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Annu, Rev, Fluid Mech. 1992.24:167-204Copyright 1992 by Annual Reviews Inc. All rights reserved

FINITE ELEMENT METHODS FORNAVIER-STOKES EQUATIONSRoland GlowinskiDepartment of Mathematics, University of Houston, Houston,Texas 77004Olivier PironneauLaboratoire dAnalyse Num6rique,Universit6 Paris 6, Paris 75005, FranceKEYWORDS:operator splitting, conjugategradient, method f characteristics,upwinding, meshrefinement

INTRODUCTIONThe main goal of this article is to address the finite element solution ofthe Navier-Stokes equations modeling compressible and incompressibleviscous flow. It is the opinion of the authors that most general and reliableincompressible viscous flow simulators arc based on finite clement method-ologies; these simulators, which can handle complicated geometries andboundary conditions, free surfaces, and turbulence effects, are well suitedto industrial applications. On the other hand, the situation is much esssatisfying concerning compressible viscous flow simulation, particularly athigh Reynolds and Mach numbers and much progress must still be madein order to reach the degree of achievementobtained by the incompressibleflow simulations.In this article, whose cope has been voluntarily limited, we concentrateon those topics combining our own work and the work of others withwhich we are familiar. The paper has been divided into two parts: In thefirst part (Sections 1 to 5) we discuss the various ingredients of a solutionmethodology or incompressible viscous flow based on operator splitting.Via splitting one obtains, at each time step, two families of subproblems:one of advection-diffusion type and one related to the steady Stokes

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168 GLOWINSKI & PIRONNEAUproblem.Their solutions are discussed with somedetails and illustratedwith the results of numerical xperiments.Thesecond art (Sections 6 to 1_0) s concernedwith compressible iscousflow calculations. As n Part I, wedescribe time discretization methods yoperator splitting and then the solution of the advection and diffusionsteps. Wehave also included a section on adaptive refinement. Again, weillustrate the presentationby the result of numerical xperiments.The authors apologize in advance o those colleagues whoseworkhasnot been mentioned ere but the scientific communityas been so prolificconcerning he topics addressedhere that keeping rack of all the relatedpublications has become formidable ask.

1. THE NAVIER-STOKES EQUATIONS FORINCOMPRESSIBLE VISCOUS FLOWUnsteady lows of incompressible viscous Newtonian luids are modeledby the following Navier-Stokes quations:

911t3t ~V211"q-(11 V)u+Vp fin f~ (momentumquation), (1.I)V- u = 0 in f) (incompressibility ondition). (1.2)In (1.1) and (1.2), f~(ca, d--2, 3 in practice) denotes theflowegion;its boundarywill be denoted by F and we shall denote by x = {xi}~- 1 ageneric point of Rd. Also, in (1.1) and (1.2) (and in the following)

(a) u = {ul}~a=l s the velocity, p is the pressure,coefficient;(b) V ~x~

dII ~1 ~ E llil)ii=1

v(> 0) is a viscositydV2= zX = ~ ~x~;i= I

=Ivl= = vv, IVvl= = Vv. Vv;

(c) ~~= ~, ~, (~~)~= v~=~ , W,~;j~ 1 OX])i=(d) f = {~}~= is a density of externalforces.


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FINITE ELEMENTS FOR NAVIER-STOKES 169Relations (1.1) and (1.2) are not sufficient to define a flow; we haveconsider further conditions such as the initial conditionu(x, 0) = uo(x) (with V" uo (1.3)

and the boundary condition(1.4)

where n denotes the unit vector of the outward normal at F. The boundarycondition (1.4) is of Dirichlet type; more complicated boundary conditionsare described in, e.g. Glowinski (1984), Bristeau et al (1985, 1987),Pironneau (1989). These authors used the following mixed boundary con-ditions

u = go on Fo, an = gl on F1, (1.5)where F0 and FI are two subsets of F satisfying F0c~Fl =ff andF0 ~ F 1 = F, and where the (stress) tensor a is defined

a = 2vD-pI with Dij = 2\Oxj + OxiJ" (1.6)Another mixed boundary condition, which occurs often in applications,

is given byu = go on Fo, V~nn -np = gl on 1-1, (1.7)

with

e-~ =(~,),=,(={vui-.}~=);(1.7) is less physical than (1,5), but like (1.5), it is quite useful to implementdownstream boundary conditions for flow in unbounded regions.The existence and possible uniqueness of solutions for problem (1.1)-(1.4) is discussed in, e.g. Temam1977), Girault & Raviart (1986),senskaya (1969), Lions (1969), Tartar (1978), and Kreiss & Lorenz (1989).SolvingEquations(1.1)-(1.4) [or (1.1)-(1.3) and (1.5), or (1.1)-(1.3)(1.7)] numerically s not trivial at all for the following reasons1. The above equations are nonlinear;2. the incompressibility condition (1.2);


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170 GLOWINSKI & PIRONNEAU3. the above equations are systems of partial differential equations,coupled through the nonlinear term (u" V)u, the incompressibility con-dition V- u = 0, and sometimes through the boundary conditions [as in

the case of (! .5)].In the following sections, we show hat a time discretization by operatorsplitting will partly overcomehe above difficulties; in particular, we shall

be able to decouple hose difficulties associated with the nonlinearity fromthose associated with the incompressibility condition.2. TIME DISCRETIZATION OF THEINCOMPRESSIBLE NAVIER-STOKES EQUATIONSBY OPERATOR SPLITTING METHODS2.1 Generalities on Operator Splitting Methods forInitial Value ProblemsWe ollow here the approach in Bristeau et al (1985, 1987) and Glowinski(1985) (see also Deanet al 1989, Glowinski1989); let us consider thereforethe following initial value problem

d~- +A(~p) = 0, q~(0) = (2.1)where A is an operator (possibly nonlinear) from a Hilbert space H intoitself, and where ~o0 e H. Suppose now hat operator A has the followingnontrivial decomposition

A = Aj+A2 (2.2)(by nontrivial, we mean hat A j and A 2 are individually simpler than A).It is then quite natural to integrate the initial value problem (2.1)numerical methods taking advantage of the decomposition property (2.2);such a goal can be achieved by the operator splitting schemes discussedbelow [for further information concerning operator splitting and relatedmethods, see Yanenko (1971), Marchuk (1975), Strang (1968), BealeMajda (1981), Leveque & Oliger (1983), Glowinski & LeTallec (1989),the references therein].The first scheme that we consider is the Peaceman-Rachford scheme[introduced by Peaceman and Rachford (1955)]. The fundamental ideabehind this scheme s quite simple and is outlined below.Let k(> 0) be ti me discretization st ep, and denote by~0"+~an approxi-mation of q~[(n +~)k], whereq~ is the solution of the initial value problem(2.1). Next, divide the time interval [nk,(n+ 1)k] into two subintervals,


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FINITE ELEMENTS FOR NAVIER-STOKES 171using the midpoint (n+ 1/2)k. Then the approximate solution ~ beingknown t nk, first compute~0n+ t/2 over Ink, (n+ 1/2)k] using a schemeofbackward Euler type with respect to A l and of forward Euler type withrespect to A2; proceed similarly over [(n+ 1/2)k, (n+ 1)k], switchingroles of A l and A 2. The following scheme(due precisely to Peaceman ndRachford) realizes this program

~0 = ~o0. (2.3)Then for n >_ 0, assuming hat qg" is known,compute uccessively q9n ~/2and ~on+~ via

~/)n+ 1/2 __ (~n -q-A 1((/9 n+ I/2)-]- ~4 2(@n) = 0, (2.4)k/2(~0n+1 __(fin+ 1/2 }- A l((p n I/2) d- .4 2((pn ) = 0. (2.5)k/2The convergence of scheme (2.3)-(2.5) has been proved in LionsMercier (1979), Godlewski (1980), and Gabay (1983) under quite generalhypotheses concerning the properties of A1 and A2; indeed, A~ and/orA 2 can be nonlinear and even multivalued. The main drawback of thePeaceman-Rachfordcheme s [as shown n e.g. Bristeau et al (1985, 1987),Glowinski (1986)] that it is not well suited (unless k is very small)simulate fas* transient phenomena nd to efficiently capture the steadystate solution of (2.1) (i.e. the solutions of A(~o)= 0) if operator stiff .In order to improve the stability properties of the Peaceman-Rachfordscheme, and more particularly its asymptotic behavior and its ability tosimulate fast transient phenomena,a so-called 0-scheme has been intro-duced by Glowinski (1985, 1986) and Bristeau et al (1985). This schemea variation of schemes discussed in Strang (1968), Beale & Majda (1981),

and Leveque & Oliger (1983) and is discussed with more details in Glow-inski & LeTallec (1989). Indeed, it is also a variant of the Peaceman-Rachford scheme described above.Let 0 be a number f the open interval (0, 1/2) [in practice 0 ~ (0, 1/3)];the 0-scheme, applied to the solution of the initial value problem (2.1)whenA = A I+A2, is described as follows:go = ~Oo. (2.6)

Then or n _> 0, rpn being knownwe compute on+, qY+~-0, qg,+ ~ as followsq~+0_~pnOk + Al(~O"+)+ Az(p") = 0, (2.7)


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172 GLOWINSKI & PIRONNEAU

~o.+ -o_ oo+oA,()+A (~P+,-o) = 0, (2.8)(1 - 20)k(Dn+1 __ f~n+ 1 --0Ok +A,(~o.+)+&(~o.+-0) (2.9)Theoretical analysis and numerical experiments show hat if 0 is properlychosen then the above schemehas muchbetter properties than the Peace-man-Rachford scheme concerning asymptotic behavior and the time inte-gration of stiff systems of differential equations. In the context of theincompressible Navier-Stokes equations, the value 0 = 1 - 1/w/~ seems tobe near optimal.Another operator splitting scheme that has been widely used (par-ticularly by the Russian applied mathematicians) is the one defined as

follows:~ = q~0. (2.10)

Then for n > 0, ~on being knownwe compute~b"+ and ~o"+ ~ by(~n+ 1 __ (jgn

k +A,(~b "+) = 0, (2.11)q)n+ 1 __ ~bn+k +A2(~o "+) = 0. (2.12)Scheme 2.10)-(2.12) is first-order accurate and quite stable; itsdrawback s that it is not well suited to capture steady state solutions,unless k is sufficiently small. A popular (and more accurate) variantof the above scheme s the one obtained by symmetrizationof (2.10)-(2.12);

it is defined by (2.10) and~n+l__ ~nk/2 +A~(q3+) = 0, (2.13)~n+l__ ~n+l

k ~-A2(q3+ ) = 0, (2.14)q"+-~a"+A,(~o"+= 0. (2.15)k/2Its application to the numerical simulation of incompressible viscous

flow is discussed in Bcale & Majda 1981).


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FINITE ELEMENTS FOR NAVIER-STOKES 1732.2 Application to the Navier-Stokes EquationsWe discuss now the application of the time discretization schemesdescribed in the above sections to the solution of the time-dependentNavier-Stokes equations [(1.1) and (1.2)], with the initial-value condition(1.3); we suppose that the boundary conditions are of the mixed ype givenby (1.7) in Section 1. IfF~ =ffwe need to have jrg0"ndF =Weshall consider application of the 0-scheme only, since it producesthe best results with regard to accuracy and convergence o steady states.Weobtain therefore the following scheme

u = u0. (2.16)Then or n _> 0 and starting from u" we solve

un+O__unO~-- ~vAu~++ VP"+= f,+o + flvAu~_ (u~. V)u~ in ~,V" u"+= 0 in fl,u.+O= g~+O n Fo,Un+1 --0 -- un+O

(1 -- 20)k

OU+ 0 OU~V~n --np "+ = g]+--flV~n n on [1,

flvAun+ 1-o+(tln+ i-o. V)tln+ 1-o = fn+ l-O

u,+ t-0 = g,~+ t-0 on Fo,

iln+ 1 __ un+ 1-0Ok

(2.17a)(2.17b)(2.17c)

+~vAu"+-Vp"+inO, (2.18a)olin+ 1--0flv~b~_n = g~. ~-o

~ll+Onp"+--c~v--~-n on F~,

Vu"+1 = Oin~,u~+ --- g~+ on F0,

(2.18b)

Asensible choice for ~ and fl is to take a = (1 - 20)/(1 - 0), fl = 0/(1with such a choice many computer subprograms are common o both thelinear and nonlinear subproblems, thereby saving a substantial amount of


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174 GLOWINSKI & PIRONNEAUcore memory. Concerning 0, numerical experiments show that 0-- 1-1/.~f~ seems o produce the best results, even in those situations where theReynolds number is large.Weobserve that using the above 0-scheme we have been able to decouplethe nonlinearity and the incompressibility in the Navier-Stokes equations[(1.1) and (1.2)]. Wenote that "+ and un+ l are obtained fr om the solutionof linear problems very close to the steady Stokes problem.To conclude this section, we would like to mention that there is prac-tically no loss in accuracy and stability by replacing (u"+ ~-0. V)un+-0 by(u~+0.V)u,+l 0in (2.18a).Remark 2.1: For 0 = 1/4, the convergence of scheme (2.16)-(2.19)proved in Fernandez-Cara & Beltran (1989).3. NUMERICAL TREATMENT OF THEADVECTION3.1 IntroductionThe following advection equation~ +u. V~b =f (3.1)is a model problem for the Navier-Stokes equations because it is a sig-nificant part of the full nonlinear operator of this equation. It also arisesas a separate problem in fluid mechanics in convection dominated prob-lems such as pollutant transport by fluids.Even though the advection equation is linear in ~b and much simplerthan the Navier-Stokes equations, it is still a challenge to scientific com-puting because it has no built in mechanism o smooth discontinuities indirections normal to the flow. Thus simple methods like the backwardEuler scheme~(~b"+l--qS")+u+ V~b"+ =f"+~ (3.2)odo not work because a space discretization of (3.2) by the finite elementmethodgives an ill-conditioned linear system if k is not small enough.In the finite element framework here are several methods to overcomethe difficulty, including, among thers, Taylor-Galerkin methods, Petrov-Galerkin or least-square Galerkin upwinding, upwinding via dis-continuous finite element approximations, and Lagrangian/Eulerianschemes based on the methodof characteristics [see Pironneau (1989) fora review]. We hall survey three methods, namely, a third-order upwinding


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FINITE ELEMENTS FOR NAVIER-STOKES 175technique, the least square Galerkin method, and the characteristic-Galer-kin method.

3.2 A Third-Order UpwindingSchemeStability analysis of finitc difference methods shows that upwindingschemes for the discretization of uV~b are much superior to centeredschemes. The same idea can be used with the finite element method:u- V~b(x) is approximated by [~b(x)-~b(y)]/2 where y = x-2u, 2third-order method of this type has been studied in Tabata & Fujima(1991) and can be described as follows.1. Choose 4 points x-2, x ~, x~, x2 on the line [x,x+u(x)]; -z, x-~

upwind of x and x~, x2 downwindof x with x-~ and x~ being thenearest upwindand downwind oints to x, respectively.2. Denote by _ 2,..., 2 their coordinates on the line on a scale such thatxi has coordinate i -= i, i - -2, - 1 ..... 2; call h = Ix- ~-xl and

[ ~o~(~- ~~)vi = l-l(,-J) , 7&0, 270 "~- i =~OE Zi (3.3)jvi

3. Define,~ ~,(x)[uvqs(x)]~I"(x)lg~7 (3.4)

It can be shown hath%[u.Vq~(x)]h= u(x)- V~b(x)+ a ] (x) + O(h4 (3.5)

Thus the approximation is fourth-order accurate if e = 0 and third-orderaccurate otherwise. If x-+~ are symmetricalwith respect to x, the approxi-mation is centered; if ~ > 0 then the approximation is upwinded. Whenthe x~ are uniformly distributed around x and ~ = 6, the finite differencescheme n Kawamura t al (1985) is recovered.To approximate the advection equation (3.1) one replaces u. VqS(x)[u" VqS(x)]~, and uses a high order explicit Runge-Kutta scheme(RK)the time derivatives. For example with RK2 quation (3.1) is approximatedby


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176 GLOWINSKI & PIRONNEAU~[~n+ i(x ) __ ~bn(x)] _[_ [u(x, t n+ 1/2). V(Dn+ ,/2(X)] h = f.+ 1/2,

(3.6)

where~bn+ 1/2 is given by1 n+ 1/2k~ [q~ (x) - q~"(x)] + [u"" Vq~"(x)]h

Ona triangulation of 1) good choices for the auxiliary points i are t heintersections nearest to x of the line [x,x+u(x)] with the edges of thetriangles. The parameter ~ can be chosen anywhere from 0 to 6.3.3 The Characteristic-Galerkin MethodThe characteristic-Galerkin methods (Benqu6 et al 1982, DouglasRussell 1982, Pironneau 1982) are derived from the analytic solution ofthe advection equation (Chorin 1973)

-- +uV~b =f in Q = ~ (0, T),4~(x, ) = 47(x),(~(x, ) =9(x,t), Vx F-(t) = {x:u(x, )" n(x)< 0, x

(3.7)(3.8)

Vte(O, T), (3.9)whereu, f, q~0, g, fl, F = c~fl are given; n is the outer unit normal vectorto F.In simple words he idea is to discretize the total derivative D~b/Dtnsteadof OO/Ot.Thus (3.7) could be diseretized in time

1 {~b,+ u"(x)k]} (3.10)~ (x)- q~"[x- =f",To justify (3.10) let {X(O}o

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FINITE ELEMENTS FOR NAVIER-STOKES 177

~ ~b[X(x, t; z), ~11,=, (x,t)+uV49(x,t), (3.13)Equation (3.7) is rewritten

~4, d~- +u" Vq5= ~ qS[X(x, ; z), z]l~=, =f(x, (3.14)3.3.1 APPROXIMATIOYN TIME By using the fact that X[x,(n+l)k;(n+ 1)k] = x, we can say that

+nv4~ {4n l(x) - 4n[Xn(x)l}, (3.15)where Xn(x) is an approximationof X[x, (n + 1)k; nk].Let us denote by X~ an O(k2) approximation of Xn(x) [the differencebetween he subscript of 3[ and the precision order is due to the fact thatXn is an approximation of X on a time interval of size k; an O(k~) schemegives a precision of O(k~+l)].

For instance, choose p and {kq}PO- ~ with Y, kq ~- k; computeXq(x) xPwithx = x;xq+~ = x q~ lln(xq)kq, q =0 .. .. . p--1. (3.16)

Wewill choose {k~} so that Xq (x)~ fL i.e. X] (f~) ~ ~. Then we derivescheme for (3.7), namely:1k((9~+~-(9"oxq) =f"+ ~, (3.17)

where ~boXdenotes the function x ~ q~[X(x)].The following result is easy to showProposition3.1:If u is regular and ifXq(O) ~ $-2for all n, scheme 3.17) is L2-stable.

is 0 (k) accurate when he distance between 2 andX[ (12) is 0 (k~).Proof in the case u" n = O.Wemultiply (3.17) by "~ and in tegrate ov er YLDenote by Iq~ theL~

normof qS; thenI~bn12 __ lff+ [k ~b"oX,)1 q~"+ (3.18)

Now y the change of variable y = Xq(x) and starting from


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178 GLOWINSKI & PIRONNEAU[ ~bnX]2 = fu ~b"[X~x)]2dx, (3.19)

it is found hat:Id?"oX~II2 = f d?"(y)2det[VX]]- y

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FINITE ELEMENTS FOR NAVIER-STOKES 179Proposition 3.2Scheme 3.24) is L2(I2) stable.The proof is done as in Proposition 3.1. Wealso haveProposition 3.3If Hob is the space of continuous piecewise affine functions on atriangulation of g2 then the L2(O)error between $"h the solution of (3.24)

and the solution ~b" of (3.17) is O(h2/k+h). Therefore the schemeO(h2/k+h+k).

Proof. Let us subtract (3.17) from (3.24) to obtain 2 erro r. Denotee~+, = q~+,_iih~bn+1, (3.27)

whereHhq5+ 1 is an interpolation in Ho~,of ~bn+ 1. Weobtain

- f~(d?"-IIh4")oXqwhdX. (3.28)From 3.28), with w~ = + ~ wein d

+(1 +clulg,~k)l .... +1so that

~ II ~ (~ +clul~,~k)l~l+C(h~+vkh). (3.29)3.3.3 IMPLEMENTATIONRObLeMS wo points remain to be clarified,namely the computation of X] (x), and the computation

I" = f, ~oX~w~ x. (3.30)Computationof (3.30):The following Gauss quadrature formula is used~" ~ Z ~[xz(")]w~(~); (3.3 )

for instance with piecewise linear elements we can take {~*} = all themidpoints of the edges and ~ = a~/3 in two dimensions (a,/4 in threedimensions) where a~ is the area (volume) of the elements that contain


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180 GLOWINSKI & PIRONNEAUComputation f X" (x)."To compute ~[Xn(~k)] one must address the following problem:

Given (k find 1 such that X"(~k) ~ T~ a triangle of the triangulation~ of f~".This is not a simple problem. A possible solution is as follows:Compute ll intcrscctions of {~, X"(~)) with the edges of the triangu-lation, starting from ~ in the dircction u"(~k) and proceeding from trianglc

to neighboring triangle until X"(~k) is reached (see Figure 1).The xp in (3.16) are these intersections; thus the numerical schemeforX"(~k) is an explicit Euler method applied within an element and whenanother element is reached the value of u~, on the new element is used. Tocompute the intersections, the barycentric coordinates {2~} of ~ may beused. First on each element computep~ such that

u(q ~) = Z #~q, ~ #, = 0, (3.32)i= l,..,d+ I i=1 .... d~l

where{q~} s in the set of the vertices of the element, q~ is its barycenter,and d is the dimensionof the space.Then lind m and p such that,V,,=,~m+P~,m~l--[,~j=O, ~;_>0, V~-. (3.33)

JTo find m we assume that rn is the index of the 2;, that is zero, thencompute p

Figure 1 Computation of XT, by following the streamline in the triangulation. On eachtriangle for a given entry point ~ (on this figure j = 2) the exit point ~z must be computed(here l - 3).


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FINITE ELEMENTS FOR NAVIER-STOKES

P - #r~" (3.34)If 2~ >__ 0, Vi, m is the correct index. Thusmost of the work goes intocomputing he/~ for all the elements. Because of round off errors, it maybe difficult at times to decide which s the next triangle that the charac-teristic will cross, for instance if X"(~k) is on a vertex; so careful pro-gramming s needed.3.4 A Least-Square Galerkin MethodConsider again the advection equation (3.17). Choose a small parameterz and a smooth function w(x, t), multiply (3.17) by w+~ ~w/~t+u.Vw),and integrate over Q -- f2 x (0, T) to obtain:

3.4.1 DISCRETIZATIONhoose a time step k and construct a quad-rangulation T~, of f~; denote tn = nk. In this way one obtains a properquadrangulation of the space time slabQ"+ = {(x, t):x~(t), t~(t", "+ )}. (3.36)

Finally define the space of piecewise linear functions in each componentof the vector (x, t)WT,+ ~ = {wh~C(Q+~): h piecewise affine i n x~, affine i n tonQT,+ 1},

(3.37)W~-- (Wh~WT~~: Wh= 0 on F}. (3.38)

Notice that this construction gives an approximation for q~ that is con-tinuous in x but has jumpdiscontinuities in time at the interfaces (t", "+ ~),therefore q~ has two values at t", which we have denoted by q~(x, t") and4~~~(x,The space-time least-square Galerkin method defines the approximatesolution ~b~+ t as a function in W~l equal to 9h on F and such that for all

,, ot, wc havc:

+ f~o[47,+-4glw~- o. (3.39)In the last integral the domainof integration fg is to be understood as


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182 (3LOWINSKI & PIRONNEAUQ;+I ~ {t = t"}. The scalar z is a positive parameter of O(h+k). Noticethe last term, which is a sort of penalty term to make sure thatq~7,+ 1 ~ q~. Thenwe have

Proposition 3.4Problem 3.39) has a unique solution.Proof:Equation 3.39) is a linear system with respect to the values of q~;,+ 1the vertices of the quadrangulation of Q"+1. There are obviously as manyequations as unknownsso we only need to check that the kernel of thesystem is zero, i.e. that f = 9 = ~0 = 0 implies ~+ ~ = 0 for all n. With

these zero data and wh = ~+ ~, (3.39) reduces

fQ.+,[~(~+~)2+2UV(~+)~Wf~,,[(~+I)2--~+~]=Oot (3.40)

Let us integrate by parts the first two terms

L =--fn.(~+)2+f~"+ f(~+~)2u.d, (3.4 1)

Now he last integral is positive because ~+ ~ is zero when u" n < 0. So(3.41) yields

~ ,+,(~+ ~ ,,(~+ + ,[(~+; 0. (3.42)

Define

then, using Schwartz inequality, (3.42) can also be written as a,2+~+2 -a2,+(b,+ -a,) 2 < 0.b~+~ 0, so (3.42) impliesthat a, = 0.Concerning convergence the following result can be found in Johnson(1989):


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Proposition 3.5FINITE ELEMENTS FOR NAVIER-STOKES 183

When r is O(h+k) and g = gh the method is O[(k+h)3/2] accurate inL2(Q).3.4.2 NUMERICALMPLEMENTATIONt every time step a new non-symmetric inear system needs to be solved; its size is twice the numberofvertices for first order elements. It can either be solved by Gauss elim-ination or an iterative method like the Generalized MinimumResidualalgorithm (GMRES) escribed in Saad & Schultz (1986). It may appearunnecessarily expensive to use a space-time formulation and indeed themethodalso works very well without it, but we have chosen to present itthis way for two reasons. First the mathematical results are available inthis formulation and second this formulation generalizes without modi-fication to movingdomains and free boundaries. The elements are slantedinstead of straight (see Tezduyaret al 1990).

4. NUMERICAL TREATMENT OF THE STOKESSTEPS4.1 GeneralitiesThe operator splitting methods discussed in Section 2 (and also othermethods) imply the solution at each time step of one or several systems ofpartial differential equations of the following type:

~u-vAu+Vp - fin f~, (4.1)V u = 0 in [2. (4.2)Using the notation of previous sections we shall take

~Uu = g0onF0, Vffnn-nP = g~ on F~, (4.3)as boundaryconditions. Here a and v are two positive constants (a ,-~ l/k)and f, go, and g~ are given functions. It can be shown hat if f, go, and g~are sufficiently smooth, then problem (4.1)-(4.3) has a unique solutionF~ ,if(if F~ = .if, i.e. F = F0, u is still unique, but p is defined to withinan arbitrary constant). Due to the incompressibility condition V-u = 0,problem (4.1)-(4.3) is a nontrivial one. However, uppose thatp is knownin L2(f~), then we obtain u via the solution of an elliptic system whosevariational formulation is given by


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184 GLOWINSKI & PIRONNEAUu~V~; ~veVowehave~f.uvdx+vf.VuVvdx=f.,.vdx

wherev0= {vlv ~[H(n)]~,von

Vg = {vlv~[H(f~)la, v = go on F0}. (4.5)Problem 4.4) can be easily solved by finite element or finite differencemethods. This observation is the basis of powerful iterative methods for

solving the Stokes problem (4.1)-(4.3). One of these methods willdiscussed in the following paragraph. Indeed these methods are sophis-ticated variants of the following very simple algorithm:pO ~ L 2(f~) is given; (4.6)

then for m ~_ O, assuming that pm s known, we computeum and pm+ viaUrn6 Vg; Vv6Vowehave~fnumvdx+v;nVumVvdx:;nfvdx

p,,~ l :pm__pV.u,n"Problem 4.7) is clearly equivalento~um- vAum : f--Vp m in ~; IIm = go on Fo,

(4.7)(4.8)

Concerning now the convergence of algorithm (4.6) (4.8), it followsfrom, e.g. Appendix3 of Glowinski (1984) that0 < p < 2~, (4.10)

thenlim {u",p"} = {u,p} in [H(~~)] d L2(~"~), (4.11)

where{u,p} is a solution of (4.1)-(4.3).Algorithm (4.6)-(4.8) maybe slow in practice, particularly for flowlarge Reynolds numbers where ~ ~ 1/k is taken very large (to follow the

V~-~ ~- gl+llpmon ~1-(4.9)


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FINITE ELEMENTS FOR NAVIER-STOKES 185fast dynamicsof such flow) and where v is small. In the following Section4.2 we shall describe a preconditioned conjugate gradient variant of algo-rithm (4.6)-(4.8) whoseconvergence properties are quite good and uniformwith respect to the values of v/~.In practice, algorithm (4.6)-(4.8), and the one to be discussed in Section4.2, are applied to finite element or finite difference, or spectral) approxi-mations of the Stokes problem(4.1)-(4.3); we shall address in Sectionthe finite element approximation of problem(4.1)-(4.3).A Preconditioned Conjugate Gradient Algorithm forSolving the Stokes Problem (4.1)-(4.3)For the mathematical justification of the preconditioned conjugate gradi-ent variant of algorithm (4.6)-(4.8) described here, see, e.g., CahouetChabard (1988), Bristeau et al (1987), Glowinski & LeTallec (1989),Glowinski (1990). The key idea (which is due to J. Cahouet) is to linearlycombine two residuals, one active for ~/v >> 1 (large Reynolds numbers),while the other is well suited to ~/v _ O, assuming that pro, Um, r% g", wm are known, compute

pro+ 1, Urn+ 1, Fro+ I, gin+ 1, Win+ I as follows:


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186 GLOWINSKI & PIRONNEAUSolve

grimtm--vAtm-- --VWm in f~; .tim = 0 on F0, v~-n = nwmon FI, (4.18)and set

?" = V.t. (4.19)Compute

pm=~tangmdx/~f~fmwtndx, (4.20)and then

pm + 1 -- mp -prow , (4.21)Urn+ 1 = ilm__pmtm, (4 22)r~+ = r~-pm?L (4.23)

Next, solveO~_ ACorn= ?m in f~; ~n-- = 0 on Fo, q3m= 0 on F~, (4.24)

and computegin+, =_ y,, __ pm(Vfm + ~m). (4.25)

lf ~f~rm+ gin+, dx/~.rmgmdx ~ ~, take p : pm+ , U= .m+ 1; ~not compute7~= fnW+lg~+l dx/ f Wg~dx, (4.26)

and thenwm+ l = ~m+ 1+ 7mW. (4.27)

Do m = m+ and go back to (4.18).Algorithm (4.12)-(4.27) has proven to be quite efficient for solvingNavier-Stokes problems at quite large Reynolds numbers. To conclude

this section, we would ike to make the following remarks:Remark4.1: In the case where Fo = F, we should replace (4.13), (4.15),(4.18), (4.24)


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FINITE ELEMENTSFOR NAVIER-STOKES 187~u-vAu = f-Vp0 in f~; u = go on F, (4.13)-ACp = r in

=on ,o,Px --o, (4.15)~_ltn--~m~lm= --Vwm in ~; fi" = 0 on F, (4.18)- A~~ = Y" in ~; O~On 0 on F, fn qS" dx = 0, (4.24)

respectively.Remark .2: Each teration of algorithm (4.12)-(4.27) requires the solutionof one elliptic system for the operator 51- vA. As already mentioned, forflow at large Reynolds numberwhere ~ ~ 1/k is large and v is small thediscrete analogues of operator ~I- vAare fairly well-conditioned matrices,making he iterative solution of these elliptic systems quite inexpensive.We lso have to solve the Poisson problems(4.15) and (4.24) [or (4.15)and (4.24)]; we shall discuss this aspect of the numerical implementationin Section 4.4.4.3 Finite Element Approximation of the Stokes Problem(4.1)-(4.3)4.3.l G~NERALTT~ESe have discussed in Sections 2 and 3 the timediscretization of the Navier-Stokes equations (1.1) and (1.2) coupledconvenient initial and boundary conditions. The finite element treatmentof the advection has also been treated in Section 3. Therefore, we still haveto address the finite element approximation of the Stokes problem [(4.1)and (4.2)]. We ssume that the boundary conditions are still those givenby (4.3). The literature concerning the finite element approximationthe Navier-Stokes equations is quite large (indeed, every issue of theInternational Journal of Numerical Methods in Fluids contains at leastone article on these topics); concentrating on books, we shall mentionGlowinski (1984), Thomasset (1981), Peyret & Taylor (1983),(1977), Cuvelier et al (1986), Girault & Raviart (1986), Pironneau (1989),and Gunzburger (1989). In this article we concentrate on triangularelements since they are easier to implement nd better suited to complicatedgeometries.It is a well accepted fact that the main difficulty related to the spaceapproximation of the Navier-Stokes equations, in the pressure-velocityformulation, is the treatment of the incompressibility condition V- u = 0.In Section 5.2 of Glowinski 1990) it was shown, using Fourier Analysis,that the mechanismroducing spurious oscillations for the discrete velocity


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188 GLOWINSKI & PIRONNEAUand pressure is the fact that some discrete variant of the operator A,defined by

Aq = -V-[(~I-vA) lVq], (4.28)too strongly damps those discrete pressure modes whose wave number isat least half the maximalwavenumberof the discrete velocity modes. Anobvious cure to this trouble is to define the discrete pressure on a (finitedifference or finite element) grid twice coarser than the one used to dis-eretize the velocity. Indeed a finite element method or the Stokes problemintroduced in Bercovier & Pironneau (1979) was (implicitly) based onidea. In the above reference one proves the convergenceof a finite elementapproximation of the Stokes problem (4.1)-(4.3) (with ~z - 0 and F0where one uses a continuous piecewise linear pressure on a triangulation~,/2, twice finer than ~; this finite element approximationwill be describedin the following Section 4.3.2.4.3.2 FUNDAMENTALISCRETE PACESWe suppose that ~ is a boundedpolygonal domainof R2. With ~ as a standard finite element triangulationof~ [sce, e.g. Ciarlet (1978) and Raviart & Thomas 1983) for this notion],and h as the maximal ength of the edges of ~, we introduce the followingdiscrete spaces (with Pk = space of the polynomials in two variables ofdegree _ 0. (4131b)

If we are in the situation associated with (4.31b) it is of fundamentalimportance to have the points at the interface of Fo and Fl(= F\Fo)vertices of ~.Three useful variants of Vh (and Voh)are obtained as follows: either

Vh = {v~,lv~eC(~) x C(O),v~lr~ei PI,VTe~-~/2}, or (4.32)V~, : {vhlvheC(~) C(~),v~,lreP~*r P~r, VT e~-h}, or (4.33)


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FINITE ELEMENTS FOR NAVIER-STOKES 189

Figure Toobtain he triangulation orthe velocity each triangle of the tri-angulationsed or the pressures dividedinto 4 subtrianglesy he mid-sides.

Vh = {v~Iv~,6C(~) C((~),vhIreP, PI,T~-~}. (4.34)In (4.32), ~/2 is the triangulation of f~ obtained from ~, by joining themidpoints of the edges of T~ ~,, as shown n Figure 2. In (4.34), ~, is the

triangulation of~ obtained from ~ by joining the vertices to the centroidin Te ~, as shown n Figure 3. Finally, in (4.33), P~r is the subspace ofP3 defined as follows

P~r = {qlq = q~ + 2~r, with q~ ~ Pi,2~R,~reP3, 9~ = 0 on ~L~r(Gr) = 1}, (4.35)

where, n (4.35), G~ s the centroid of T. A function like ~r is usually calleda bubb&function.The spaces V~ defined by (4.32), (4.33), and (4.34) have been introducedin Bercovier & Pironneau (1979), Arnold et al (1984), and Pironneau(1989), respectively.4.3.3 APPROXIMATION F THE BOUNDARYONDITIONS If the boundaryconditions are defined by

Fi#ure 3 To obtain the triangulation forthe velocity each triangle of the tri-angulation used for the pressure is dividedinto 3 subtriangles by the center.


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190 GLOWINSKI & PIRONNEAUu = g on F, with frg "ndF = 0, (4.36)

it is of fundamental importance to approximate g by gh such thatfgh n dF = 0. (4.37)Let us discuss the construction of such a g~, [we follow here Appendix3

of Glowinski (1984)]. For simplicity, we suppose that g is continuous overF. We irst definc thc space 7 Vhas7V~= {/~l/~ = v~lr, vh~ Vh}, (4.38)

i.e., ~ Vh s the space of the traces on F of those functions vh belonging toV~. Actually, if V~ is defined by (4.38), then yV~ s also the space of thosefunctions defined over F (taking their values in R2), continuous over F,and piecewise quadratic over the edges of ~ contained in F.Our problem is to construct an approximation g~, of g such that

gt,~ yV/,, frg~,n dF = 0. (4.39)If ~r~,g is the unique element of 7V~,, obtained by piecewise quadraticinterpolation ofg over F, i.e., obtained from the values taken by g at thosenodes of~ belonging to F, we usually have jr Trig" n dF ~ 0. To overcome

this difficulty we mayproceed as follows:(i) Wedefine an approximation nh of n as the solution of the followinglincar variational problem n y V~:

n~V~; Irnh/~dF = frn/~hdF, V/t~TV~. (4.40)Problem 4.40) is equivalent to a linear system whosematrix is sparse,symmetric positive definite, extremely well conditioned, and easy tocompute also, problem (4.40) needs to be solved only once].(ii) Define gh by

gh= g~g-(fv~g.ndI-/frn.n~dr)n~. (4.41)It is easy to check that (4.40) and (4.41) imply (4.39).

Now, f the boundary conditions are defined by either


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FINITE ELEMENTSFOR NAVIER-STOKES 191U = go on Fo, an = g1 on FI, (4.42)

o18uu = go on F0, v~-np = g~ on Fl, (4.43)

a well chosen variational formulation will automatically satisfy theboundary condition on F~; we shall see in the next paragraph that thereis no difficulty associated with the Dirichlet condition on F0.4.3.4 APPROXIMATIONF THE STOKES ROBLEMhe problem that weconsider first is defined by (4.1)-(4.3) with Fo = F and F~ = in (1.3);using the spaces Ph, Vhand Vow,defined by (4.29), (4.30) [or (4.32), (4.33),or (4.34)] and (4.31 a), respectively, we approximate 4.1) (4.3)Find{u,,, Ph} ~ VhXPh such that

(4.44)

~f~u~,.v~ ~+~f~w~ Vv,,

fv" uhqhdx = 0, Vqhe Ph, (4.48)Uh= g0hon F0. (4.49)

dx- fp~Vv~dx= If~v~dxja+ ~ gl~v~dF, Vv,~: Voh,

There is no particular difficulty solving the above discrete Stokes prob-

(4.47)

faY"~,qh = 0, Vq~, ~ Ph, (4.45)xu~ = g~ on F (with g~ ~ ?V~). (4.46)In (4.44) and (4.45), f~ and gh are convenient approximations of fg, respectively (see Section 43.3 for the construction of g~,).If the boundary conditions are given by (4.3) with jr, dF > 0,

approximate the problem (4.1)~4.3) by the following variant of (4.44)-(4.46):Find{uh, Ph} e Vh Ph such that


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192 GLOWINSKI & PIRONNEAUlem by discrete variations of the conjugate gradient algorithm discussedin Section 4.2.4.4 Remarks Concernin 9 the Computer Solution of theDiscrete ProblemsTo summarize, the practical implementation of the Stokes solversdescribed in Section 4.2 and 4.3 requires the solution of two ypes of ellipticproblems. If, for example, the boundary conditions for the velocity are ofthe Dirichlet type (i.e. if 0 =F), weshall hav e to solve a problem of thetype

eu-- vAu= f in ~, u = g on F,and a problem of the type

--Atp = f~ in ~, ~ = gp on F.

(4.50)

(4.51)Paradoxically, solving (4.50) is not very expensive for flows at highReynolds numbers; for such flows, the viscosity v is small, and their fastdynamicsrequires small At, i.e., large values of e. The spectral analysis

done in Dean et al (1989) and Glowinski (1990) shows that undercircumstances standard iterative methods (like overrelaxation and con-jugate gradient) will have a very fast convergence when applied to thesolution of the linear systems approximating problem (4.50). On the otherhand, solving (4.51) may be quite time consuming, particularly fordimensional problems, despite the fact that (4.51) has to be solvedthe discrete pressure space whose dimension is much smaller than thedimension of the discrete velocity space. Fortunately iterative methodssuch as multigrid and preconditioned conjugate gradient algorithms haveproven to be quite efficient for solving the discrete variants of the problem(4.51). The same conclusion holds for the other type of boundary con-ditions considered in this article. For more details and complements, seethe above references.

5. NUMERICAL RESULTSIn the above sections we have discussed three upwinding techniques andthree classes of finite element approximations. As examples of the utilityof finite element methods, we shall nowpresent somenumerical results forcomplicated geometries.5.1 Flow Around a CarThe automobile industry is interested in cars with low drag because theyrequire less gas. Also, for example, the design of the rear window s


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FINITE ELEMENTS FOR NAVIER-STOKES 193critical against raindrops coming from the road. Even though the Reynoldsnumber for this problem is high, since the flow is detached and complexbehind the vehicle, a direct simulation of Reynolds number around 1000already gives a lot of information on the main eddies and on the drag.

The result shown in Figure 4 is a pressure contour on the plane ofsymmetry of the vehicle. The computation was performed with the finiteelement method in the spaces defined by (4.29) and (4.33) and the Charac-teristic-Galerkin method. The mesh was generated by a Delaunay-Voronoimesh generator. It has approximately 50,000 vertices. The computing timefor 100 time steps is a few hours on a CRAY 2.5.2 Flow Over a Periodic Array of CylindersSome computer industries try to improve the cooling of circuit boards byintruding over the board an array of parallel rods perpendicular to thecooling-fan flow. For this problem the Reynolds number is moderate anda direct simulation is possible. After Re w 00 the flow becomes threedimensional. The program has been optimized for the Alliant FX80-8 andruns in 217 minutes for 40,000 vertices; the memory required is about 20Mbytes.The results shown in Figure 5 have been obtained with the finite elementapproximation associated with the spaces P, nd V I , efined (in Section4.3) by (4.29) and (4.32), respectively; the resulting nonlinear problemshave been solved by one step of Newtons method [see, e.g. Dennis &Schnabel (1983) for a fairly complete description of Newtons method]mixed with a step of the characteristic-Galerkin method (see Buffat 1991).

Figure 4visualized at the plane of symmetry of the car. (Courtesy of F . Hecht.)Flow around a car a1 Re = 5000 and a viscosity of 2.5 x The pressure is


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FINITE ELEMENTSORNAVIER-STOKES 1956. THE COMPRESSIBLE NAVIER-STOKESEQUATIONSThe Navier-Stokes equations describing the motion of Newtonian com-pressible perfect fluids are given by:

~3~- +V(pu) = (6.1)1-- + V" (pu u) + Vp- #Au- ~/~V(V u) (6.2)~t

~(p~)-- + V- (upE) + V. (pu) =

+V{tcVO+Ig(Vu+Vur)-~glVu]u }, (6.3)with the constitutive equation

E = ~- + 0, (6.4)where/z is the viscosity, x the thermal diffusivity, fthe external bodyforces,I the identity tensor, and 7 the adiabatic constant (= 1.4 for air).A standard set of boundary conditions are:

u, 0 given on the boundaryF; p given on F- = {x ~ F : u(x)" n(x)

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