SOME IMPLEMENTATIONS OF PROJECTION METHODS FOR NAVIER STOKES EQUATIONS

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1 SOME IMPLEMENTATIONS OF PROJECTION METHODS FOR NAVIER STOKES EQUATIONS JEAN-LUC GUERMOND Abstract. Tis paper is concerned wit te implementation of spatially discrete versions of Corin Temam s projection metods. Te empasis is put on te projection step, wic enforces incompressibility. Tree types of variational approximations are reviewed. In te first one, te projection step is solved as a div-grad problem wit velocity test functions satisfying (at first glance) a paradoxical Diriclet condition. In te second metod, te projection step is still solved as a div-grad problem but te velocity test functions satisfy a boundary condition only for te normal component. In te tird approac te projection step is solved in te form of a Poisson equation supplemented wit a Neumann boundary condition. Te first metod is sown to be legitimate and economical for finite element approximations, wereas te second is sown to be useful for spectral approximations. Te tird one is probably te easiest to implement since it avoids te problem of te mass matrix occurring in te two oters. Toug te second and tird approaces do not directly involve a inf-sup condition, tis condition is pointed out to be necessary to establis convergence and rule out possible spurious pressure. Finally some links between tese algoritms and some preconditioning tecniques of te Uzawa operator are sown. Résumé. Dans cet article on s intéresse à quelques approximations spatiales des métodes de projections du type Corin Temam. On s intéresse plus particulièrement à l étape de projection, qui sert à imposer l incompressibilité. Trois types d approximations variationnelles sont étudiées. Dans la première on résout l étape de projection sous la forme d un problème de Darcy avec des vitesses test satisfaisant une condition de Diriclet (à première vue) paradoxale. Dans la seconde approce, le problème est encore résolu sous sa forme div-grad (ie. Darcy) mais les vitesses test satisfont cette fois-ci une condition à la frontière portant uniquement sur la composante normale. Dans la troisième métode, l étape de projection est résolue sous la forme d une équation de Poisson avec une condition de Neumann. On montre que la première métode est légitime pour des approximations par éléments finis, alors que la seconde a de l intérêt aussi pour des approximations spectrales. La troisième métode est probablement la plus aisée à mettre en œuvre puisque qu elle permet d éviter l inversion d une matrice de masse qui est obligatoire pour les deux autres. Bien que les deux dernières métodes n imposent pas directement de compatibilité entre les espaces de vitesse et de pression, on montre qu une telle condition est nécessaire pour assurer la convergence de la métode. Finalement on montre quelques liens entre ces algoritmes et certaines tecniques de préconditionnement de l opérateur d Uzawa. Key words. Projection metod, Fractional step metod, Navier Stokes equations. AMS(MOS) subject classifications. 35A40, 35Q10, 65J15 1. Introduction. In tis paper we consider discrete approximations of a class of fractional step tecniques known as Corin Temam projection metods [8] [23]. Tese tecniques ave been proposed for approximating in time te unsteady incompressible Navier Stokes equations. Tey are devised to turn around te coupling between te pressure and te velocity tat is implied by te incompressibility constraint: divu = 0. Te basic idea consists in devising time marcing procedures tat uncouple viscous and Modél. Mat. Anal. Numér. M2AN, 30 (1996) Laboratoire d Informatique pour la Mécanique et les Sciences de l Ingénieur, C.N.R.S., BP 133, 91403, Orsay, France (guermond@limsi.fr) 1

2 incompressibility effects. Tese tecniques are very efficient and ave probably been te first numerical scemes enabling a cost-effective solution of tree-dimensional timedependent problems ( cf. Quartapelle [19, p. 177]). Teir simplicity and sometimes surprising efficiency render tem particularly attractive to te CFD community (see e.g. [2], [9], [11], [12], [25]). Altoug tese tecniques ave long been used for calculating steady-state solutions to Navier Stokes equations, tey are now regaining teir status as true time marcing procedures for calculating time-dependent incompressible viscous flows. Tis renewed interest for time-dependent solutions to Navier Stokes equations is prompted by te increasing capacities of computers and te success of large eddy teories wic recognize tat unsteadiness of large eddies sould be well predicted wereas smaller scales can (reasonably) be filtered. Since its initial appearance, te projection metod as been implemented wit various types of spatial approximations and te fractional step as been modified in order to improve te overall accuracy of te sceme. Toug te stability of tis metod and its modified versions can generally be proven quite easily wen space variables are continuous, te stability and convergence of teir discrete counterparts are often overlooked in te literature. For instance, te projection step may be put into two different forms. One possibility consists in solving a so-called div-grad or Darcy problem as follows (1.1) u + φ = ũ u = 0 u n Ω = 0. Te second possibility consists in obtaining from (1.1) a Poisson equation supplemented wit a omogeneous Neumann boundary condition on te pressure (1.2) 2 φ = ũ φ n Ω = 0. Altoug (1.1) and (1.2) are equivalent in some sense, teir discrete counterparts are not in general. On te one and, discrete variational approximations of te Darcy problem yield pressure equations of type B I 1 Bt Φ = F, were I is a mass matrix and B is a matrix associated wit te divergence operator. In tis case te omogeneous Neumann boundary condition on te pressure is not enforced, toug it is implicitly accounted for by te velocity tests functions wic ave te normal component vanising at te boundary. On te oter and, discrete variational approximations of (1.2) yield equations of type D Φ = F, were D is an approximation of te Laplace operator and te omogeneous Neumann boundary condition is explicitly, toug weakly, enforced by te variational formulation. At tis point, one may ask oneself wic procedure is correct? If bot are correct wat are teir respective range of application? It is 2

3 sown in tis paper tat bot approaces are correct, and eac of tem as its own advantages witin its respective functional framework. Te oter point tat is discussed in tis paper concerns te appropriate boundary condition tat sould be imposed on te end-of-step velocity. Tanks to teorem 2.1 (see below), it is clear on (1.1) tat, as far as te spatially continuous problem is concerned, only te normal component of te end-of-step velocity sould be constrained. However, wen te problem is discretized in space, te answer is no longer clear: a full Diriclet condition on te end-of-step velocity is sometimes advocated by some autors (cf. e.g. Greso and Can [12, part II]), wereas oter autors impose a, more natural, condition on te normal component of te velocity (cf. Donea et al. [9], Azaiez et al. [1]). Wic solution is correct? Wat are teir respective advantages? It is te purpose of tis paper to sow tat bot solutions are suitable if applied in te correct functional frameworks. Actually, we sow in tis paper tat te intermediate velocity and te final velocity sould be approximated in two different spaces. Tis paper is organized as follows. In 2, we review non incremental and incremental projection scemes in te space continuum. In 3, we analyze a discrete projection sceme in wic te provisional velocity and te corrected one are approximated in te same space; tat is, te end-of-step velocity satisfies a Diriclet condition. Tis sceme is sown to be efficient for finite element approximations. A discrete projection sceme wit an approximation space enforcing a boundary condition only on te normal component of te end-of-step velocity is analyzed in 4. Tis functional framework is sown to be useful for spectral approximations. In 5, te projection step is formulated as a Poisson problem supplemented wit a Neumann boundary condition. Tis tecnique is probably te easiest to implement, for it turns around a mass matrix problem tat plagues te two oters. Some generalization and convergence results are presented in 6. In 7, we sow tat te tree projection algoritms are, in some sense, equivalent to some known preconditioning tecniques of te Uzawa operator for wic te preconditioner is applied only once. 2. Preliminaries Te continuous unsteady Stokes problem. In tis paper we consider numerical approximations wit respect to time and space of te time-dependent Navier Stokes equations formulated in te primitive variables, namely velocity and pressure. However, to simplify te presentation and since we are mainly concerned wit te parabolic aspect of te problem, we restrict ourselves to te time-dependent Stokes problem (2.1) u t 2 u + p = f u = 0 u Ω = 0 u t=0 = v 0, 3

4 were f(t) is a body force, and te boundary condition on te velocity is set to zero for sake of simplicity. Te fluid domain Ω is open connected and bounded in IR d (d = 2 or 3 in practical applications). Te domain boundary Ω is assumed to be smoot; say Ω is Lipscitz and Ω is locally on one side of its boundary. In te following we work witin te classical framework of te Sobolev spaces. Te set of real functions infinitely differentiable wit compact support in Ω is denoted by D(Ω), and te set of distributions on Ω is denoted by D (Ω). As usual, L 2 (Ω) denotes te space of real-valued functions te squares of wic are summable in Ω. Te inner product in L 2 (Ω) is denoted by (, ) and 0 is te associated norm; we identify L 2 (Ω) wit its dual. H m (Ω), m 0, is te set of distributions te successive derivatives of wic, up to order m, can be identified wit square summable functions. Te space H m (Ω), equipped wit te norm u 2 m = ( m α =0 D α u 2 0) 1/2, expressed in te multi-index notation, is a Hilbert space [18]. We define H m 0 (Ω) as te completion of D(Ω) in H m (Ω), and we denote H m (Ω) te dual of H m 0 (Ω). Te incompressibility condition on te velocity leads to consider solenoidal vector fields. For tis reason, we define N(Ω) = {v D(Ω), v = 0}, and we denote by H and V te completions of N(Ω) in L 2 (Ω) d and H 1 0(Ω) d, respectively. Spaces H and V are caracterized by: (2.2) H = {v L 2 (Ω) d, v = 0, v n Ω = 0}, (2.3) V = {v H 1 (Ω) d, v = 0, v Ω = 0}. See for instance Temam [22, pp ] for a proof. In te following, te space H plays an important role by means of Teorem 2.1. Under te ypoteses on Ω stated above, we ave te ortogonal decomposition (2.4) L 2 (Ω) d = H (H 1 (Ω)/IR). Proof. See for instance Girault and Raviart [10]. If one assumes tat f L 2 (0, T ; H 1 (Ω) d ) and u 0 H, ten it is well-known tat (2.1) is well-posed (look for u L 2 (0, T ; V) C 0 (0, T ; H) and restrict te time evolution problem to L 2 (0, T ; V ) were V is te dual of V and apply Lions s teorem [18, p. 257]). Furtermore, one may verify tat p L 2 (0, T ; L 2 (Ω)/IR). It is ereafter assumed tat te data are regular enoug and satisfy all te compatibility conditions tat are needed for a smoot solution to exit. We now turn te attention to te time approximation of (2.1) by means of projection metods. To make te presentation self-contained, we begin by recalling te main features of some projection scemes Te non-incremental and incremental projection scemes. Projection metods ave been introduced by Corin [8] and Temam [23]. Tey are time marcing procedures based on a fractional step tecnique tat may be viewed also as 4

5 a predictor-corrector strategy aiming at uncoupling viscous diffusion and incompressibility effects. Te time interval [0, T ] on wic te solution is sougt is partitioned into K + 1 time steps tat are ereafter denoted by t k = k for 0 k K, were = T/K. In te algoritm originally devised by Corin and Temam, eac time step is decomposed into two substeps as follows. For eac time step, solve first (2.5) ũ k+1 u k ũ k+1 Ω = 0, 2 ũ k+1 = f k+1 ten project te provisional velocity ũ k+1 onto H; in oter words, solve u k+1 ũ k+1 + p k+1 = 0 (2.6) u k+1 = 0 u k+1 n Ω = 0. Te series (u k ) is initialized by u 0 = v 0. Te velocity ũ k+1 is a prediction of u(t k+1 ), and u k+1 is a correction of ũ k+1. One possible improvement of te algoritm above consists in predicting a better value of te provisional velocity ũ k+1 by putting te gradient of te pressure tat as been calculated at te time step t k in te rigt-and side of (2.5). Tis algoritm, ereafter referred to as te incremental form of te projection tecnique [11], consists in te following. Initialize te series (u k ) and (p k ) respectively by u 0 = v 0 and p 0 = p t=0, assuming tat p C 0 (0, T ; L 2 (Ω)/IR). For eac time step solve (2.7) ũ k+1 u k ũ k+1 Ω = 0, 2 ũ k+1 = f k+1 p k and project ũ k+1 onto H (2.8) u k+1 ũ k+1 u k+1 = 0 u k+1 n Ω = 0. + (p k+1 p k ) = 0 Note tat tis algoritm assumes more regularity tan te non incremental one since it requires an additional condition, ie. p 0 = p t=0, tat was not specified in te original Stokes problem (2.1) so tat some regularity on p as t 0 needs to be assured. Steps (2.6) and (2.8) are called projection steps since, according to teorem 2.1, tey are equivalent to u k+1 = P H ũ k+1 and eiter p k+1 = (ũ k+1 P H ũ k+1 )/ or (p k+1 p k ) = (ũ k+1 P H ũ k+1 )/, were P H is te ortogonal projection of L 2 (Ω) d 5

6 onto H. In bot cases, te velocity ũ k+1 is a prediction of u(t k+1 ) tat satisfies te correct boundary condition but is not divergence free. Tis defect is corrected by projecting ũ k+1 onto H (tis step as given its name to te metod). However, te end-of-step velocity, u k+1, does not satisfy exactly te correct boundary condition since its tangential component is not necessarily zero. Note tat fractional step tecniques (2.5) (2.6) and (2.7) (2.8) uncouple viscous diffusion and incompressibility. In practice tese tecniques require solving Helmoltz problems and performing projections onto H, wereas classical coupled tecniques usually involve a Uzawa operator: (I d σ 2 ) 1 were σ is proportional to and I d is te identity. Tis operator, called after Uzawa s algoritm in wic it is implicitly used (see for instance Temam [22, p. 138]), is non-local and ill-conditioned as te time step tend to zero (see also 7 for oter details). Wen it comes to analyzing te convergence of projection algoritms (2.5) (2.6) and (2.7) (2.8), it is sufficient to restrict te analysis to tat of te incremental algoritm, for te error equations of te non-incremental one can be put into an incremental form as follows ẽ k+1 e k ẽ k+1 Ω = 0, 2 ẽ k+1 = R k+1 p(t k+1 ) and e k+1 ẽ k+1 e k+1 = 0 e k+1 n Ω = 0. + (δ k+1 p(t k+1 )) = 0 Were R k+1 is te integral Taylor residual, and we ave defined te error functions e k = u(t k ) u k, ẽ k = u(t k ) ũ k and δ k = p(t k ) p k. As a consequence, in te following we only consider te incremental form of te projection algoritm. Global accuracy of projection scemes can be furter improved by replacing te one-step backward Euler sceme in (2.7) by a Crank Nicolson approximation as Van Kan did [25] or by a two-step backward Euler approximation. Stability and convergence of some of tese modified sceme is studied in [13], [15], [16], [20], [21], and [25], but tese considerations are out of te scope of te present paper. Te objective of te work presented erein is to bring some answers to questions concerning spatially discrete approximations of te projection step (2.6) or (2.8) Te spatially discrete unsteady Stokes problem. Let X and M be convergent, internal, and stable approximations of H 1 0(Ω) d and L 2 (Ω)/IR. It is ereafter assumed tat X and M are finite dimensional vector spaces. We define X te dual of X ; X is identical to X in terms of vector space but is equipped wit te dual norm induced be te scalar product of L 2 (Ω) d. We identify M wit its dual space, for te natural norm of M is tat of L 2 (Ω). 6

7 We now introduce te continuous bilinear form b : X M IR so tat b (u, p ) = (divu, p ), and we associate wit b te continuous linear operator B : X M and its transpose B t : M X so tat for every couple (u, p ) in X M we ave (B u, p ) = b (u, p ) and (u, Bp t ) = b (u, p ). We assume tat B : X M is onto. An important consequence of te surjectivity of B is summarized by te following well-known result wic is a corollary of Banac s closed range teorem. Lemma 2.1. Let E and F be two Hilbert spaces, and T L(E, F ). Te following propositions are equivalent: (i) T : E F is onto. (ii) T t : F ker(t ) o is one to one 1 and tere is β > 0, so tat T t p E β p E, for all p E. For a proof, te reader is referred to Brezzi [5] or Girault Raviart [10, pp ]. As a result, tere is β > 0 so tat (2.9) q M, B t q X β q 0. Te constant β is sometimes referred to as te inf-sup constant or te LBB constant (LBB being for Ladyzenskaya Babǔska Brezzi). A large coice of discrete spaces X and M satisfying suc a condition is available in te literature. A review of compatible spaces in te framework of finite elements may be found in Girault Raviart [10]. For spectral approximation see, for instance, Bernardi Maday [4]. Te null space of B playing an important rôle in te following, we set V = ker(b ), and we equip V wit te norm induced by tat of X. We also define by H = V in term of vector space and we equip H wit te norm of L 2 (Ω) d (in some sense H plays te role of te completion of V in L 2 (Ω) d ) Let us also introduce te continuous bilinear form a : X X IR so tat a (u, v ) = ( u, v ), and recall tat, tanks to te Poincaré inequality in H 1 0(Ω) d, a is X -elliptic; tat is, (2.10) α > 0, u X, a (u, u ) α u 2 1. We associate wit a te linear continuous operator A : X X so tat for all (u, v ) X X, a (u, v) = (A u, v ). In te functional framework defined above, te spatially discrete version of te time-dependent Stokes problem can be formulated as follows. For f L 2 (0, T, X ) and v 0, H, find u L 2 (0, T ; X ) C 0 (0, T ; H ) and p L 2 (0, T ; M ) so tat (2.11) du dt + A u + B t p = f B u = 0 u t=0 = v 0,, 1 Recall tat te polar set of a space V E is defined by V o = {e E, e, v = 0, v V } 7

8 were f and v are suitable approximations of f and v 0 in X. Te data, f and v 0, are assumed to be as smoot as needed, and in te rest of te paper we focus on time approximations of (2.11). Tis problem as a unique solution (u, p ), and tis solution is stable (in te appropriate norms) wit respect to te data. Since X and M are convergent and stable internal approximations of H 1 0(Ω) d and L 2 (Ω)/IR respectively, te solution to (2.11) converges in an appropriate sense to tat of te continuous unsteady Stokes problem (2.1). Our main concern now consists in approximating te time derivative in (2.11). In wat follows, we are exclusively concerned wit time approximations of problem (2.11) by means of projection tecniques similar to (2.5) (2.6) or (2.7) (2.8). 3. A full Diriclet boundary condition on te end-of-step velocity Te functional framework. In tis section we build a discrete projection algoritm in wic we take te provisional velocity ũ k and te end-of-step velocity u k in te same approximation space. In order to build an analogy between te discrete framework and its continuum counterpart, we introduce a subspace of M tat is te analogue of H 1 (Ω) L 2 (Ω). For tis purpose we define te positive bilinear form (p, q) M 1 = (B t p, B t q). According to (ii) in lemma 2.1, it is clear tat (, ) M 1 is a scalar product, and p M 1 = B t p 0 is a norm. We now define M 1 so tat M 1 = M in terms of vector space, but we equip M 1 wit te norm M 1. Furtermore, we define Y = X in term of vector space and we equip Y wit te norm of L 2 (Ω) d. Te introduction M 1 and Y is justified by Corollary 3.1. We ave te stable, ortogonal decomposition: (3.1) Y = H B t (M 1 ). Proof. Let l Y and define P H : Y H te ortogonal projection onto H. We ave (l P H l, v ) = 0 for all v in H, in oter words we also ave (l P H l, v ) = 0 for all v in V ; tat is to say, (l P H l ) V o. From (ii) of lemma 2.1, we infer tat tere is a unique p M so tat Bp t = l P H l. Furtermore, it is clear tat (P H l, Bp t ) = 0; as a result, P H l 2 0 = (l, P H l ) and Bp t 2 0 = (l, Bp t ), from wic we infer tat te decomposition is stable: P H l 0 l 0 and p M 1 l 0. Note tat te above decomposition of Y is te spatially discrete counterpart of te classical decomposition: L 2 (Ω) d = H (H 1 (Ω)/IR) Te discrete projection algoritm. Wit te functional framework introduced above, te logical implementation of te viscous step (2.7) consists in looking for ũ k+1 in X so tat (3.2) ũ k+1 u k + A ũ k+1 = f k+1 B t p k. Tis problem is well posed tanks to te X -ellipticity of A. Note tat ũ k+1, being approximated in X, satisfies te Diriclet condition ũ k+1 Ω = 0. We now turn te attention to te discrete projection step. 8

9 Te projection step of te incremental algoritm can be implemented as follows. Find u k+1 in Y and p k+1 p k in M so tat (3.3) u k+1 ũ k+1 B u k+1 = 0. + B t (p k+1 p k ) = 0 According to corollary 3.1 tis problem is well posed. Actually, te couple (u k+1, (p k+1 p k )) is te decomposition of ũ k+1 in H B(M t ); 1 tat is, u k+1 = P H ũ k+1. Note, owever, tat tis way of setting te discrete projection step may not seem te most appropriate since te velocity test functions involved satisfy a Diriclet boundary condition. In order to empasize tis point we can reformulate (3.3) as follows: find u k+1 in X and p k+1 in M so tat (3.4) v X H 1 0(Ω) d, 1 (u k+1 ũ k+1, v ) ( v, p k+1 p k ) = 0 q M L 2 (Ω)/IR, ( u k+1, q ) = 0. X being an internal approximation to H 1 0(Ω) d, u k+1 satisfies u k+1 Ω = 0 and te velocity test functions satisfy v Ω = 0, wereas it migt seem more appropriate to enforce only u k+1 n Ω = 0 and v n Ω = 0 as suggested by te continuous projection step (2.6) or (2.8). We sow in te following tat tis coice as some consequence on te condition number of te pressure operator involved te linear system (3.4) Te condition number issue. If te velocity u k+1 is eliminated from (3.3), te projection step reduces to solving te following pressure problem (3.5) B B(p t k+1 p k ) = B ũ k+1. We now turn te attention to te influence of te end-of-step boundary condition on te condition number of te pressure operator B B t. We analyze tis influence for finite element approximations and for spectral approximations. It is sown tat te full Diriclet boundary condition on te end-of-step velocity is optimal, in term of condition number of B B t, for finite element approximations, wereas it is not for spectral approximations. In te following we assume tat X and M are composed of finite elements based on a uniformly regular triangulation T of Ω, te caracteristic mes size of wic is denoted by. We also assume tat tese two spaces are uniformly compatible in te sense tat te inf-sup constant β is independent of. Recall tat for uniformly regular triangulation we ave te inverse inequality Lemma 3.1. (cf. Girault Raviart [10, p. 103]) Tere is a constant c > 0 independent of so tat (3.6) v X, v 0 c 1 v 0. 9

10 (3.7) We now give an upper bound on te condition number κ(b B t ) Teorem 3.1. Tere is a constant c > 0 so tat κ(b B t ) c β 2 2. Proof. Let p be in M and define g in M so tat g = B B t p. By setting u = B t p we ave u + B t p = 0 B u = g. (a) From tis system we deduce te bound u 2 0 = (B u, p ) g 0 p 0. Furtermore, (ii) in lemma 2.1 yields u 0 = Bp t 0 β p 0. As a result, we ave β p 2 0 g 0, from wic we deduce tat te smallest eigenvalue of B B t is bounded from below by β. 2 (b) From te system above we also deduce u 2 0 = (divu, p ), wic togeter wit te inverse inequality (3.6), yields u 0 c p 0. Furtermore, we ave g 2 0 = (divu, g ), wic for te same reason as above yields g 0 c u 0. By combining te two bounds above, we obtain tat te largest eigenvalue of B B t is bounded from above by c/ 2. (c) Te operator B B t being symmetric, its condition number is equal to te ratio of its largest to its smallest eigenvalues. Tis bound on κ(b B) t is likely to be optimal since te underlying partial differential equation is a Poisson problem supplemented wit a Neumann boundary condition, and it is known tat approximating suc a problem by finite elements yields an operator te condition number of wic is equivalent to 1/ 2. Hence, te important conclusion of tis section is tat altoug approximating te projection step by means of test functions satisfying v Ω = 0 does not seem natural, it neverteless remains optimal in te framework of finite element approximations since it yields a pressure operator wit an optimal condition number. Now assume tat Ω =] 1, +1[ 2, and let N K > 0 be two integers. Let X = IP N (Ω) H 1 0(Ω) d be te space of te polynomial velocities on Ω vanising on Ω and te partial degree of wic are less tan or equal to N. Likewise, we define M = P K (Ω) L 2 (Ω)/IR te space of te pressures on Ω tat are polynomials wit partial degree less tan or equal to K. We assume tat M is compatible wit X, and we denote by β NK te corresponding inf-sup constant (cf. Bernardi Maday [4] for a review on suc approximations). Te counterpart of lemma 2.1 for spectral approximations is te following. Lemma 3.2. (cf. Canuto Quarteroni [7] lemma 2.1, p. 73) Tere is a constant c > 0, independent of N, so tat (3.8) v n P N ([ 1, +1]), v n 0 cn 2 v n 0. 10

11 We now give an upper bound on te condition number of te pressure operator. Teorem 3.2. Tere is c > 0 so tat (3.9) κ(b B) t cn 4. β 2 NK Proof. Proceed as for te proof of teorem 3.1 In general te exponent 4 is optimal for spectral Poisson problems supplemented wit eiter Diriclet or Neumann boundary conditions. It may be furter reduced to 3 troug te action of a mass matrix if te base functions are conveniently weigted (cf. Bernardi Maday [4, Lemma 5.5, c. III]). Note tat te present functional framework is intrinsic (ie. no particular base is cosen); tat is, we deal wit operators instead of matrices. In tis context, te intrinsic counterpart of te mass matrix is te identity operator. Tis reason explains wy we obtained te exponent 4 instead of 3. Te bound (3.9) would be optimal if β NK were bounded from below by a constant wen N and K tend to infinity. Unfortunately, in general β NK tends to zero. For instance, for K = N 2 (ie. te (IP 0 N, P N 2 ) approximation) we ave (see Bernardi- Maday [4, p. 147]) (3.10) c N β N,N 2 c N. As a result, κ(b B) t cn 5 wic is no longer optimal. Tis default to te optimality is even worse if te pressure is taken in P N (Ω), for in tis case (once te spurious modes are discarded) te inf-sup constant beaves like 1/N. In conclusion, enforcing v Ω = 0 on te test functions for te projection step may not be optimal in te framework of spectral approximations. Tis default to te optimality comes from te inf-sup constant tat vanis as te number of modes increases. It is sown in te next section tat if te boundary condition on te velocity test functions is relaxed (ie. v n Ω = 0), ten te corresponding inf-sup constant can be uniformly bounded from below by a strictly positive constant. 4. Te normal trace of te end-of-step velocity enforced Te functional framework. In order to take into account te boundary condition v n Ω = 0, we introduce H div 0 (Ω) = {v L 2 (Ω) d, divv L 2 (Ω), v n Ω = 0}. Equipped wit te norm ( v divv 2 0) 1/2, H div 0 (Ω) is a Hilbert space, and we ave H 1 0(Ω) d H div 0 (Ω) L 2 (Ω) d L 2 (Ω) d H div 0 (Ω) H 1 (Ω) d, were te embeddings are dense and continuous. Let Y be a finite dimensional, internal and convergent approximation of H div 0 (Ω). Toug it may seem natural to equip Y wit te norm of H div 0 (Ω), we equip it wit te norm of L 2 (Ω) d for it will be sufficient for our purposes as sown below. A discrete divergence operator C : Y M M and its transpose C t : M Y are defined so tat for all (v, q ) in Y M we ave (C v, q ) = (divv, p ) = (v, Cp t ). We 11

12 assume also tat C : Y M is onto in te sense tat C t satisfies te following inf-sup condition (4.1) β > 0, q M C t q 0 β q 0. Now, in order to build some discrete counterpart to H, define H = ker(c ) and equip H wit te L 2 norm. Introduce also te norm q M 1 = Cq t 0, and define M 1 so tat M 1 = M in terms of vector space, and equip M 1 wit te M 1 norm. We are now in measure of establising Corollary 4.1. We ave te decomposition: (4.2) Y = H C t (M 1 ). Tis decomposition is ortogonal and stable wit respect to te L 2 norm. It is also assumed tat X Y. We denote by i : X Y te natural injection of X into Y and i t : Y Y X its transpose. Note tat i t can be identified to te L 2 projection of Y onto X ; in particular, we ave (4.3) i t v 0 v 0. Te relationsip between B and C is brougt to ligt by Proposition 4.1. C is an extension of B and i t C t = B t. Proof. (a) For all (v, q ) in X M, we ave (C i v, q ) = (divv, q ) = (B v, q ) since X Y ; tat is to say, C i v = B v for all v X. (b) From (a) we ave C i = B, from wic we easily infer i t C t = B t Te discrete projection algoritm. In te framework defined above te viscous step reads as follows: find ũ k X so tat (4.4) ũ k+1 i t u k + A ũ k+1 = f k+1 B t p k. Tis problem is well posed tanks to te ellipticity of A. Note tat u k must be projected onto X since it naturally belongs to Y. Te discrete projection step reads: find u k+1 in Y and p k+1 in M so tat (4.5) u k+1 i ũ k+1 C u k+1 = 0. + C t (p k+1 p k ) = 0 Corollary 4.1 implies tat tis linear system as a unique solution. Note tat te solvability of te viscous and projection steps (4.4) (4.5) do not require B t to be into. In oter words, te pressure is uniquely defined (in a seemingly stable manner) by te projection step (4.5). At tis point it may come to one s mind tat requiring B to be onto is an un-necessary stringent condition. As a result, one may tink of coosing X and M wit no reference to any inf-sup condition so 12

13 tat B t may possibly ave spurious modes. However, wen it comes to studying te convergence in time of te global sceme, it is sown below (see proof of teorem 6.2) tat te discrete pressure of interest is tat wic is rid of te possible spurious modes of B, t and te global stability constant on te pressure is not tat of C t : M Y but tat of B t : M X. In order to illustrate te definitions above, let us give an example in te framework of spectral approximations (tis example is borrowed from Azaiez Bernardi Grundmann [1]). Define Ω =] 1, +1[ 2, and let X = IP N (Ω) H 1 0(Ω) d be te space of te polynomial velocities on Ω vanising on Ω and te partial degree of wic are less tan or equal to N. Likewise, define Y = IP N (Ω) H div 0 (Ω) te space of te velocities wic are polynomials wit partial degree less tan or equal to N and te normal component of wic vanis on Ω. Wen te pressure space is P N (Ω)/IR and te velocity space is Y, it can be sown tat te gradient operator as tree spurious pressure modes (cf. [1, Lemma 4.1]): span L N (x), L N (y), L N (x)l N (y), were L N is te Legendre polynomial of order N. We get rid of tese unwelcome modes by defining N as te ortogonal of te modes in question in P N (Ω)/IR. Tus defined, N and Y are compatible and tere is a inf-sup constant, β, independent of N and strictly positive (cf. [1, Lemma 4.2]) so tat inf q N (divv, q ) sup β. v Y v 0 q 0 In tis spectral framework, te condition number of te pressure operator C C t satisfies te optimal bound (4.6) κ(c C t ) cn 4. Furtermore, it is sown in Bernardi Maday [4, Proposition 5.2, p. 135] tat te discrete gradient B t : N X as still four spurious modes: S = span L N(x)L N(y), L N(x)yL N(y), xl N(x)L N(y), xl N(x)yL N(y). Tis unwelcome modes sould be discarded by defining M as te ortogonal of S in N. Hence, not only te pressure space sould be rid of te tree modes span L N (x), L N (y), L N (x)l N (y) so tat C t is into, but it sould also be rid of te four spurious modes of S. In practice, te regularization of te pressure can be done as a post processing. Furtermore, it can be sown (cf. [4, p. 135]) tat te resulting stability constant of B t, β, is equivalent to 1/N. In conclusion, introducing velocity test functions satisfying te boundary condition v n Ω = 0 may be justified for some spectral approximations, for it yields a pressure operator wit an optimal condition number. Note, owever, tat tis procedure is less economical tan te previous one since it involves two approximation spaces for te velocity and it involves two gradient operators (ie. two matrices), namely B t and C t. It is not clear weter te present approac sould be preferred to te previous one as far as finite element approximations are concerned (see also Quartapelle [19, pp ] for oter details on tis tecnique). Furtermore, altoug te algoritm (4.4) (4.5) does not explicitly require B t to be into, te pressure of interest (ie. te 13

14 one on wic we ave some stability) is tat wic is rid of te spurious modes of B t (if suc modes exist). 5. Te projection step as a Poisson problem Motivation. Assume as in te previous sections tat te projection step is formulated in te form of a Darcy problem wit velocity test functions satisfying eiter a Diriclet condition or a boundary condition only for te normal component. In practical implementations we ave to coose particular bases of X (or Y ) and M. Eac of tese coices yields a mass matrix I and a matrix B associated wit te divergence operator B. For a velocity field u in X (or Y ) and a pressure field p in M, we denote by U and P te vectors of te components of u and p in te bases in question. In tis context, te projection step (3.3) (or (4.5)) yields te following linear system in terms of te pressure unknowns (5.1) B I 1 Bt (P k+1 P k ) = B Ũk+1. Toug tis approac is quite natural, for it is based on te definition of te projection operator (cf. teorem 2.1), te presence of te inverse of te mass matrix may amper its practicability in some circumstances. For instance, for finite element approximations te mass matrix is not diagonal, and te direct solution of te pressure problem may not be feasible, especially wen a large number of unknowns is involved. In practice, alternative approaces consist in lumping te mass matrix (cf. Greso and Can [12, part II] or Quartapelle [19, pp ]). Toug tis tecnique may work, no stability result as yet been proven. It is te purpose of tis section to sow tat te mass matrix problem may be circumvented if te projection step is recast in te form of a Poisson problem (1.2), as advocated in Temam [24] Te functional framework. As in te previous sections we denote by X te Hilbert space in wic te provisional velocities are approximated. Recall tat X is an internal approximation of H 1 0(Ω) d. Tanks to te Poincaré-Wirtinger inequality in H 1 (Ω)/IR, we equip H 1 (Ω)/IR wit te equivalent norm p 0. We now introduce M 1 a stable and internal approximation of H 1 (Ω)/IR and we equip it wit te norm of H 1 (Ω)/IR defined above. To make a clear difference between te H 1 norm in M, 1 te L 2 norm, and te dual norm, we introduce M and M 1. Tese spaces are identical to M 1 in terms of vector space and tey are respectively equipped wit te L 2 norm and te dual norm induced by te L 2 scalar product. Note tat we now impose M H 1 (Ω)/IR, wereas in te previous sections we only needed M L 2 (Ω)/IR. In te following, te definitions of A : X X, and B : X M remain uncanged. In order to build a discrete Poisson problem for te pressure, we introduce te continuous bilinear form d : M 1 M 1 IR so tat (5.2) (p, q ) M 1 M 1, d (p, q) = ( p, q ). 14

15 Tis bilinear form is obviously M-elliptic. 1 We associate wit d te linear continuous operator D : M 1 M 1 so tat d (p, q ) = (D p, q ) for all (p, q ) in M 1 M. 1 Te ellipticity of d implies tat D : M 1 M 1 is one to one. We introduce also te vector space Y = X + M 1 and we equip it wit te norm of L 2 (Ω) d. It is clear tat X Y. We respectively denote by i : X Y and by i t : Y X te natural injection of X into Y and its transpose. Note tat we ave te following stability inequality (5.3) v Y, i t v 0 v Te discrete projection algoritm. In a way muc similar to tat in te oter sections, te viscous step reduces to finding ũ k+1 in X so tat (5.4) ũ k+1 i t u k + A ũ k+1 = f k+1 B t p k. Te ellipticity of A guarantees tat tis problem is well posed. Note tat for reasons explained furter, u k must be projected onto X for u k naturally belongs to Y = X + M 1 (see (5.6)). We now solve te projection step (2.8) as a Poisson problem supplemented wit a omogeneous Neumann condition (5.5) D (p k+1 p k ) = B ũ k+1. Given te ellipticity of D, tis discrete problem as a unique solution. Note tat tis problem is very classical, and te CFD and applied matematics communities ave spent a lot of energy devising efficient numerical algoritms permitting to solve it. In some sense tis problem is more attractive tan its div-grad counterparts (3.3) or (4.5) since its matrix formulation does not involve te inverse of te mass matrix. Te last step of te algoritm consists in correcting te velocity. Tis is done by setting (5.6) u k+1 = ũ k+1 (p k+1 p k ). Note tat u k+1 belongs to X + M 1 wic is a subset of L 2 (Ω) d. Strictly speaking, tis velocity is not divergence free since tere is no reason for divũ k+1 to be equal to 2 (p k+1 p k ). For instance, if P 1 finite elements are used for approximating te pressure, te Laplacian of te pressure increment is a H 1 (Ω) measure, wereas te divergence of te provisional velocity is in L 2 (Ω); ence, te divergence of u k+1 is a H 1 (Ω) measure. However, we ave Proposition 5.1. u k+1 is weakly divergence free in te sense tat (5.7) q M, (u k+1, q ) = 0, and its divergence and normal trace converge to zero (in some weak sense) as 0 if ũ k+1 converges in H 1 0(Ω) d wen 0. 15

16 Proof. (a) By definition, (5.5) is equivalent to q M, ( (p k+1 p k ), q ) + (divũ k+1, q ) = 0. Furtermore, by taking te inner product of (5.6) by q we obtain q M, (u k+1, q ) = (divũ k+1, q ) ( (p k+1 p k ), q ), from wic (5.7) is easily deduced. (b) Assume for sake of simplicity tat Ω is a polygon and ũ k+1 converges in H 1 0(Ω) d to some function ũ, ten te series of functions (p k+1 p k ) converges in H 1 (Ω)/IR to te solution of te Poisson problem 2 φ = divũ/ and φ/ n Ω = 0. As a result, u k+1 converges to u = ũ φ in L 2 (Ω) d ; te limit function u is divergence free and its normal trace is zero. Indeed, te sceme (5.4) (5.5) can be put formally in a form quite similar to tat of te two oter tecniques described in te sections above. A discrete divergence operator C : Y M can be defined by (5.8) v Y, q M, (C v, q ) = (v, q ). By setting H = ker(c ), we clearly ave (5.9) Y = H C t (M 1 ). Furtermore, te relation between C and B is brougt to ligt by Proposition 5.2. C is an extension of B and i t C t = B. t Te particular coice we ave made on Y implies tat Proposition 5.3. C t is te restriction of to M. Proof. For all (v, q ) in Y M, we ave (Cq t, v ) = ( q, v ); tat is to say (Cq t q, v ) = 0. But q is in Y by definition and Cq t is in Y (= Y in terms of vector space), ence Cq t = q. Corollary 5.1. look for u k+1 in Y and p k+1 (5.10) Te projection step (5.4) (5.5) is equivalent to te problem: in M so tat u k+1 i ũ k+1 + C(p t k+1 p k ) = 0 C u k+1 = 0. Remark tat te algoritm described above does not explicitly involve any compatibility condition between X and M ; tat is, no inf-sup condition is required to ensure wellposedness of any of te fractional steps (5.4), (5.5), or (5.6). For instance, (IP 1, P 1 ) finite elements would be perfectly suited to te algoritm above. Actually a compatibility condition sows up wen we are interested in te stability and te convergence of te sceme (see proof of teorem 6.2). After all, we are interested in approximating te velocity and pressure wic are solution to (2.11), but uniqueness and stability on te pressure in (2.11) is ensured only if B t as no spurious modes. Hence, if suc modes exist, tey ave eventually to be discarded. If for some reason one is interested in solving te problem by means of continuous finite elements of degree one, a possible coice consists in using (IP 1 iso IP 2,P 1 ) finite elements (cf. Bercovier Pironneau [3]). 16

17 6. Generalization and error bounds Generalization. We sow in tis section tat te tree projection algoritms described above can be put into a unified framework. Te definitions of te spaces X and M togeter tat of te operators A and B are uncanged. We define Y a finite dimensional subspace of L 2 (Ω) d and endow Y wit te norm of L 2 (Ω) d ; we assume also tat X Y (in terms of vector space) and we denote by i te continuous injection of X into Y ; te transpose of i is te L 2 projection of Y onto X. Note tat Y is an internal approximation of L 2 (Ω) d, for X is an approximation of H 1 0(Ω) d and H 1 0(Ω) d is dense in L 2 (Ω) d. In addition, we assume tat tere is an operator C : Y M so tat C is an extension of B ; in oter words we assume tat we ave te following commutative diagrams: X B M i C Y X i t Y B t C t Since B is onto and C is an extension of B, C is necessarily onto. One consequence of tis (togeter wit (ii) of lemma 2.1) is tat C t q 0 is a norm; tis norm is ereafter denoted by q M 1 = C t q 0 and we denote by M 1 te vector space M equipped wit tis norm. Te null space of C is denoted by H. Te definitions above enable us to build a discrete counterpart of te aforementioned ortogonal decomposition L 2 (Ω) d = H (H 1 (Ω)): M (6.1) Y = H C t (M 1 ). Te tree algoritms described above can be put into te following unified form. Te diffusion step reads: find ũ k X so tat (6.2) ũ k+1 i t u k + A ũ k+1 = f k+1 B t p k. Te projection step consists in looking for u k+1 in Y and p k+1 in M so tat (6.3) u k+1 i ũ k+1 C u k+1 = 0. + C t (p k+1 p k ) = 0 Te decomposition (6.1) implies tat tis linear system as a unique solution. Tis step is a projection step in te sense tat u k+1 is te projection of ũ k+1 onto H. We now turn our attention to te convergence issue. 17

18 6.2. Te convergence issue. We prove in tis section tat te solution to te projection algoritm composed of te viscous step (6.2) and te projection step (6.3) converges in some sense to te solution of (2.1). However, since a complete proof of convergence is out of te scope of te present paper we only give te main ideas. For complete proofs of convergence in te context of te non-linear Navier Stokes equations, te reader is referred to Guermond Quartapelle [16]. For sake of simplicity, we sall compare te solution of te projection sceme (6.2) (6.3) to tat of te following coupled sceme: set w 0 = v o, and for k 0 solve (6.4) w k+1 w k B w k+1 = 0. + A w k+1 + B t q k+1 = f k+1, We sall assume tat for some k 0 large enoug, te solution to tis algoritm satisfies te following local in time convergence result: (6.5) [ ] max u(t k ) w k k p(t k ) q k 0 c( + ), k K were c is a generic constant tat does not depend on but possibly depends on te regularity of te data of (2.1). We will restrict our analysis to t k t k 0 in order to avoid a possible blow up of te error estimates at t = 0 due to a possible lack of smootness of te solution to (2.1) (and/or incompatibility of te data at t = 0 [20]). We assume also tat te extended gradient operator, C t, is somewat stable in H 1 (Ω) (a precise meaning of tis stability is given in Guermond Quartapelle [16]) so tat we ave (6.6) max k 0 k K Ct (q k+1 q) k 0 c, Of course (6.5) and (6.6) can be proved under reasonable ypoteses on X and M and on te data of (2.1); te reader is referred to Guermond Quartapelle [16] for furter details. In order to initialize our fractional step tecnique, we assume tat we ave carried out k 0 steps of (6.4); tat is to say, we set u k 0 = wk 0 and pk 0 = qk 0 and te projection algoritm (6.2) (6.3) is implemented for k k 0. Let us denote by e k = w k u k, ẽ k = w k ũ k and ɛ k = q k p k te error functions. For conciseness we introduce te notation δ t z k+1 = z k+1 z k for any function z. Te ability of te solution to (6.2) (6.3) to approximate tat of (2.1) for k 0 k K is stated in Teorem 6.1. If is small enoug, te solution to te projection sceme (6.2) (6.3) satisfies: (6.7) max k 0 k K u(tk ) u k 0 + K u(t k ) ũ k 2 1 k=k 0 1/2 c( + ). 18

19 Proof. (a) By subtracting (6.2) from te first equation of (6.4), we derive te equation wic controls te error ẽ k+1 (6.8) were we ave set ẽ k+1 i t e k + A ẽ k+1 + B t ψ k = 0 ψ k = q k+1 p k = δ t q k+1 + ɛ k. Note tat we ave used i t w k = w k since w k is in X. Furtermore, noticing tat w k+1 is in X, B w k+1 = 0, and C is an extension of B, we obtain te system of equations wic controls e k and ɛ k (6.9) e k+1 i ẽ k+1 C e k+1 = 0. + C t (ɛ k+1 ψ k ) = 0, (b) In order to obtain a bound on ẽ k+1, we take te inner product of (6.8) by 2 ẽ k+1. Using te ellipticity of A (te ellipticity constant is denoted by α) togeter wit te classical relation 2(a, a b) = a 2 + a b 2 b 2 we obtain: ẽ k ẽ k+1 i t e k α ẽ k (ẽ k+1, B t ψ k ) = i t e k 2 0. Te stability of i t yields (6.10) ẽ k α ẽ k (ẽ k+1, B t ψ k ) e k 2 0. (c) In order to obtain some control on Bɛ t k+1, we take te inner product of te first equation of (6.9) by 2 2 Cψ t, k and using te fact tat C is an extension of B we obtain 2(ẽ k+1, B t ψ k ) + 2 C t ɛ k e k+1 i ẽ k = 2 C t ψ k 2 0. Wit te elp of (6.6), te rigt and side of tis equation is bounded from above as follows. tat is to say 2 C t ψ k 2 0 = 2 δ t q k+1 + ɛ k 2 0, 2 (1 + ) C t ɛ k (1 + ) C t δ t q k+1 2 0, 2 (1 + ) C t ɛ k c 3, (6.11) 2(ẽ k+1, B t ψ k ) + 2 C t ɛ k e k+1 i ẽ k (1 + ) C t ɛ k c 3.

20 (d) We obtain some control on e k+1 by taking te inner product of (6.9) by 2e k+1 (6.12) e k e k+1 i ẽ k ẽ k = 0. (e) After summing up (6.10) + (6.11) + (6.12) we obtain e k C t ɛ k α ẽ k e k (1 + ) 2 C t ɛ k c 3. By taking te sum from k = k 0 to some integer n K we obtain e n C t ɛ n α n k=k0 ẽ k e k C t ɛ k c 2 By our particular coice of te initial conditions we ave + e k 0 0 = 0, and C t ɛ k 0 0 = 0. As a result, te discrete Gronwall lemma yields n k=k 0 [ 2 C t ɛ k 2 0] (6.13) e n n k=k 0 ẽ k c 2. Te final result is a consequence of u(t k ) u k = e k + u(t k ) w k, u(t k ) ũ k = ẽ k + u(t k ) w k, togeter wit te convergence ypotesis (6.5). Te ability of δ t u k+1 / to approximate δ t w k+1 /dt is explicited by Proposition 6.1. If is small enoug, te solution to te projection sceme (6.2) (6.3) satisfies: (6.14) max δ te k k k K K δ t ẽ k 2 1 k=k 0 1/2 c 2. Proof. (a) In a first step we control δ t ẽ k 0+1, δ t e k 0+1, and δ t ɛ k0+1. We note first tat We obtain te bound ẽ k i t e k (ẽ k 0+1, B t ψ k 0 ) 2(ẽ k 0+1, B t [δ t q k ɛ k 0 ]) 1 2 ẽk c 4. ẽ k c 2, 20

21 wic yields (6.15) δ t ẽ k c 2. Furtermore, from te projection step (6.9) we obtain e k ẽ k 0+1 0, C(ɛ t k 0+1 ψ k 0 ) 0 ẽ k /. Te first bound easily yields (6.16) δ t e k c 2. Te oter bound yields C t (ɛ k 0+1 ɛ k 0 ) 0 ẽ k / + C t δ t q k c. In oter words, we ave (6.17) Cδ t t ɛ k c. (b) Now we proceed as in te proof of teorem 6.1. For k k 0 + 1, te equation wic controls te error δ t ẽ k+1 is (6.18) δ t ẽ k+1 i t δ t e k + A δ t ẽ k+1 + B t δ t ψ k = 0, and te system of equations wic controls δ t e k and δ t ɛ k is found to be (6.19) δ t e k+1 i δ t ẽ k+1 C δ t e k+1 = 0. + C t [δ t ɛ k+1 δ t ψ k ] = 0, (c) By reasoning as in te proof of teorem 6.1, we see tat te final bound is a consequence of (a) and (b). We are now in measure of establising a convergence result on te pressure. Teorem 6.2. Te approximate pressure given by te projection sceme (6.2) (6.3) satisfies: (6.20) 1/2 K p(t k ) p k 2 0 c( + ). k=k 0 21

22 (6.21) Proof. By summing (6.8) and i t (6.9), and using te relation i t C t = B t, we obtain Bɛ t k+1 = it δ t e k+1 A ẽ k+1. Te inf-sup condition on B t : M X implies tat c 1 ɛ k 0 δ te k c 2 ẽ k+1 1. Te final bound is a consequence of tis inequality and (6.13), (6.14) togeter wit te convergence ypotesis (6.5) and te identity p(t k ) p k = ɛ k + p(t k ) q k. Note tat in order to obtain an error estimate on te pressure we ave reconstructed an approximation of te momentum equation (6.21), by combining te viscous step (6.8) and te projection step (6.9). Te discrete momentum equation (6.21) clearly sows tat te discrete gradient operator wic comes into play is not C t : M Y but B t : M X. Hence, altoug te fractional steps (6.2) (6.3) do not seem to require B t to be onto, convergence in time is ensured only if X and M satisfy a inf-sup condition. 7. Relationsip wit some Uzawa preconditioning. We sow in tis section ow te tree projection algoritms presented above can be interpreted as particular preconditioning tecniques. Our starting point is still te time-dependent Stokes problem (2.11). Assume we wis to approximate it by means of te one step backward Euler sceme (6.4). Te formulation (6.4) is classical; it couples te pressure and te velocity by means of te kinematical constraint B w k+1 = 0. One way of looking at tis problem consists in eliminating te velocity by means of a Gauss elimination, for te first pivot in te system (6.4), I d / + A, is invertible. As a result te problem (6.4) consists in finding te pressure field q k+1 so tat (7.1) B (I d A ) 1 B(q t k+1 ) = 1 B (I d A ) 1 (f k+1 + w). k In te literature, te operator B (I d A ) 1 B t is frequently referred to as te Uzawa operator; it is ereafter denoted by U. Note tat tis operator as a condition number of O(1) if is of O(1). However, is bound to tend to zero. As a result, te condition number of U is very large in practice, and iterative solutions of (7.1) are possible only if U is preconditioned. If a preconditioner were at and, one crude iterative tecnique would consist of Picard iterations. In te following we sow ow suc an algoritm can be implemented. q k+1 (7.2) Assume tat q k is te first guess of te pressure, ten te pressure increment, q, k is solution to U (q k+1 q) k = B ũ k+1, 22