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9 "r,rrtered Element-by-Element Computationsfor Fluid Flow

J. Liou and T. E. Tezduyarl

Abstract. It is shown that the clustered element-by-element method, together

with the preconditioned generalized minimal residual method, can be effectively

used to solve compressible and incompressible flow problems. Discretizations of

the governing equations of these flow problems involve the entropy variable for-

mulation for compressible flows and the space-time formulation for incompressible

flows. The clustered element-by-element method is a generalized version of the

standard element-by-element method. In this method, the elements are partitioned

into clusters of elements, with a desired number of elements in each cluster, and

the iterations are performed in a cluster-by-cluster fashion. The method is highly

vectorizable and parallelizable if used with proper clustering and element-grouping

schemes. To demonstrate the effectiveness of the proposed methods, and to eval-

uate the convergence rates achieved, several test computations are performed on

model problems with regular and irregular meshes.

9.1 Introduction

The clustered element-by-element method (CEBE) was first proposed in fliougl]to be used with the conjugate gradient method for solving problems with sym-

metric spatial operators (e.g., for problems governed by the Poisson equation). In

this chapter) we use the CEBE method in conjunction with the generalized mini-

mal residual (GMRES) method lSaSch86] to solve compressible and incompressible

flow problems. These problems involve numerical difficulties such as nonlinearities,

nonsymmetric spatial operators, shocks, and the incompressibility constraint. In

the CEBE method the elements are merged into clusters of elements. Any number

of elements can be brought together to form a cluster, and the number should be

viewed as an optimization parameter to minimize the computational cost. The

CEBE preconditioners are defined as series products of the cluster matrices, and

the computations are performed in a cluster-by-cluster fashion. Each cluster ma-

trix is formed by assembling together the element matrices corresponding to the

elements in that cluster. To facilitate vectorization and parallel processing, just like

the way it is done in the grouped element-by-element (GEBE) method [Tez89], the

lDepartment of Aerospace Engineering and Mechanics, Army-High Performance ComputingResearch Center, and Minnesota Supercomputer Institute University of Minnesota, Minneapolis,MN 55415. This research wa*s sponsored by NASA-.Iohnson Space Center under grant NAG 9-449,and by NSF under grant MSM-8796352.

tezduyar

Text Box

Parallel Computational Fluid Dynamics (ed. H.D. Simon), MIT Press, Cambridge, Massachusetts (1992) 167-187.

168 Chapter 9. Clustered Element-by-Element Computations for Fluid Flow

clusters can be grouped in such a way that no two clusters in any group have shared

nodes. Fbrthermore, depending on the cluster size (i.e., the number of elements in

the cluster), elements within each cluster can again be grouped in the same way. In

the two limit cases, the CEBE method becomes equivalent to the GEBE method

(when the cluster size is equal to one) and the direct solution method (when the

cluster size is equal to the total number of elements).

The space-time finite element formulation has been successfully used for various

problems (see, for example,[Shak88, HanSze90, Hugh88]). In this chapter we em-

ploy this formulation for discretization of the equations governing incompressible

flows. Equal-order interpolation functions are used for velocity and pressure, and

the variational formulation includes Galerkin/least-squares stabilization [Hugh89].Because the finite element interpolation functions are discontinuous in time, the

fully discrete equations can be solved one space-time slab at a time. However, the

memory needed for the global matrices involved in this method is very substantial,

and this could be a major obstacle for the extension of the method to practical

Iarge-scale problems. For example, in twedimensions, the memory needed for the

space"time formulation (with piecewise linear interpolation functions) of a problem

is approximately four times more compared to using the finite element method only

for spatial discretization. We employ the CEBE iteration scheme to reduce the cost

involved in solving the linear equation systems arising from the space-time finite

element discretization.

For compressible flow problems, we solve the Euler equations by using the entropy

variabies formulation [Hugh86a]. With properly designed streamline.upwind/Petrov-

Galerkin (SUPG) and shock capturing operators, this formulation has been success-

fully applied to various challenging problems (see, for example, [Shak88, Mall85]).

Explicit computations, though economical in memory requirements, are limited by

stability conditions, and thus involve constraints in terms of the time step size that

can be chosen for the computation. Implicit computations, on the other hand, allow

larger time steps, but need much more memory for the global matrices. F\rrther-

more, the implicit computations take much more computing time per time step.

By using the CEBE iteration method to solve the equation systems involved, we

can essentially keep the desirable stability properties of the implicit method while

reducing the memory and computing time needed.

In section 9.2, the governing equations and the finite element formulations will

be discussed for incompressible and compressible flow computations. In section 9.3,

implementation of the CEBE method with the GMRES method will be discussed.

In section 9.4, the proposed methods are tested on several compressible and incom-

pressible flow problems. Benchmark computations are performed to compare the

9.2. The Governing Equations and the Finite Element Formulation

convergence rates achieved. The concluding remarks are given in section 9.5.

9.2 The Governing Equations and the Finite El-ement Formulation

In this section we discuss the discretization techniques for the governing equations

of incompressible and compressible flows.

9.2.L Incompressible Flows

Let O and (0,T) denote the spatial and temporal domains, with x and t representing

the coordinates associated with O and (0,T). We consider the velocity-pressure

formulation of the Navier-Stokes equations:

. d uc { f i + u ' V u ) - V ' a : 0 o n Q x ( 0 , ? ) ,

V . u : 0 o n Q x ( 0 , ? ) ,

where p is the density, and a is the stress tensor defined as

o: -p I - l21 le (u) ,

with1 -

e (u) : l lVu + (V") t ) .'2

169

(e .1)

(e.2)

(e.3)

(e 4)

(e 5)

(e.6)

llere p represents the viscosity while I denotes the identity tensor. Both the Dirich-

let and Neumann type boundary conditions are taken into account:

u : 9 o n f n x ( 0 , ? ) ,

n . o : h o n f l x ( 0 , ? ) ,

where fn and f ;, are complementary subsets of the boundary f . The initial condi-

tion consists of a divergence-free velocity field specified over the entire domain:

u(x,0) : qo on Q. (e.7)

In the space-time finite element formulation, we partition the time interval (0,T)

into subintervals 1, : (tn,tr11), where f, and fral belong to an ordered series

of time levels 0 : to ( tr ( ... ( tru - T. We define the space-time slab Q, as

the domain enclosed by the surfaces 0,,Q,+1, and Pr, where P' is the surface


described by the boundary I as t traverses Ir. Pn can be decomposed into (P,)nand (P,)7, with respect to the type of boundary conditions being applied. The finiteelement interpolation functions are discontinuous in time, and the fully discretizedequations are solved one space-time slab at a time.

Let e denote the set of elements resulting from the finite element discretizationof the space-time slab. For each slab we define the following finite element functionspaces:

Hrh (Q ; : {Oh ldh e Co (e ) ,6h le " ^ ep t (O) x p i ( t 1 , f : 0 o r r , ye i e e } , ( 9 .8 )

s f i : { uh lun e lH tn (Q) ] ' " , , u / ' : ! on (p , )g } ,

yff : {wrlwh e lHrh(e; ] '" , , *h : 0 on (p-)n},

s t :v ! : {qh lsh e Hrh(e; } ,

where n"4 is the number of space dimensions. In time domain, we can use piecewise

constant (, : 0) or piecewise linear (z : 1) functions.The space-time finite element formulation of 9.1 - 9.7 can be written as follows:

given (uh);, find uh e Sff and ph e 3!, such that , Vwh e Vfi Vqh eV!

Ie^ nh .p(T -uh .vuh ) dadt * Io^ur*n)

, o(ph.uh) dedt

+ ' i , r o ( o * n ^ A 'T r J e ; ; + u h ' v w h )

- v o ( q h ' * h ) i

, . ? u h Alo(; + uft . vuh) _ v . o(ph,uh71 anat

+ t shv . , rh ded t+ [ oo ) j ( (uh ) f - (uh ) ; ) aoJ q " J a ^

t

I * n h d P ,J e")n

where n"; is the number of elements in the space-time slab, and

(.tn)* : I im6-ouh (t. + 6).

(e.e)

(e .10)

(e .11 )

(e.12)

(e.13)

9.2. The Governing Equations and the Finite Element Formulation

The formulation is applied sequentially to all space-time slabs Qr,Qz,...,Qn-r.The computations start with

(,rn); : , t3. (9.14)

The coefficient r used in 9.12 is obtained by a multi-dimensional generalization

of the optimal r given in [Hugh86b] for the space'time formulation of a one-

dimensional advection-diffusion problem:

, - , ,2 l l } 'h l l ) , * (*)r) ' ' '

. (e.rs), - \ \ h

, ' \ h 2 t ,

Here h is the spatial "element length", and z is the kinematic viscosity.

Remarks:

1. It can be shown that with functions piecewise constant in time, and with

the definition of r as given by equation 9.15, the steady-state solution is

independent of the time step size. F\rrthermore, as the time step size goes to

infinity, the steady-state solution obtained with functions piecewise linear in

time becomes identical to the one obtained with functions piecewise constant

in time. Therefore, one can use very large time steps to reach the steady-state

solution.

2. The last term on the left-hand-side of equation 9.12 enforces, weakly, the

continuity of the solution in time.

9.2.2 Compressible Flows

For the equations given in the remaining part of this section, repeated indices

imply summation over the range of the spatial dimension (n"a). The dynamics

which governs two-dimensional, inviscid, compressible flows are described by the

Euler equations. In terms of conservation variables, U : (p,Put,?uz,pe)", these

eouations are

L7l

where Fa's are the Euler fluxes. Alternatively, equation 9.16 can be written as

T * P : o o n o x ( o , T ) .iJt iJxt

T * o , P : o o n o x ( o . z - ) .dt

'dr t

where

(e .16)

(e.17)


An: # , , i : r , . . . , n "7 . (e .18)

write the Euler

The following boundary and initial conditions are assumed to be given:

B U : G o n l , x ( 0 , ? ) , ( 9 . 1 9 )

U(x,0) : Uo(x) on f , ) , (e.20)

where B is a general boundary condition operator (see [LeBeau] for the implicittreatment of impermeable boundary conditions), and G and Us(x) are given func-trons.

Based on a transformation of variables, U : U(V), one canequations in terms of entropy variables [Hugh86a]:

. a v . a vAo - * A1 | : 0 on f ) x (0 ,7 ) ,

ot or;

where

Ao : U ,V and A r : A rAo , ' i : I , . . . , n "4 .

(e .21)

(s.22)

As is r symmetric and positive-definite matrix, and A;'s are symmetric matrices.For the finite element formulation of the entropy variables form of the Euler

equations, we define the following function spaces:

Sh : {v/ ' lvh e [alh1o;lo"t ,vnln" e fpl(n";1","r, q(Vh) : sft) on In], (9.23)

yh : {whlwfr € [Hlh(Q)]nd"r , whlo" € lp l (o") ] ' , " r ,q(wh) : o on ls ] , (9.24)

where n6o1 is the number of degrees of freedom and q(V h ) is the boundary condition

operator for the entropy variables lShak88]. The finite element formulation of theEuler equations can be written as follows: find Vh € Sh, such that, VWh € VA

- Avh +A,a=vni ,o\- /

. awh : avh - AvhJr, *" ' {,{o

-dl dri ,

?, Jn", a't jn f ao

it + Ar a; ) d0

(e.25)

where z (a symmetric and positive-definite matrix) is the generalized SUPG oper-ator [Hugh86b], defined as follows:

+ 21,"X Qa-v.)Asff on:,,

9.3. The Clustered Element-by-Element Methodl a a

(e.26)

The term (ra-r,) is the shock capturing diffusivity [Mall85], defined as foliows:

,: (#L,r;'ffirx1-rn

. , . - , L n , " " e Y * ' L o ' 4 # t r 1 ,v d : \ m ) ,

i a v h - i a V A,n i -6 i l '

I ^ jA t i t r /2

v r - \ A V h i a v h /

T i o ' ^ s - _

Note that the contribution of the generalized SUPG operator is subtracted from

the shock capturing operator to avoid excessive diffusion around the shocks'

9.3 The Clustered Element-by-Element Method

After linearization of the fully discretized equations, the following equation system

needs be solved for the nodal values of the unknowns:

A x : b .

We rewrite 9.29 in a scaled form

A i : b ,

where

[ : 1 ry - i 4q1 - ] ,

* : W l x ,

u : w - i u .

The scaling matrix W is defined as

W : d,i,agA.

With this definition of W,diagir becomes an identity matrix'

For the flow problems considered in this chapter, the matrix A is not in general

symmetric and positive.definite. Therefore, the proposed CEBE preconditioner

(e.27)

(e.28)

(e.2e)

(e.30)

(s.31)

(e.32)

(e.33)

(e.34)

I74 Chapter 9. Clustered Element-by-Element Computations for Fluid FIow

will be used in conjunction with the GMRES method. For completeness, we brieflydescribe the GMRES method used for solving 9.30:0. Set the iteration counter m : 0, and start with an initial guess *s.

1. Calculate the residual scaled with the preconditioner matrix F:

i - : P- t (A*- - 6) . (9.35)

2. Construct the Krylov vector space :

r r \ I ^€.-, : r.l-a-----iir

l l rm l l(e.36)

J - r

g ( j ) : p - t [ e ( i - l ) _ f { f - t A " t j - t ) , " ( t ) ; " ( i ) , 2 1 j 1 n 1 " n , ( 9 . 3 7 )

/ r ) 1 ( r )e\J / - (9 .38)-

l i f ( r ) l l 'where n; r , is the d imension of the Kry lov space, and "( i ) , i :1 ,2, . . . ,? l f re, are thebasis vectors.

3. Update the unknown vector:

X m + r : x ^ l l s 5 e \ , ( 9 . 3 9 )J : 7

where s : {sj} is the solution of the equation system

Qs: z , (9 .40)

with

e : leu l : [ (p - lAe( , ; ) , p - t f r ( r ) )1 , | < i , , j 1 r l kg , (9 .41)

, : { " n } : { ( F - l A e ( o ) , - i - ) } , I - - i z - n p n . @ . 4 2 )

4. For next iteration, set rn +rnl l and go to 1.

The iterations continue until the ratio ll i^ ll I ll ie ll falls below a predeterminedvalue. It should be noted that the matrix Q is symmetric and positive-definite.Remark:

9.3. The Clustered Element-bv-Element Method

3. The convergence rate of this algorithm depends on the condition number

of the matrix p-14. Therefore We would like to select a preconditioner

that involves minimal inversion cost, and provides, within cost limitations,

an optimal representation of A. For example, the Jacobi conjugate gradient

method, in which p : d,iagi^(: I), involves minimal cost for the inversion

of F; however this representation of A is rather a poor one, and therefore it

usually takes too many iterations to converge.

With the CEBE method, the set of elements e is partitioned into subsets (clusters

of e lements) €J,J: I ,2 , . . . ,N"1 , where N"r is the number of c lusters, such that

r | (9 .43), : U r t ,

(e.44)

The global coefficient matrix A can then be expressed as

A : \ - A, . (e.45)/ . ' - " ,r - 1

with the cluster matrix A.r defined as

Ar: t A", (e.46)P F r I

where A" is the element level matrix.

Consider the factorization of the matrix (I + B.7) ,

( I + BJ) : L i lJ r , J : 1 ,2, . . . , N. r , (g '47)

where

i) t : At - !Vr, (9 '48)

and L.r and o"r are the Iower and upper triangular factors of (I + 8.71. rne CEBE

preconditioner is defined as

N"r 1

P: II i, fl u,. (e.4e)J:r J:N"t

N'r( h - � i " ," - | | " ' '


Remarks:

4. The convergence of the algorithm depends on the numbering of the clustersbut not on the numbering of the elements within each cluster. By treatingeach cluster as a super-element, we can identify the clustered element-by-element procedure as a generalization of the standard element-by-elementmethod.

5. We have the option of storing the cluster matrices and their inverses, orrecomputing them as they are needed.

9.4 Numerical Examples

We tested the proposed CEBE/GMRES method on several two-dimensional modelproblems governed by the incompressible Navier-Stokes and compressible Eulerequations. We compared the convergence rates achieved in computations usingdifferent numbers of GMRES basis vectors. In the space-time formulation of allincompressible flow problems considered, for velocity and pressure we use functionswhich are piecewise linear in time. Also for all incompressible flow problems, a verylarge time step (100,000.0) is used to reach the steady-state solution.

9.4.L Lid-driven cavity flow at Reynolds number 400 and 1000

In both of these problems the cavity has a square shape, and the Reynolds numberis based on the dimension of the cavity and the velocity of the lid. For a squaremesh with n x n elements and m x m clusters, the memory needed for the globalcoefficient matrices is approximately 36n3 f m. We note that there are six unknownsassociated with each node in space. Figure 9.1 shows (for Re : 400 and a mesh with32 x 32 elements) the residual ratio vs the number of outer iterations for variouscluster sizes and different numbers of GMRES basis vectors (nrg). It can be seenfrom Figure 9.1 that, as expected, the larger the cluster size is, the better theconvergence rate achieved. Of course, when the cluster size is equal to the numberof elements, the method reduces to a fully implicit one, and only one iteration isneeded to reach the solution. When the cluster size is equal to one, on the otherhand, the method becomes a standard element-by-element method. Regardless ofthe cluster size, the CEBE preconditioner gives better convergence rates than theJacobi preconditioner.

Figures 9.2 and 9.3 show the computed flow field for Re : 400 and 1000, respec-tively. In both cases, we use a mesh with 64 x 64 elements and 4 x 4 clusters.

9.5. Concluding Remarks

Both figures show the velocity components along the vertical and horizontal center

lines, pressure, vorticity, and stream function.

g.4.2 Incompressible flow past a circular cylinder at Reynolds number

100

With this test problem, we evaluate the performance of the CEBE method on ir-

regular meshes. The dimensions of the computational domain, normalized by the

cylinder diameter, are 30.5 and 16.0 in the flow and cross-flow directions, respec-

tively. The free-stream velocity is 0.125. Reynolds number based on the uniform

free-stream velocity and the cylinder diameter is 100. Symmetry conditions are

imposed on the upper and lower boundaries, and the traction-free condition is im-

posed at the outflovr boundary. First, to study the convergence rates achieved,

we use a finite element mesh with 1280 elements and 1360 nodes (see Figure 9'4)'

Figure 9.5 shows the residual ratio vs the number of outer iterations for various

cluster sizes and different numbers of GMRES basis vectors. Figure 9.6 shows, for

a more refined mesh (with 5400 elements, 5510 nodes, and 72 clusters), pressure!

vorticity, stream function and stationary stream function'

g.4.3 compressible flow past a circular cylinder at Mach number 3.0

In this problem, we impose symmetry conditions along the horizontal axis of the

cylinder, and use a computational domain containing only half of the cylinder' The

dimensions of the computational domain, normalized by the cylinder diameter, are

16 and 8 in the horizontal and vertical directions, respectively. The mesh consists

of 4800 elements and 4941 nodes. With four unknowns associated with each node'

the number of equations for this mesh is 19,478. Impermeable boundary conditions

are specified on the cylinder. Figure 9.7 shows, at a typical time step, the residual

ratio vs the number of iterations for various cluster sizes and different numbers

of GMRES basis vectors. Figure 9.8 shows the mesh, and pressure and density

contours for the steady-state solution'

9.5 Concluding Remarks

We have demonstrated that the clustered element-by-element preconditioners can

be efiectively used with the generalized minimal residual method to solve large-

scale compressible and incompressible flow problems. We employed these iteration

methods in conjunction with discretizations involving the entropy variable formula-

tion for compressible flows and the space-time formulation for incompressible flows.

In the CEBE method, the elements are partitioned into clusters of elements' with

177


a desired number of elements in each cluster, and the iterations are performed ina cluster-by-cluster fashion. The clustering concept is very similar, in philosophy,to the domain decomposition concept. with this approach, we can select an al-gorithm anywhere in the spectrum of algorithms ranging from the direct solutiontechnique to the standard element-by-element iteration method. The method ishighly vectorizable and parallelizable.


n k g = 5

nkg = 15

179

L

nkg = 10

nkg = 20

Figure 9.1Lid-drir.. cavity flow at R.eynolds number 400: residual ratio versus number of

orrter iterations for various cluster sizes and different numbers of GMRES basrs

vecf,ors.

l o '

l o "

t 0 '

t o '

l o '

l 0 '

l 0 '

l o "

l 0 '

l o o

l o '

l o '

l o "

l 0 '

l o r

1 0 "

1 0 "

180

pr€ ssurs

vo r t i c i i y s t r s o m f u n c l i o n

Figure 9.2Lid-driven cavity flow at Reynolds number 400: velocity components along the

vertical and horizontal center lines, pressrrre. vorticity, and strearn function.

;

;I

Chapter 9. Clustered Element-by-Element Computations for Fluid Flow

u ( o r x = u . J , v ( o t y = 0 . 5 )

- 0 . 1 - 0 . 2 0 . 0 0 . 2 1 . O 0 . E 1 . 00 , 6o . l0 . 20 - 00 . 60 . 6

9.5. Conchrding Remarks

u ( o t x = 0 . 5 )

v o r l i c i t y

Figure 9.3Lid-driven cavity flow at Reynolds

vertical and horizontal center lines.

v ( o t y - 0 , 5 )

pressure

s l r e o m f u n c l i o n

nrrmher 1000: velocity components alongpressure, vorticity. and stream function.

181

c;I

I

I


Figure 9.4Incompressible flow pa^st a circular cylinder at Reynolds number 100: the finite

element mesh used in the benchmark comprrtations (1280 elenents and 1360 nodes)


n k g = 5

183

L

t t 0 l 5

number of iterations

nkg = 15

OL

nkg = 10

t t 0 l J

number of i terat ions

nkg = 20

Figure 9.5Incompressible flow past a circular cylinder at Reynolds number 100: residual ratio

versus number of outer iterations for various cluster sizes and different numbers of

GMRES basis vectors.

l o '

l o '

t o '

1 0 '

l o '

l 0 '

l 0 '

t 0 '


p r € s s u r e

s l r e o m f u n c l i o n s t r e o m f u n c l i o n

Figure 9.6Incompressible flow past a circular cylinder at Reynolds number 100: pressure,

vorticity, streani function, and stationary stream function.

D(v o r l i c i t y v o r l i c i t y

s i o l i o n o r y s l r e o m f u n c l i o n s t o l i o n o r y s t r e o m f u n c l i o n

:>

9.5. Concludine Remarks

n k g = 5

l o n

l o '

l o '

l 0 '

l 0

nkg = 15

185

o

L

t

nkg = 10

i t 0 I t

number of iterations

nkg = 20

t o t

1 0 "

l o '

l 0 -

l 0

l 0

J l 0 1 5

number of i terat ions

Figure 9.7compressible flow past a circular cylinder at Mach number 3.0: residual ratio

versrrs number of orrter iterations for various cluster sizes and different nrtmbers of

GMRES basis vectors.

l o t r

l o ' l

l o ' '

l 0 '

l o '

1 0 "

l o '

l 0 "


m 6 s h

p r€ssu re p 1 6 s s u r e

d e n s i l y

d € n s i i y o i y = 0

d e n s i i y

>" i=co

- 3 . 0 - 2 , O

d e n s i i y o t l h € c y l i n d e r s u r f o c e

o . oX

co

! q

- 1 , O

Figure 9.8Compressible flow past a circular cylinder at Mach number 3.0: finite element

mesh (4800 elements and 4941 nodes), pressure and density.

9.6. Bibl iography 187

9.6 Bibliography

IHanSze90]

IHugh8e]

IHugh86a]

IHugh86b]

IHugh88]

ILeBeau]

lLiou9l l

IMall85]

ISaSch86]

IShak88]

ITez89]

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for computational fluid dynamics: VIII. the Galerkin/least squares method for ad-

vective diffusive equations. Comp. Meth. in Appl. Mech. and' En7., 73:173 189'

1989.

T.J.R. Hughes, L. P. Franca, and M. Mallet. A new finite element formulation for

computational fluid dynamics: I. symmetric forms of the compressible Euler and

Navier-Stokes equations and the second law of thermodynamics. Comp. Meth. i,n

Appl. Mech. and Eng., 54:223 - 234, 1986.

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tional fluid dynamics: III. the generalized streamline operator for multidimensional

advective-difiusive systems. Comp. Meth. i'n Appl. Mech. and Eng-' 58:3O5 328,

1986.

T.J.R. Hughes and G. M. Hulbert. Space-time finite element methods for ela^stody-

namics: Formulations and error estimates. Comp. Me'th. in AppI. Mech. and Eng.,

66:339 - 363. 1988.

J. LeBeau. The finite element computation of compressible flows. Master's thesis.

Univers i ty of Minnesota. 1991.

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finite element computations. Chapter 13 in Domatn Decomposition Methods forPart ia l Di f ferentzal Equat ions, edi tors R. Glowinski et a l . , SIAM, 140 -150' 1991.

M. Mallet. A Finite Element Method for Computational Fluid Dgnamics. PhD

thesis, Stanford University. 1985.

Y. Saad and M. Schultz. GMR.ES: A generalized minimai residual algorithm for

solv ing nonsymmetr ic l inear systems. SIAM J. Scient . Stat . Comput. ,7:856 869'

1986.

F. Shakib. Fi.nzte Elernent Analysis of the compress'ible Euler and Naaier- stokes

Equations. PhD thesis, Stanford University, 1988.

T. E. Tezduyar and J. Liou. Grouped element-by-element iteration schemes for

incompressible flow computations. Computer Phllsics Comm., 53:441 453, 1989'

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