dominant. As a consequence iterative matrix solvers will converge much faster than for SGA.

• Finally we show some results of classical benchmark problems to investigate the behavior of various schemes.

Error in max-norm of convection-diffusion problem for various values

of ε. Solution by SUPG. Linear and quadratic triangles.





Number of iterations by a preconditioned CGS solver for the solution of the rotating cone problem . SGA and SUPG.

- in the table means that no convergence was possible.






Advection-diffusion in 1D

SGA 40 linear elements, lumped mass matrix

– exact solution, + numerical solution

SGA 40 linear elements

consistent mass matrix – exact solution, numerical solution

Lumping drastically decreases the accuracy of the numerical solution






Advection-diffusion in 1D

SUPG with stationary upwind parameter 40 linear elements

– exact solution, + numerical solution for this moderate Peclet number, the standard Galerkin method performs a little bit

better than SUPG.




Incompressible flow equations • We consider fluids with the following properties:

– The medium is incompressible, – The medium has a Newtonian character, – The medium properties are temperature independent

and uniform, – The flow is laminar.

• For a 3D flow field the basic equations of fluid flow under the above restrictions, can be written as:

• The continuity equation



The incompressible flow equations • The Navier-Stokes equations

• In component form:

• For an incompressible and isotropic medium the stress terms can be written as

p denotes the pressure, I the unit tensor e the rate of strain tensor, d the deviatoric stress tensor and μ the viscosity of the fluid



The incompressible flow equations • The components eij of the strain rate tensor e are

defined by • So the stress components are:

• If μ is constant it is possible to simplify as:

• Or in dimensionless form:



Initial and boundary conditions • The initial velocity is needed (and it must be incompressible!)

• Since the N-S equs is a system of 2nd order PDEs in space, it is

necessary to prescribe boundary conditions for each velocity component on the complete boundary of the domain.

• At high Re numbers the convective terms dominate the stress tensor and as a consequence the boundary conditions at outflow must be such that they restrict the flow as little as possible.

• For incompressible flows no explicit boundary conditions for the pressure must be given. Usually boundary conditions for the pressure are implicitly given by prescribing the normal stress.



Boundary conditions • The following types of boundary conditions are commonly used

for the 2D incompressible Navier-Stokes equations (the extension to 3D is straight forward):

• Typical examples of these boundary conditions are: – At fixed walls: no-slip condition u = o. – At inflow the velocity profile given: u = g. Typical inflow profiles are ut = 0, un parabolic or ut = 0 and un constant.

• At outflow one may prescribe the velocity. However, for convection dominated flows, such a boundary condition may lead to wiggles due to inaccuracies of the boundary conditions. Less restrictive outflow boundary conditions are for example ut = 0 and σnn = 0 or σnt = 0 and σnn = 0.

• ut = 0, σnn = 0 prescribe a parallel outflow with zero normal stress.

1 u given (Dirichlet boundary condition) 2 un and σnt given 3 ut and σnn given 4 σnt and σnn given,



Boundary conditions • From we can compute:

• As a consequence for high Reynolds numbers σnn is approximately equal to −p. So σnn = 0 implies that implicitly p is set equal to zero.

• The boundary condition ut = 0, σnn = 0 is correct for channel flow.

• If you do not have a channel flow σnt = 0, σnn = 0, although incorrect, may be a good choice in numerical computations.



The weak formulation

• We write one weak form for the incompressibility equ and one weak form for the N-S equs.



The weak form

• No derivatives of p and q are necessary. Hence it is sufficient that p and q are integrable. • For u and v but also their first derivatives must be integrable. As a consequence we do

not need continuity of p and q in the Galerkin formulation, but the functions u and v must be continuous over the element boundaries.

• If we demand that both u and v are divergence free, then the first equation vanishes and the pressure disappears from the 2nd equ.



The standard Galerkin form • Introducing separate interpolation functions for u

and p, we can write:



System of equations

• U denotes the vector of unknowns u1i and u2i,

• P denotes the vector of unknowns pi,

• KU denotes the discretization of the viscous terms,

• N(U) the discretization of the nonlinear convective terms

• LU denotes the discretization of the divergence of u and −LTP the discretization of the gradient of p. The right-hand side F contains all contributions of the source term, the boundary integral as well as the contribution of the prescribed boundary conditions.

• The solution of the system of equations introduces two difficulties. Firstly the equations are non-linear. Secondly equation (b) does not contain the unknown pressure P.

( ) ( )0 ( )

TKU N U L P F aLU b

+ − ==



Treating of the non linear terms • In order to solve the system of non-linear equations, an iterative

procedure is necessary (Newton methods). make an initial estimation while (not converged) do

– linearize the non-linear equations based on the previous solution – Solve the resulting system of linear equations

• Suppose we have computed the solution uk at a preceding iteration level k. We write this solution

as uk. First we shall derive the Newton linearization. To that end we define f(u, ∇ u) as



Treating the non linear terms

• This is the standard Newton-Raphson. • Picard approximations:

• Numerical experiments have shown that only the 2nd approximation produces a good convergence.

• Newton is in fact a linear combination of the 3 Picard approximations. • After linearization of the convective terms the standard Galerkin method

may be applied, resulting in a system of linear equations.



Iterative process • A possible strategy to converge to the final solution is the following: - start with some initial guess, - perform one step Picard iteration in order to approach the final

solution, sometimes more than one step, - use Newton iteration in the next steps. • An initial guess may be e.g. the solution of the Stokes problem, which is

formed by the Navier-Stokes equations where the convective terms have been neglected.

• If the Re number is too high it is possible that the distance between the solution of Stokes and Navier-Stokes is too large. In that case the solution of Navier-Stokes with a smaller Reynolds number might be a good choice. A process in which the Re number is increased gradually is called a continuation method.

• The iteration process no longer converges as soon as the flow becomes transient or turbulent.



Element construction • The velocity over the element-sides must be

continuous, whereas the pressure may be discontinuous over the element boundaries.

• The continuity equation, discretized as LU = 0, does contain only velocity unknowns. However, the number of rows in this equation is completely determined by the number of pressure unknowns.

• Suppose that there are more pressure unknowns than velocity unknowns. In that case equation (b) contains more rows than unknowns and this leads to a singular matrix.

( ) ( )0 ( )

TKU N U L P F aLU b

+ − ==



Element construction • The number of P-unknowns should never exceed

the number of u-unknowns. This should be valid independently of the number of elements.

• This demand restricts the number of applicable elements considerably.

• In order to satisfy this criterion, a general accepted rule is that the order of approximation of P must be one lower than the order of approximation of u. – So if u is approximated by a linear polynomial, then P is

approximated by a constant per element and so on.

( ) ( )0 ( )

TKU N U L P F aLU b

+ − ==



Element construction • This rule is not sufficient to guarantee that the number of P-unknowns is

not larger than the number of u-unknowns independently of the number of elements.

• Consider the mesh in the fig. with linear elements for u and constant elements for P. In this example the mesh contains 8 P nodes and 9 u nodes.



Element construction • Suppose we have Dirichlet boundary conditions for the

velocity, which means that all boundary velocities are prescribed.

• The pressure is unique except for an additive constant. To fix this constant one of the pressure unknowns is given. So finally we have 2 velocity unknowns and 7 pressure unknowns.

• We thus have a singular matrix. The corresponding element is not admissible. – The situation is not improved even as we add more elements.



Element construction • This is an example of an admissible element.

The velocity unknowns are not positioned in the vertices of the triangle but in the midside

points. • The velocity approximation is linear but not

continuous over the element boundaries. Such an element is called nonconforming and introduces for that reason extra problems with the approximation.

• The element has more velocity unknowns than pressure unknowns. For the given mesh, the number of velocity unknowns is equal to 16 and the number of pressure unknowns equal to 7 in the case of Dirichlet boundary conditions.



Brezzi-Babuska condition • The derivation of the admissibility condition

given above helps to identify non-admissible elements.

• The exact admissibility condition is known under the name Brezzi-Babuska condition (or BB condition).

• The BB condition is rather abstract and in practice it is very difficult to verify whether

is satisfied or not.

• Fortin (1981) has given a simple method to check the BB condition on a number of elements.



Fortin condition • An element satisfies the BB condition, whenever, given a continuous

differentiable vector field u, one can explicitly build a discrete vector field such that:

• Midside velocity points in 2D and centroid velocity points on surfaces in 3D make it possible to control the amount of flow through a side (2D) and through a surface (3D) of an element, without altering the amount

of flow through other sides or surfaces. Such nodal points make it easier to satisfy the continuity equations.

• It is sufficient that the normal component of the velocity in these centroid points is available as unknowns.

• In the literature frequently elements are used, that do not satisfy the BB condition. Such elements cannot be used with the standard Galerkin method, however other methods (e.g. the penalty function method to be discussed later) permit the use of these elements.



Admissible elements

Taylor-Hood element (P2 − P1).

Taylor-Hood element (Q2 − Q1)





Discontinuous pressure elements

Crouzeix-Raviart element (p1 − p0)





Admissible elements

Crouzeix-Raviart element (P+2 − P1)





Admissible elements

Crouzeix-Raviart quadrilateral (Q2 − P1)




Solution of the system of equations • Here Ku denotes the discretization of both the viscous terms and the linearized convective terms. • If the unknowns are numbered in the sequence: first all velocity

unknowns and then all pressure unknowns it is clear that the system of equations gets the shape as sketched in provided an optimal nodal point numbering is applied. Unfortunately this numbering (velocity first, pressure last) is far from optimal. The total profile is still very large.

A much smaller profile may be achieved if pressure and velocity unknowns are intermixed.

0

TKu L P FLu

− ==



Solution of the system of equations • With intermixed velocity and pressure DOF:



Solution of the system of equations • Because of the boundary conditions, the first degrees of freedom are

pressures, which do not appear in the continuity equation.

• In order to prevent zeros on the main diagonal, partial pivoting must be applied. Unfortunately, partial pivoting reorders the sequence of the equations and increases the profile or band width.

• Therefore a large amount of extra computing time and computer memory is required. However, it still remains cheaper than the earlier discussed numbering.

• It is possible to define a numbering which produces a nearly optimal profile and prevents the appearance of zeros at the start of the main diagonal. Such a numbering, however, goes beyond the scope of this lecture.



The penalty method • Consider the stationary linear Stokes equation in dimensionless form:

• The pressure p is unique up to an additive constant. The idea of the penalty method is to perturb the continuity equation by a small term containing the pressure. An obvious choice is



The discrete penalty method • In the discrete penalty function method, the Navier-Stokes equations

are discretized before applying the penalty function method. So we start with the formulation:

• The continuity equation is perturbed by a term εMpp, where Mp is the so-called pressure mass matrix, defined by

0

TKu L p FLu

− ==

11 TpK L M L u F

ε− + =



The continuous penalty method • For the Stokes problem we have:

1Ku Au Fε

+ =



The penalty method for the transient N-S



Transient N-S equations with the penalty method



The penalty method 1/λ ε=

115

γ =

http://mpdc.mae.cornell.edu/Publications/PDFiles/REPORTS/ns.pdf




The penalty method



Penalty method – Time integration





Penalty method: Lead driven cavity






Flow passed a cylinder






Pressure correction method

• Here, n denotes the old time level and n + 1 the new time level. In the first step of the algorithm, the momentum equation is solved using p at the old time level. This yields an intermediate velocity field u∗ satisfying

u∗ is provided with the boundary conditions at level n + 1. In order to solve this, the term u∗ ・grad u∗ must be linearized with respect to the old solution un.



Projection method

• It can be shown that the second term is of the same order as the truncation error of the method and hence may be neglected. As a consequence the last equation reduces to

• In the second step u∗ is projected onto the space of divergence-free vector fields by applying the divergence operator:



The projection method in discrete form

0

TM u Ku L p FLu

+ − ==

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