AN OVERVIEW OF PROJECTION METHODS FOR INCOMPRESSIBLE FLOWS

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1 AN OVERVIEW OF PROJECTION METHODS FOR INCOMPRESSIBLE FLOWS J.L. GUERMOND, P. MINEV 2, AND JIE SHEN 3 Abstract. We discuss in tis paper a series of important numerical issues related to te analysis and implementation of fractional step metods, often referred to in te literature as projection metods, for incompressible flows. We classify a broad range of projection scemes into tree classes, namely te pressure-correction metods, te velocity-correction metods, and te consistent splitting metods. For eac class of scemes, we review te teoretical and numerical convergence results available in te literature as well as associated open questions. We summarize te essential results in a table wic could serve as a reference to analysts and practitioners. Contents. Introduction 2 2. Notations and preliminaries 3 3. Te pressure-correction scemes Te nonincremental pressure-correction sceme Te standard incremental pressure-correction scemes Te rotational incremental pressure-correction scemes Generalization Implementation Relation wit oter scemes Numerical tests 9 4. Te velocity-correction scemes Te nonincremental velocity-correction sceme Te standard incremental velocity-correction scemes Te rotational incremental velocity-correction scemes Implementation Numerical experiments Relation wit oter scemes 5 5. Consistent splitting scemes Te key idea Standard splitting sceme Consistent splitting sceme Numerical experiments Relation wit te gauge metod 9 Date: Draft version: June, Matematics Subject Classification. 65N35, 65N22, 65F5, 35J5. Key words and prases. Navier Stokes equations, Projection metods, fractional step metods, incompressibility, Finite elements, Spectral approximations. LIMSI (CNRS-UPR 352), BP 33, 943, Orsay, France (guermond@limsi.fr). Te work of tis autor as been supported by CNRS and ICES under a TICAM Visiting Faculty Fellowsip. 2 Dept. of Mat. & Stat. Sci., University of Alberta, Edmonton, Canada T6G 2G (minev@ualberta.ca). Te work of tis autor is supported by a Discovery Grant of NSERC. 3 Department of Matematics, Purdue University, West Lafayette, IN 4797, USA. (sen@mat.purdue.edu). Te work of tis autor is partially supported by NFS grant DMS-395.

2 2 J.L. GUERMOND, P. MINEV, AND JIE SHEN 6. Inexact factorization scemes Te matrix setting Inexact factorization enforces a Neumann B.C Inexact factorization is one viewpoint among many oters Inexact factorization is as accurate as PDE-projection Interpretation of convergence tests On te importance of norms A numerical illustration Effect of te inf-sup condition Te naive point of view Te functional analysis point of view Numerical illustrations Is te Neumann B.C. essential or natural? Te Neumann B.C. is a priori natural Essential Neumann B.C. limits te convergence in standard forms Te treatment of te Neumann B.C. does not matter in rotational forms 3. Open boundary conditions 33.. Pressure correction metods Inexact factorization Numerical results 35. Open questions and concluding remarks 36.. Stability of tird- and iger-order scemes: Diriclet boundary conditions Second-order scemes for problems wit open boundary conditions Stability analysis of an equivalent Stokes problem Concluding remarks 37 References 37. Introduction A major difficulty for te numerical simulation of incompressible flows is tat te velocity and te pressure are coupled by te incompressibility constraint. Te interest in using projection metods to overcome tis difficulty in time-dependent viscous incompressible flows started in te late 96 s wit te ground breaking work of Corin and Temam [8, 43]. Te most attractive feature of projection metods is tat, at eac time step, one only needs to solve a sequence of decoupled elliptic equations for te velocity and te pressure, making it very efficient for large scale numerical simulations. Altoug projection metods can be viewed as fractional/splitting step metods, te usual metodology developed for fractional step metod (see e.g., Yanenko [49]) does not apply directly, since te pressure is not a dynamic variable, i.e., te Navier Stokes equations are not of Caucy Kovalevskaya type. As a consequence, it is non-trivial to develop and analyze iger-order projection metods, and over te last tirty five years, a large body of literature as been devoted to te construction, analysis, and implementation of projection-type scemes, and te searc for optimal projection scemes as been a pre-occupation of many researcers worldwide. Over te years many valuable results as well as a significant amount of erroneous or misleading statements ave been produced. In te last couple of years, several new developments (cf. [23, 24, 22] among oters) emerged and addressed, collectively, many important teoretical and implementation issues wic ad been elusive for a long time. We now believe tat a rater clear picture of te situation is emerging, and we feel tat te time as come for a compreensive overview of projection metods. Te goal of tis paper is fourfold: (i) present te best approximation results available to date for eac class of scemes; (ii) review important implementation issues scarcely discussed in te

3 NUMERICAL ISSUES ON PROJECTION METHODS 3 literature; (iii) correct some erroneous or misleading statements made in te literature (including some made by ourselves); and (iv) list some open questions for future researc. Tis paper is organized as follows. We introduce some notations and recall some preliminary results in Section 2. In sections 3, 4 and 5 we are only concerned wit time discretizations; we present te pressure-correction, velocity-correction, and consistent splitting scemes, respectively, and we review available results for eac of tem. In Sections 6 to 9, we discuss various issues related to space discretization. Ten, in Section, we consider te problem wit open boundary conditions. Some open questions and concluding remarks are reported in te final section. 2. Notations and preliminaries We consider te time-dependent Navier-Stokes equations on a finite time interval [,T]andin an open, connected, bounded domain Ω R d (d =2, or 3) wit a sufficiently smoot boundary Γ. Since te precise definition of te functional settings is very important for stating stability and convergence results, we introduce te standard Sobolev spaces H m (Ω) (m =, ±, )wose norms are denoted by m. In particular, te norm and inner product of L 2 (Ω) = H (Ω) are denoted by and (, ) respectively. To account for omogeneous Diriclet boundary conditions we define H (Ω) = {v H (Ω) : v Γ =}. Owing to te Poincaré inequality, for v H (Ω) d, v is a norm equivalent to v.wealsoave (2.) v 2 = v 2 + v 2, v H (Ω) d. We introduce two spaces of solenoidal vector fields (2.2) H = {v L 2 (Ω) d ; v =; v n Γ =}, (2.3) V = {v H (Ω) d ; v =; v Γ =}. Te following well-known L 2 -ortogonal decomposition (2.4) L 2 (Ω) d = H H (Ω), plays a key role in te analysis of projection metods. We denote by P H te L 2 -ortogonal projector onto H, and recall tat P H is stable in H r (Ω) d if Ω is of class C r+ wit r, i.e., (see Remark.6 in [44]) (2.5) P H u C(r, Ω) u r. Since te nonlinear term in te Navier-Stokes equations does not affect te convergence rate of te splitting error, we ereafter sall mainly be concerned wit te time-dependent Stokes equations written in terms of te primitive variables, namely te velocity u and te pressure p: t u ν 2 u + p = f in Ω [,T], (2.6) u = inω [,T], u Γ = in [,T], and u t= = u in Ω, were f is a smoot source term and u H is an initial velocity field. We empasize tat all te results stated in tis paper are applicable to te full nonlinear Navier-Stokes equations provided sufficient regularity on te solution olds (see, for instance, [38, 36, 4, 9]). For te sake of simplicity, we mostly consider omogeneous Diriclet boundary conditions on te velocity. Te situation were a natural boundary condition is prescribed on a part of te boundary is considered in Section. Let t > be a time step and set t k = k t for k K =[T/ t]. Let φ,φ,...φ K be some sequence of functions in a Hilbert space E. Wedenotebyφ t tis sequence, and we define te following discrete norms: ( /2 K (2.7) φ t l2 (E) := t φ k E) 2 (, φ t l (E) := max φ k 2 ) E. k K k=

4 4 J.L. GUERMOND, P. MINEV, AND JIE SHEN We denote by c a generic constant tat is independent of ɛ, t, and ( being te messize), but possibly depends on te data and te solution. We sall use te expression A B to say tat tere exists a generic constant c suc tat A cb. 3. Te pressure-correction scemes Pressure-correction scemes are time-marcing tecniques composed of two substeps for eac time step: te pressure is treated explicitly or ignored in te first substep and is corrected in te second one by projecting te provisional velocity onto te space H. i.e., H. 3.. Te nonincremental pressure-correction sceme. Te simplest pressure-correction sceme as originally been proposed by Corin and Temam [8, 43]. Using te implicit Euler time stepping, te algoritm reads as follows: Set u = u,tenfork compute (ũ k+,u k+,p k+ ) by solving (3.) t (ũk+ u k ) ν 2 ũ k+ = f(t k+ ), ũ k+ Γ =. { t (3.2) (uk+ ũ k+ )+ p k+ =, u k+ =, u k+ n Γ =. Te first substep accounts for viscous effects and te second one accounts for incompressibility. Te second substep is usually referred to as te projection step, for it is a realization of te identity u k+ = P H ũ k+. From te point of view of accuracy, te following olds Teorem 3.. Assuming tat (u, p), solving (2.6), is sufficiently smoot, te solution of (3.) (3.2), satisfies te following error estimates: u t u t l (L 2 (Ω) d ) + u t ũ t l (L 2 (Ω) d ) c(u, p,t) t, p t p t l (L 2 (Ω)) + u t ũ t l (H (Ω) d ) c(u, p,t) t /2. Proof. See, for instance, Sen [38] and Rannacer [36]. Remark 3.. (i) From (3.2), we observe tat te boundary condition p k+ n Γ = is enforced on te pressure. Tis artificial Neumann boundary condition induces a numerical boundary layer tat prevents te sceme to be fully first-order on te te velocity in te H -norm and on te pressure in te L 2 -norm; see Rannacer [36]. (ii) Tis sceme as an irreducible splitting error of order O( t). Hence, using a iger-order time stepping sceme for te operator t ν 2 does not improve te overall accuracy Te standard incremental pressure-correction scemes. Since te pressure gradient is obviously missing in (3.), it was (probably first) observed by Goda in [2] tat adding an old value of te pressure gradient, say p k, in te first substep and ten accordingly correcting te velocity in te second substep increases te accuracy. Tis idea was made popular by Van Kan wo proposed a second-order incremental pressure-correction sceme in [46]. Using te Backward Difference Formula of second-order (BDF2) 2 to approximate te time derivative, te incremental pressure-correction sceme reads as follows (3.3) 2 t (3ũk+ 4u k + u k ) ν 2 ũ k+ + p k = f(t k+ ), ũ k+ Γ =, Te analysis of [46] is owever rater formal insofar as it sows tat te error is of c() t 2, te constant c() being a mes-dependent factor tat beaves like O(/ 2 ) 2 Te coice of a particular time discretization is not important, trougout te paper, we sall use BDF scemes altoug Adams scemes are perfectly acceptable

5 NUMERICAL ISSUES ON PROJECTION METHODS 5 (3.4) { 2 t (3uk+ 3ũ k+ )+ (p k+ p k )=, u k+ =, u k+ n Γ =. Te second substep is again a projection since it is equivalent to u k+ = P H ũ k+. For reasons tat will become clear later, we ereafter refer to tis algoritm as te incremental pressure-correction sceme in standard form, te term incremental will be omitted wen no confusion can arise. Te sceme needs to be initialized wit (ũ,u,p ) and we make te following ypotesis Hypotesis 3.. (ũ,u,p ) is computed suc tat te following estimates old: u( t) ũ c t 2, u( t) ũ c t 3/2, p( t) p c t. Note tat te above ypotesis olds (cf. [24]) if (ũ,u,p ) are computed as follows. Set u = u and p = p(), were p() is determined by using (2.6) at t =, and evaluate (u, ũ,p ) from te following first-order pressure-correction sceme: t (ũ u ) ν 2 ũ + p = f(t ), ũ Γ =, and t (u ũ )+ (p p )=, u =, u n Γ =. Te accuracy of te above algoritm is stated in te following Teorem 3.2. Under te Hypotesis 3., if te solution to (2.6) is smoot enoug in space and time, te solution to (3.3) (3.4) satisfies te following error estimates: u t u t l 2 (L 2 (Ω) d ) + u t ũ t l 2 (L 2 (Ω) d ) c(u, p,t) t 2, p t p t l (L 2 (Ω)) + u t ũ t l (H (Ω) d ) c(u, p,t) t. Proof. See Sen [4] for te semi-discrete case using te Crank-Nicolson time stepping, E and Liu [9] for an asymptotic analysis in a periodic cannel, and Guermond [6] for te fully discrete case using BDF2 to marc in time. Tese results are valid in fairly general domains provided te H 2 regularity of te Stokes operator olds. Remark 3.2. (i) Toug te sceme (3.3) (3.4) is second-order accurate on te velocity in te L 2 -norm, it is plagued by a numerical boundary layer tat prevents it to be fully second-order on te velocity in te H -norm and on te pressure in te L 2 -norm. Actually, from (3.4) we observe tat (p k+ p k ) n Γ = wic implies tat (3.5) p k+ n Γ = p k n Γ = p n Γ. It is tis non-pysical Neumann boundary condition enforced on te pressure tat introduces te numerical boundary layer referred to above and consequently limits te accuracy of te sceme. (ii) Teorem 3.2 is expected to old if te algoritm is implemented wit any A-stable secondorder time stepping. Since tis sceme as an irreducible splitting error of O( t 2 ), using a iger tan tan second-order time stepping for approximating te operator t ν 2 does not improve te overall accuracy. A related aspect of tis sceme is studied by Strikwerda and Lee [42] wo used a normal mode analysis in te alf-plane and sowed tat te pressure approximation in a standard pressure-correction sceme can be at most first-order accurate.

6 6 J.L. GUERMOND, P. MINEV, AND JIE SHEN 3.3. Te rotational incremental pressure-correction scemes. To overcome te difficulty caused by te artificial pressure Neumann boundary condition (3.5), Timmermans, Minev and Van De Vosse proposed in [45] to sligtly modify te algoritm as follows. Wile retaining te viscous step (3.3) uncanged (3.6) 2 t (3ũk+ 4u k + u k ) ν 2 ũ k+ + p k = f(t k+ ), ũ k+ Γ =, tey proposed to replace te second step (3.4) by { 2 t (3.7) (3uk+ 3ũ k+ )+ φ k+ =, u k+ =, u k+ n Γ =. (3.8) φ k+ = p k+ p k + ν ũ k+ To understand wy te modified sceme performs better, we take te sum of (3.6) and (3.7). Noticing from (3.7) tat ũ k+ = u k+,weobtain { 2 t (3.9) (3uk+ 4u k + u k )+ν u k+ + p k+ = f(t k+ ), u k+ =, u k+ n Γ = wit u k+ n Γ =. We observe from (3.9) tat n p k+ Γ =(f(t k+ ) ν u k+ ) n Γ, wic, unlike (3.5), is a consistent pressure boundary condition. Te splitting error now manifests itself only in te form of an inexact tangential boundary condition on te velocity. In view of (3.9), were te operator plays a key role, te algoritm (3.6) (3.7) (3.8) is referred to in [2, 24] as te incremental pressure-correction sceme in rotational form. Teorem 3.3. Assume tat te initialization Hypotesis 3. olds. Provided te solution to (2.6) is smoot enoug in time and space, te solution (u k, ũ k,p k ) to (3.6) (3.7) (3.8) satisfies te estimates: u t u t l 2 (L 2 (Ω) d ) + u t ũ t l 2 (L 2 (Ω) d ) t 2, u t u t l2 (H (Ω) d ) + u t ũ t l2 (H (Ω) d ) + p t p t l 2 (L 2 (Ω)) t 3 2. Proof. See Guermond and Sen [24]. Remark 3.3. In [6], Brown, Cortez and Minion sowed, using a normal modes analysis in a semi-infinite periodic cannel, tat te pressure approximation in te rotational formulation of te incremental pressure-correction algoritm is second-order accurate. Numerical experiments reported in [24] sow tat tis result is valid in a periodic cannel only, and tat te convergence rate of 3 2 for te pressure is likely to be te best possible for general domains by using rotational incremental pressure-correction algoritms. In general, te normal modes analysis cannot be used to prove convergence estimates wen more tan one space direction is not periodic. For instance, te normal mode analysis cannot account for sarp corners in polygonal domains Generalization. Te above algoritms generalize to a large class of time-marcing algoritms. For instance, assuming v to be a smoot function, denote by t (β qv k+ q j= β jv k j ) te qt-order backward difference formula (BDFq) tat approximates t v(t k+ ). To simplify te notation, for any sequence φ t := (φ,φ,...) we set q (3.) Dφ k+ = β q φ k+ β j φ k j. In particular, (3.) Dv k+ = j= { v k+ v k if q =, 3 2 vk+ 2v k + 2 vk if q =2.

7 NUMERICAL ISSUES ON PROJECTION METHODS 7 Likewise, we denote by r (3.2) p,k+ = γ j p k j te r-t order extrapolation for p(t k+ )werep(t) is a smoot function. In particular, if r =, (3.3) p,k+ = p k if r =, 2p k p k if r =2. Now, te pressure-correction scemes can be rewritten into te following form: ( q (3.4) t β q ũ k+ β j u k j) ν 2 ũ k+ + p,k+ = f(t k+ ), ũ k+ Γ =, j= j= (3.5) { βq t (uk+ ũ k+ )+ φ k+ =, u k+ =, u k+ n Γ =. (3.6) φ k+ = p k+ p,k+ + χν ũ k+, were χ is a user-defined coefficient tat may be equal to or. Te coice χ = yields te standard forms of te algoritm, wereas χ = yields te rotational forms. Remark 3.4. If one cooses r = q, ten te formal consistency error for te velocity in H - norm (resp. te pressure in L 2 -norm) is of order q (resp. of order r = q ). Stability and convergence results are only available for (q, r) =(, ) and (2, ). If one cooses r = q, ten te formal consistency errors for te velocity in H -norm and te pressure in L 2 -norm are bot of te same order. However, stability and convergence results are only available for q = r =. Te issues related to algoritms using r 2 are discussed in Implementation. One question often raised in te literature and often clouded in controversy is: wic of ũ k+ or u k+ is te correct velocity? It is often argued tat te end-of-step velocity, namely u k+, sould be te one to be retained in actual computations since it is divergence free. However, tis argument is biased since, altoug u k+ is divergence free, its tangential trace is not enforced to be zero. Hence, te situation is te following: u k+ is divergence free but does not satisfy te appropriate boundary condition, wile ũ k+ satisfies te Diriclet condition but is not divergence free. Wen looking at Teorems 3.2 and 3.3, we realize tat u t and ũ t yield te same error estimates; tat is, from te accuracy point of view tere is no objective reason for preferring one field to te oter. From te implementation point of view, tere are two arguments in favour of eliminating u t. First, as argued in Guermond [5], wen implementing te metod using finite elements and solving te projection step as a weak Poisson problem, te discrete field u k+ is discontinuous at te interface between elements; ence, u k+ is an awkward quantity to compute. Second, te field u t can be entirely removed from te algoritm by simple algebraic manipulations. Indeed, from (3.5) it is clear tat for j k β j t (uk j ũ k j )+ βj β q φ k j =. Hence, substituting ũ k j into (3.4) yields te following algoritm ( D (3.7) tũk+ ν 2 ũ k+ + p,k+ + q j= β j β q φ k j) = f(t k+ ), ũ k+ Γ =,

8 8 J.L. GUERMOND, P. MINEV, AND JIE SHEN (3.8) 2 φ k+ = βq t ũk+, n φ k+ Γ =. (3.9) φ k+ = p k+ p,k+ + χν ũ k+, were Dũ k+ is defined in (3.). It is now clear tat u t is a field tat one can completely avoid to compute. Wen solving te nonlinear equations, te above conclusion needs to be tempered since one faces te following alternative: wic of ũ k and u k sould be used to compute te nonlinear term? Tree semi-implicit forms of te nonlinear term are usually advocated: te so-called skewsymmetric form (introduced by Temam [43]), te so-called divergence form, and te rotational form. Tese forms yield unconditional stability witout requiring te approximate advection field to be exactly divergence-free. It is sown in [2] tat using ũ k for te advection field does not spoil te overall splitting error of te algoritm. Te same result as been proved to old in [] if a metod of caracteristics is used to evaluate te nonlinear term using ũ k. If instead of ũ k te field u k is used to compute te caracteristics, te asymptotic error estimates ave been proved to remain uncanged in [7]. Numerical tests reported in [7] sow owever tat, wen using a metod of caracteristics, te error is somewat smaller if instead of ũ k te field u k is used to approximate te caracteristics Relation wit oter scemes. We sow in tis section tat a sceme introduced by Kim and Moin in [28] is equivalent to te rotational form of te pressure-correction metod up to an appropriate canges of variables Te Kim and Moin sceme; strong setting. Altoug te sceme originally proposed in [28] uses te Crank Nicolson time stepping, we encefort adopt te BDFq approximation of t u(t k+ ) to simplify te presentation. Tis coice does not cange our conclusions. Ten, te sceme proposed in [28] can be written as follows: Initialize adequately (u j ) j=,...,q,tenfor k q compute û k+ solving ( (3.2) t β q û k+ q β j u k j) ν 2 û k+ = f(t k+ ), j= ten correct û k+ by computing u k+ and ψ k+ as follows { βq t (3.2) (uk+ û k+ )+ ψ k+ =, u k+ =, u k+ n Γ =, û k+ Γ = t β q ψ,k+ Γ, were ψ,k+ is some approximation of ψ(t k+ ), ψ(t) being a quantity related to te pressure p(t) via p(t) =ψ(t) ν t/β q 2 ψ(t). Altoug in [28] only te case ψ,k+ = ψ k is considered, te following coices can be used (eac of tem being labelled by an integer r): if r =, (3.22) ψ,k+ = ψ k if r =, 2ψ k ψ k if r =2. Te problem (3.2) is solved in practice as a Poisson equation supplemented wit te Neumann boundary condition n ψ k+ Γ = n ψ,k+. Te fact tat te viscous step involves te trace of a gradient as a Diriclet condition renders te metod quite inconvenient for finite element discretization. As a result, successful implementations of tis metod are only reported wit spectral or finite difference approximations were te trace of derivatives are easily available. We now sow tat by rewriting te above algoritm in an adequate L 2 setting, we recover te rotational form of te pressure-correction algoritms described above.

9 NUMERICAL ISSUES ON PROJECTION METHODS Te Kim Moin sceme; L 2 weak setting. We introduce te following canges of variables (3.23) (3.24) (3.25) ũ k+ =û k+ t β q ψ,k+, p k+ = ψ k+ ν t β q 2 ψ k+, p,k+ = ψ,k+ ν t β q 2 ψ,k+. Te boundary condition in (3.2) implies ũ k+ Γ =. Moreover, using (3.23) to substitute û k+ into te momentum equation (3.2) and taking into account (3.25), we obtain te following boundary value problem for te new velocity ũ k+ : ( (3.26) t β q ũ k+ q β j u k j) ν 2 ũ k+ + p,k+ = f(t k+ ), ũ k+ Γ =. j= By inspecting (3.2) and (3.23), we observe tat u k+ and ũ k+ differ by a gradient, ence, it is convenient to introduce a quantity φ k+ suc tat (3.27) { βq t (uk+ ũ k+ )+ φ k+ =, u k+ =, u k+ n Γ = Subtracting (3.25) from (3.24), we get p k+ p,k+ = ψ k+ ψ,k+ ν t β q 2 (ψ k+ ψ,k+ ). Ten, taking te divergence of (3.2) and using (3.23), we obtain 2 (ψ k+ ψ,k+ )= βq t ũk+. Tat is to say p k+ p,k+ +ν ũ k+ = ψ k+ ψ,k+. Moreover, substituting (3.23) into (3.2), we derive β q t (uk+ ũ k+ )+ ( ψ k+ ψ,k+) =. By comparing (3.27) and te above equation, we infer φ k+ = ψ k+ ψ,k+. Tis means tat p k+ and ψ k+ are related by (3.28) φ k+ = p k+ p,k+ + ν ũ k+. Note finally tat (3.24) and (3.25) togeter wit (3.22) imply if r =, (3.29) p,k+ = p k if r =, 2p k p k if r =2. In conclusion, te algoritm (3.2) (3.2) (3.22) is equivalent to (3.26) (3.27) (3.28) (3.29) up to te cange of variables (3.23) (3.24) (3.25). In oter words, for q = andr =, te algoritm is equivalent to te Corin Temam algoritm (3.) (3.2). Te original Kim Moin sceme, corresponding to q = 2andr =, is equivalent to te incremental pressure-correction sceme in rotational form (3.6) (3.7) (3.8) Numerical tests. We illustrate in tis section te convergence properties of te pressurecorrection algoritm using BDF2 to marc in time and te first-order extrapolation of te pressure, i.e., p,k+ = p k.

10 J.L. GUERMOND, P. MINEV, AND JIE SHEN Numerical results wit spectral approximation. We first consider a square domain Ω = ], [ 2 wit Diriclet boundary conditions on te velocity. A Legendre-Galerkin approximation [39] is used in space. Denoting by P N te space of polynomials of degree less tan or equal to N, we approximate te velocity and te pressure in P N P N and P N 2 respectively. We take te exact solution (u, p) of (2.6) to be (3.3) u(x, y, t) =π sin t(sin 2πy sin 2 πx, sin 2πx sin 2 πy), p(x, y, t) =sintcos πx sin πy. Ten te source term f is given by f = u t 2 u + p. In te computations reported erein, we take N = 48 so tat te spatial discretization errors are negligible compared wit te time discretization errors Figure. Pressure error field at time t = in a square: left, standard form; rigt, rotational form. In Figure, we sow te pressure error field at T = for a typical time step using te standard and te rotational forms of te algoritm. We observe tat for te standard form of te algoritm, a numerical boundary layer appears on te two boundaries {(x, y) :x (, ),y = ±} were te exact pressure is suc tat n p ( n p = on te oter two boundaries). For te rotational form, tere is no numerical boundary layer, but we observe large spikes at te four corners of te domain. Tis test suggests tat te divergence correction of te rotational form successfully cured te numerical boundary layer problem. However, te large errors at te four corners degrade te global convergence rate of te pressure approximation. To better understand wy tere are localized large errors at te corners of te domain, we ave also implemented te standard and rotational forms of te pressure-correction sceme in a periodic cannel Ω = (, 2π) (, ). Te cannel is periodic in te x direction and te velocity is subject to a Diriclet boundary condition at y = ±. We coose te same exact solution (u, p) as tat given above, and we use a Fourier-Legendre spectral approximation wit modes guaranteeing tat te spatial discretization errors are negligible compared wit te time discretization errors. In Figure 2, we sow te pressure error field at T = for a typical time step. Te main difference between te problem set in te square domain and tat set in te periodic cannel is tat te former as corner singularities wile te latter does not. Tus, it can be conjectured tat te large errors occurring at te corners of te square domain are due to te lack of smootness of te domain. Tis conclusion is confirmed by te numerical experiments using mixed finite elements reported in te next subsection.

11 NUMERICAL ISSUES ON PROJECTION METHODS x Figure 2. Error field on pressure at time t = in a cannel: left; standard form; rigt, rotational form Numerical results wit P 2 /P finite elements. To furter assess te influence of te smootness of te domain boundary on te accuracy of te BDF2 rotational pressure-correction metod, we ave performed convergence tests using P 2 /P finite elements. Te tests are performed using te following analytical solution (3.3) u =(sin(x + t)sin(y + t), cos(x + t)cos(y + t)), p =sin(x y + t), in te square domain ], [ 2 and in te circular domain {(x, y); x 2 + y 2.5}. We sow in figure 3 te error fields on te pressure at time T = for te square and te circular domains. Te messize is =/4 and t =.625. Te two fields are represented using te same vertical scale. Te pressure field on te circular domain is free of numerical boundary layer, wereas large errors are still present at te corners of te domain for bot formulations Figure 3. Error field on pressure in a rectangular domain (left) and on a circular domain (rigt) In Figure 4 we sow te L -norm of te error on te pressure as a function of t. Te error is measured at T = 2. One series of computation is made on te square and te oter on te circle. Te messize in bot computations is = /8. It is clear tat te errors calculated on te circular domain are O( t 2 ), wereas tose calculated on te square are only O( t.6 ).

12 2 J.L. GUERMOND, P. MINEV, AND JIE SHEN Tis result, seems to confirm tat te 3 2 convergence rate tat we establised for te pressure approximation in rotational form is te best possible for general domains. However, wy te corner singularity affects te convergence rate for a smoot solution is still not well understood. Circle Square Slope Slope 2 2 Figure 4. Comparison of convergence rates on pressure in L -norm; for te circular domain; + for te square. 4. Te velocity-correction scemes We review in tis section a class of scemes wic are referred to as velocity-correction scemes in [2, 23]. Tese scemes ave been introduced in a somewat different (altoug equivalent) form by Orszag, Israeli & Deville [32] and Karniadakis, Israeli & Orszag [26]. Te main idea is to switc te role of te velocity and te pressure in te pressure-correction scemes, i.e., te viscous term is treated explicitly or ignored in te first substep and te velocity is corrected accordingly in te second substep. 4.. Te nonincremental velocity-correction sceme. Set ũ = u, and for k compute (ũ k+,u k+,p k+ ) by solving { t (4.) (uk+ ũ k )+ p k+ = f(t k+ ), u k+ =, u k+ n Γ =. (4.2) t (ũk+ u k+ ) ν 2 ũ k+ =, ũ k+ Γ =. It is clear tat tis algoritm suffers from te dual ailments of te Corin Temam algoritm (3. 3.2), i.e., it enforces n p k+ Γ = f(t k+ ) n and 2 ũ k+ Γ =, wereas te Corin Temam sceme enforces ν 2 ũ k+ Γ = f(t k+ )and n p k+ Γ =. In terms of accuracy, te two algoritms are equivalent as stated in te following teorem. Teorem 4.. If te solution to (2.6) is smoot enoug in space and time, te solution to (4.) (4.2) satisfies te following error estimates: u t u t l (L 2 (Ω) d ) + u t ũ t l (L 2 (Ω) d ) c(u, p,t) t, p t p t l (L 2 (Ω)) + u t ũ t l (H (Ω) d ) c(u, p,t) t /2.

13 NUMERICAL ISSUES ON PROJECTION METHODS 3 Proof. Since te proof is very similar to tat of te Corin Temam algoritm, we refer te readers to Sen [38], Rannacer [36], or to te proof of second-order accuracy in Guermond and Sen [2, 23] Te standard incremental velocity-correction scemes. We now consider te counterpart of te incremental pressure-correction algoritm in standard form. Adopt te notation of 3.4, and let u,k+ = r j= γ ju k j be a r-t order extrapolation of u(t k+ ). Te standard form of te incremental velocity-correction metod is defined as follows: set ũ = u and coose ũ,...ũ q to be suitably accurate approximations of u( t),...,u(t q ), ten for k q, compute (u k+, ũ k+,p k+ ) by solving q (4.3) ( t β q u k+ j= u k+ =, u k+ n Γ =, β j ũ k j) ν 2 ũ,k+ + p k+ = f(t k+ ), β (4.4) q t (ũk+ u k+ ) ν 2 (ũ k+ ũ,k+ )=, ũ k+ Γ =. Note tat (4.3) can also be written as ( q u k+ β = P j ( H j= β q ũ k j + t ν 2ũ,k+ β q + f(t k+ ) )). Hence, te metod (4.3) (4.4) falls into te class of te projection metods. Since te projection step precedes te viscous step, one could also refer to tese metods as projection diffusion metods as in [3]. Let us assume tat te following initialization ypotesis olds if (q, r) =(2, ): Hypotesis 4.. ũ is computed suc tat te following estimates old: u( t) ũ c t 2, u( t) ũ c t 3/2, u( t) ũ 2 c t. Not tat Hypotesis 4. olds if (ũ,u,p ) are calculated by replacing te BDF2 formula in (4.3) (4.4) wit te implicit Euler formula at te very first time step. Teorem 4.2. Under te initialization Hypotesis 4. and provided tat te solution to (2.6) is smoot enoug in time and space, te solution (u k, ũ k,p k ) to (4.3) (4.4) wit (q, r) =(2, ) is suc tat u t u t l 2 (L 2 (Ω) d ) + u t ũ t l 2 (L 2 (Ω) d ) c(u, p,t) t 2, u t u t l (L 2 (Ω) d ) + u t ũ t l (L 2 (Ω) d ) c(u, p,t) t 3 2, u t ũ t l (H (Ω) d ) + p t p t l (L 2 (Ω)) c(u, p,t) t. Proof. See Guermond and Sen [23]. Remark 4.. For r =, observe from (4.4) tat 2 (ũ k+ ũ k ) n Γ = wic implies tat 2 ũ k+ n Γ = 2 ũ k n Γ = = 2 ũ n Γ. Tis in turn implies (4.5) n p k+ Γ =(f(t k+ )+ν 2 ũ ) n Γ. Tis is obviously an artificial Neumann boundary condition for te pressure, wic is responsible for a numerical boundary layer on te pressure tat limits te accuracy of te sceme, just as in te case of incremental pressure-correction scemes in standard form.

14 4 J.L. GUERMOND, P. MINEV, AND JIE SHEN 4.3. Te rotational incremental velocity-correction scemes. Temainobstacleinproving error estimates better tan first-order on te velocity in te H -norm and on te pressure in te L 2 -norm comes from te fact tat te algoritm (4.3) (4.4) enforces te non-realistic pressure Neumann boundary condition (4.5). Tis penomenon is reminiscent of te numerical boundary layer induced by te non-pysical boundary condition n p k+ Γ =... = n p Γ enforced by te pressure-correction metod in its standard form. Tis non-pysical boundary condition needs to be corrected to obtain a better approximation of te pressure. Considering te identity 2 ũ,k+ = ũ,k+ ũ,k+ and te fact tat we are searcing for divergence-free solutions, we are led to replace 2 ũ,k+ in (4.3) (4.4) by ũ,k+. Te new sceme is as follows: t (4.6) (β q qu k+ β j ũ k j )+ν ũ,k+ + p k+ = f(t k+ ), j= u k+ =, u k+ n Γ =, and β (4.7) q t (ũk+ u k+ ) ν 2 ũ k+ ν ũ,k+ =, ũ k+ Γ =. Tis sceme, introduced by Guermond and Sen in [2, 23], is referred to as te rotational form of te velocity-correction algoritm. Te rotational form yields a better pressure approximation tan te standard form as stated in te following teorem. Teorem 4.3. If te solution to (2.6) is smoot enoug in time and space, and under te initialization Hypotesis 4., te solution (u k, ũ k,p k ) to (4.6) (4.7) wit (q, r) =(2, ) satisfies te estimates: u t u t l2 (L 2 (Ω) d ) + u t ũ t l2 (L 2 (Ω) d ) c(u e,p e,t) t 2, u t ũ t l2 (H (Ω) d ) + p t p t l 2 (L 2 (Ω)) c(u e,p e,t) t 3/2. Proof. We refer to Guermond and Sen in [23]. Remark 4.2. Note tat for q = r =, Teorems 4.2 and 4.3 old wit t 2 and t 3 2 replaced by t Implementation. Note tat te projection steps (4.3) and (4.6) cannot be solved in te form of weak Poisson problems wen using H -conformal finite elements since te trace of ũ,k+ is not well defined wen ũ,k+ in te finite element space. Tis difficulty can be avoided by making suitable substitutions as sown below. Observe first tat by adding (4.6) and (4.7) we obtain D tũk+ ν 2 ũ k+ + p k+ = f(t k+ ). Ten, by subtracting te above equation at time t,k+ = r j= γ jt k j from step (4.6) at time t k+,weobtain { βq t (uk+ ũ k+ )+ D tũk+ D tũ,k+ + φ k+ = f(t k+ ) f,k+, u k+ =, u k+ n Γ =, were f,k+ = r j= γ jf(t k j )isr-t order extrapolation of f(t k+ ). Ten te projected velocity u k+ can be entirely eliminated from te algoritm by rewriting te above equation in te form of a Poisson problem: ( 2 φ k+ βq = (4.8) tũk+ D tũk+ + D tũ,k+ + f(t k+ ) f,k+), n φ k+ Γ =,

15 NUMERICAL ISSUES ON PROJECTION METHODS 5 were we ave set (4.9) φ k+ = p k+ p,k+ + χν ũ,k+, were χ = yields te standard form of te metod and χ = yields te rotational form. Te viscous velocity ũ k+ is finally updated by solving (4.) D tũk+ ν 2 ũ k+ + p k+ = f(t k+ ) ũ k+ Γ = Numerical experiments. Te numerical convergence rates of velocity-correction scemes are similar to tose of teir pressure-correction counterparts. We refer to 3.7 and to [23, 32, 26] for details Relation wit oter scemes. In tis section we sow tat te scemes proposed by Orszag, Israeli & Deville [32] and Karniadakis, Israeli & Orszag [26] can be interpreted as te rotational form of te velocity-correction metods in rotational form Te scemes in [32, 26]. Let us denote by t (β qu k+ q j= β ju k j )teqt-order BDF approximation for t u(t k+ ). Ten, te sceme originally proposed in [32] and [26] (wit a Adams-Moulton type sceme replaced by te BDF sceme; note tat tis replacement is made to simplify te presentation and does not cange te error beaviours) can be written as follows: Find û k+ and p k+ suc tat q (4.) ( t β q û k+ j= β j ũ k j) + p k+ = f(t k+ ), û k+ =, û k+ n Γ = t(ν 2 u),k+ n Γ, ten correct û k+ by computing ũ k+ as follows β (4.2) q t (ũk+ û k+ ) ν 2 ũ k+ =, ũ k+ Γ =, were ( 2 u),k+ = r j= γ j 2 u k j is a r-t order extrapolated approximate value of 2 u(t k+ ). In particular, if r =, (4.3) ( 2 u),k+ = ũ k if r =, (2ũ k ũ k ) if r =2. In practice, (4.) is solved as a Poisson equation supplemented wit te Neumann boundary condition n p k+ Γ =(f(t k+ )+(ν 2 u),k+ ) n Γ, wic is derived from (4.). Since second derivatives of te velocity are used in te Neumann boundary condition for te pressure, tis class of metods cannot be applied directly in conjunction wit a finite element metod were tese derivatives are usually not available. Tis is te main reason wy successful implementations of tese metods are only reported wit spectral or spectral-element approximations were te trace of te second-order derivatives of te velocity are available. On te oter and, it is reasonable to suspect tat, due to te explicit treatment of second derivatives of te velocity, tis type of algoritms can only be conditionally stable wit a stability condition like t c 2 for finite element approximations and t cn 4 for spectral or spectral element approximations. Actually, by rewriting te above algoritms in te L 2 weak framework, we discover tat tey are equivalent to te velocity-correction algoritms in rotational form; ence, te above scemes are indeed unconditionally stable (at least for r =, ).

16 6 J.L. GUERMOND, P. MINEV, AND JIE SHEN Teweaksetting. We now rewrite (4.) (4.2) in te standard L 2 setting. By setting (4.4) u k+ =û k+ + tν( 2 ũ),k+, and observing tat u k+ =, tanks to (4.3) and u k+ n Γ =, (4.) can be rewritten as q (4.5) ( t β q u k+ j= u k+ =, u k+ n Γ =. β j ũ k j) ν( 2 u),k+ + p k+ = f(t k+ ), Now, inserting te definition of u k+ into (4.2), we obtain β (4.6) q t (ũk+ u k+ ) ν 2 ũ k+ + ν( 2 u),k+ =, ũ k+ Γ =, Hence, wen te space is continuous and up to te cange of variable (4.4), te sceme ( ) is equivalent to te velocity-correction algoritm in rotational form ( ). Remark 4.3. It is reported in [26] tat te sceme ( ) wit q = r or q = r +forq 3 is numerically stable. However, no rigorous proof of tis fact is yet available (see a discussion in.). 5. Consistent splitting scemes We review in tis section te consistent splitting sceme introduced in Guermond and Sen [22] and we sow tat, up to an appropriate cange of variables and wen te space is continuous, tis algoritm is equivalent to te so-called gauge metod introduced in E and Liu [] (see also Wang and Liu [47], Brown, Cortez, and Minion [6]). 5.. Te key idea. By taking te L 2 -inner product of te momentum equation in (2.6) wit q and noticing tat (u t, q) = ( u t,q) =, we obtain (5.) p q = (f + ν 2 u) q, q H (Ω), Ω Ω Note tat if u is known, (5.) is simply te weak form of a Poisson equation for te pressure. Te principle of te consistent splitting sceme is to compute te velocity and te pressure in two consecutive steps: First, compute te velocity by treating te pressure explicitly, ten update te pressure using (5.). Let us use te qt-order backward difference formula (BDFq) to approximate t v(t k+ )and te q-t order extrapolation to approximate p(t k+ ). Tese approximations are denoted by t (β qv k+ q j= β jv k j )andp,k+ = q j= γ jp k j, respectively. Of course, te present teory is not restricted to tese coices. Any implicit consistent approximation of ( t ν 2 )v(t k+ )and any explicit consistent approximation of p(t k+ ) is acceptable. We ereafter adopt te notation introduced in (3.) but instead of (3.3), were (q )t order extrapolation is used, we set p k if q =, (5.2) p,k+ = 2p k p k if q =2, 3p k 3p k + p k 2 if q = Standard splitting sceme. A q-t order decoupled approximation to (2.6) is defined as follows: Let u = u t= and p = p t= (wic can be obtained by solving (5.) at t =). If q 2, ten for k q, let (u k,p k )betek-t order approximation to (u(k t),p(k t)) (wic can be obtained recursively by using te sceme described below using BDFk and te k-t order extrapolation of te pressure). Ten, for k q, seek u k+ and p k+ suc tat D (5.3) t uk+ ν 2 u k+ + p,k+ = f(t k+ ), u k+ Γ =, (5.4) ( p k+, q) =(f(t k+ )+ν 2 u k+, q), q H (Ω).

17 NUMERICAL ISSUES ON PROJECTION METHODS 7 Note tat in te second step we need to compute 2 u k+ wic may not be well defined in a finite element discretization. Hence, we sall derive an alternative formulation wic does not require computing 2 u k+ and is more suitable for finite element discretizations. To tis end, we take te inner product of te first step wit q and subtract te result from te second step. Ten, we obtain te following equivalent formulation of (5.3)-(5.4): D (5.5) t uk+ ν 2 u k+ + p,k+ = f(t k+ ), u k+ Γ =, (5.6) ( (p k+ p,k+ ), q) =( D t uk+, q), q H (Ω). Remark 5.. (i) Te two scemes (5.3)-(5.4) and (5.5)-(5.6) are strictly equivalent wen space is continuous but tey yield two different implementations wen te space variables are discretized (see 8 for furter details). (ii) Neiter sceme (5.3)-(5.4) nor (5.5)-(5.6) is a projection sceme, for te velocity approximation u k+ is not divergence-free. Neverteless, tese algoritms are similar to te pressurecorrection algoritm wit te end-of-step velocity eliminated (see ). (iii) As in a projection sceme, one only needs to solve a set of Helmoltz-type equations for u k+ and a Poisson equation (5.4) or (5.6) (in weak form) for p k+. (iv) Just as in a pressure-correction sceme in standard form [23], te equation (5.6) implies tat n (p k+ p,k+ ) Ω = wic is an artificial Neumann boundary condition not satisfied by te exact pressure. Tis boundary condition induces a numerical boundary layer wic, in turn, results in loss of accuracy. Te following result olds; see [22]: Teorem 5.. Provided tat te solution to (2.6) is smoot enoug in time and space, te solution (u t,p t ) to (5.5)-(5.6) satisfies te estimates: u t u t l2 (L 2 (Ω) d ) t 2, u t u t l (H (Ω) d ) + p t p t l (L 2 (Ω)) t. Note tat te above error estimates are of te same order as tose of te second-order pressurecorrection sceme in standard form, but tey are less accurate tan tose of te second-order pressure-correction sceme in rotational form 5.3. Consistent splitting sceme. Similarly to pressure-correction and velocity-correction scemes, te accuracy of te above splitting scemes can be improved by replacing 2 u k+ in (5.4) by u k+, leading to te following algoritm: D (5.7) t uk+ ν 2 u k+ + p,k+ = f(t k+ ), u k+ Γ =, (5.8) ( p k+, q) =(f(t k+ ) ν u k+, q), q H (Ω). Owing to te identity 2 u k+ = u k+ u k+, tis procedure amounts to removing te term u k+ in (5.4). It is sown in [23, 24] tat wen tis strategy is applied to pressurecorrection and velocity-correction scemes it yields an a priori control on te divergence of u k+, wic in turn leads to improved accuracy on te vorticity and te pressure. Once again, to avoid computing u k+ explicitly in te second step, we take te inner product of (5.7) wit q and we subtract te result from (5.8). Tis leads to an equivalent alternative form of (5.7)-(5.8): Du (5.9) k+ t ν 2 u k+ + p,k+ = f(t k+ ), u k+ Γ =, (5.) ( ψ k+, q) =( D t uk+, q), q H (Ω). (5.) p k+ = ψ k+ + p,k+ ν u k+.

18 8 J.L. GUERMOND, P. MINEV, AND JIE SHEN Note tat te complexity of te scemes (5.7)-(5.8) and (5.9)-(5.)-(5.) is te same as tat of (5.3)-(5.4) or (5.5)-(5.6). However, as ample numerical results indicate, te pressure approximation p k+ is no longer plagued by an artificial Neumann boundary condition and, consequently, tese scemes provide truly q-t order accuracy (at least for q = and 2) for te velocity, vorticity and pressure. Tus, (5.7)-(5.8) and (5.9) (5.) (5.) are encefort referred to as consistent splitting scemes. We note tat te sceme proposed in [3], were an intermediate divergence-free acceleration a := u t ν uis introduced, is quite similar to (5.9) (5.) (5.). Te analysis of te stability and te convergence of te consistent splitting sceme is more involved tan tat of te standard form. For te time being, only optimal convergence results wit q = ave been proved; see [22]. Teorem 5.2. Provided tat te solution to (2.6) is smoot enoug in time and space, te solution (u t,p t ) of (5.9)-(5.)-(5.) wit q =is unconditionally bounded and satisfies te following error estimates u t u t l (H (Ω) d ) + p t p t l (L 2 (Ω)) t. Conjecture 5.. For q =2te following olds u t u t l (H (Ω) d ) + p t p t l (L 2 (Ω)) t 2. Altoug numerical tests seems to confirm te above conjecture, its proof remains elusive Numerical experiments. To demonstrate te accuracy of te consistent splitting scemes, we perform convergence tests wit respect to t using mixed P 2 /P finite elements in space. Te analytical solution is tat given in (3.3). Te domain is Ω =], [ 2 and te messize is /8.Wemaketetestsonterange5. 4 t so tat te approximation error in space is far smaller tan te time splitting error. We ave tested te algoritms (5.5) (5.6) and (5.9) (5.) (5.) using q = 2 to substantiate Conjecture 5.. L2, std., 2nd order H, std., 2nd order 2 2 L2, rot., 2nd order H, rot., 2nd order Slope Slope 2 Slope 2 Slope Standard splitting Consistent splitting Figure 5. Convergence tests wit BDF2 and Finite elements. velocity in te L 2 -norm and in te H -norm at T =. Error on te Te error on te velocity in te L 2 -norm and in te H -norm at T = is reported in Figure 5. Te error is sown as a function of t. Te results corresponding to te standard form of te

19 NUMERICAL ISSUES ON PROJECTION METHODS 9 algoritm are reported in te left panel of te figure, and tose corresponding to te rotational form are in te rigt panel. Te standard form of te algoritm is second-order accurate in te L 2 -norm, but te convergence rate in te H -norm is rougly 3 2. One clearly observes in te rigt panel of te figure tat te rotational form of te algoritm is second-order accurate bot in te L 2 -norm and te H -norm. Note tat te saturations observed for very small time steps is due to te approximation error in space wic becomes dominant for very small time steps. We sow in Figure 6 te error on te pressure measured in te L -norm for bot versions of te algoritm. Te results clearly sow tat te pressure approximation in standard form is only first-order, wereas in te rotational formulation it is truly second-order. Te poor convergence rate in te standard form can be attributed to te presence of numerical boundary layers wic are induced by te fact tat te boundary condition enforced by te approximate pressure, namely n (p k+ 2p k + p k ) Γ =,isnotconsistent. Linfty, std., 2nd order Linfty, rot., 2nd order 2 3 Slope Slope Figure 6. Convergence tests wit BDF2 and Finite elements. Error on te pressure in te L -norm at T = wit standard splitting and consistent splitting Relation wit te gauge metod. Te gauge formulation [] of te Navier-Stokes equations consists of replacing te pressure by a so-called gauge variable ξ and defining an auxiliary vector field m suc tat m = u + ξ. Ten, te Stokes problem can be reformulated as follows (5.2) { t m ν 2 m = f, m t= = m, m n Γ =, (m ξ) n Γ = 2 ξ = m, n ξn Γ =. Te velocity and te pressure are recovered by (5.3) u = m ξ, p = t ξ ν 2 ξ. Tis type of formulation as been proposed originally to get rid of te pressure and te saddlepoint structure it implies. Unfortunately, tis goal is not quite fulfilled since te boundary condition (m ξ) n Γ = implies a coupling between te m and ξ variables tat as exactly te same complexity as tat between te velocity and te pressure in te Stokes problem.

20 2 J.L. GUERMOND, P. MINEV, AND JIE SHEN We now construct a decoupled time discretization of (5.2) using BDFq. Assuming tat we ave initialized properly (m j ) j=,...,q,fork q we compute m k+ suc tat { D t (5.4) mk+ ν 2 m k+ = f(t k+ ), m k+ n Γ =, (m k+ + ξ,k+ ) n Γ =, were ξ,k+ is an extrapolation for ξ(t k+ ) suc tat ξ,k+ n Γ =. A natural coice is: { (5.5) ξ,k+ ξ k if q =, = 2ξ k ξ k if q =2. Ten, ξ k+ is updated by (5.6) 2 ξ k+ = m k+, n ξ k+ Γ =. Te fact tat te viscous step (5.4) involves te trace of a gradient as a Diriclet boundary condition renders te metod quite inconvenient from bot te teoretical and te practical point of view: a priori energy estimates are difficult to obtain in tis form, and te metod cannot be used wit H -conformal finite element metods. In te following we sall reformulate te sceme (5.4)-(5.5)-(5.6) by making a suitable cange of variables to avoid tis difficulty Te L 2 weak setting. To rewrite (5.4) (5.6) in te L 2 setting, we introduce te following canges of variables: (5.7) ũ k+ = m k+ ξ,k+, u k+ = m k+ ξ k+, p k+ = D t ξk+ ν 2 ξ k+, p,k+ = D t ξ,k+ ν 2 ξ,k+. Using te definition of p,k+, we infer { D t (mk+ ξ,k+ ) ν 2 (m k+ ξ,k+ )+ p,k+ = f(t k+ ), (m k+ ξ,k+ ) n Γ =, (m k+ ξ,k+ ) n Γ =, wic, owing to te definition of ũ k+, yields D tũk+ ν 2 ũ k+ + p,k+ = f(t k+ ), ũ k+ Γ =. From te definitions of u k+ and ũ k+ it is clear tat u k+ ũ k+ + (ξ k+ ξ,k+ )=, wic yields 2 (ξ k+ ξ,k+ )= ũ k+. Now, using te definitions of p k+ and p,k+, we infer p k+ p,k+ = D t (ξk+ ξ,k+ ) ν 2 (ξ k+ ξ,k+ ) = D t (ξk+ ξ,k+ ) ν ũ k+. Taking te Laplacian of te above equation, we find 2 (p k+ p,k+ + ν ũ k+ )= 2 D t (ξk+ ξ,k+ )= D tũk+. Tus, we ave proved tat, up to an appropriate cange of variables and wen te space is continuous, te gauge algoritm (5.4)-(5.5)-(5.6) is equivalent to te following (5.8) (5.9) (5.2) Dũ k+ t ν 2 ũ k+ + p,k+ = f(t k+ ), ũ k+ Γ =. 2 φ k+ = Dũk+ t, n φ k+ Γ =, p k+ p,k+ + ν ũ k+ = φ k+, wic is exactly te consistent splitting sceme (5.9)-(5.)-(5.) owing to te definitions of p,k+ and ξ,k+.

21 NUMERICAL ISSUES ON PROJECTION METHODS 2 6. Inexact factorization scemes In tis section we turn our attention to te so-called inexact algebraic factorization scemes. Tis class of metods ave gained some popularity since tey do not involve, explicitly, any artificial pressure boundary condition and are believed by some to provide better convergence rates tan teir PDE counterparts pressure-correction scemes. We sall sow below tat te inexact factorization metods enforce weakly an artificial pressure boundary condition, and do not provide better accuracy tan teir PDE-based counterparts. 6.. Te matrix setting. Let X H (Ω) d and M L 2 (Ω) be two finite-dimensional spaces satisfying te inf-sup condition: Hypotesis 6.. Tere exists β > independent of suc tat Ω (6.) inf sup q v β. q M v X q v We also assume te following interpolation properties: Hypotesis 6.2. Tere exist two spaces W H (Ω) d, Z L 2 (Ω) and two continuous functions ɛ (), ɛ 2 () vanising at suc tat for all v W and q Z inf v v ɛ () v W, v X (6.2) inf v v + inf q q ɛ 2 ()( v W + q Z ). v X q M For finite elements tere is usually some positive integer s suc tat W = H (Ω) d H s+ (Ω) d, Z = L 2 (Ω) H s (Ω), ɛ () = s+,andɛ 2 () = s. Many couples of finite elements spaces satisfying te inf-sup condition are reported in Brezzi Fortin [5] and Girault Raviart []. Let N u =dim(x ), N p =dim(m ), and let {v i } i=,...,nu, {q k } k=,...,np be basis functions for X and M respectively. We define te following matrices: [ ] [ (6.3) M = φ i φ j, K = Ω ] ν φ i : φ j, D = Ω [ ] q k φ j. Ω Denoting by U R Nu and P R Np te components of u X and p M in te considered bases, using, for example, te implicit Euler time stepping for (2.6), we obtain: (6.4) t M + K DT U k+ = t MU k + F k+ D P k+ were we ave set [ ] (6.5) F k = φ i f(t k ). Ω Te main idea beind te inexact factorization consists of replacing te matrix in (6.4) by its incomplete block LU factorization. One of te simplest possibilities is (6.6) t M + K DT t M + K I tm D T D D tdm D T I Ten, (6.4) can be approximately solved as follows: Step : ( t M + K)Ũ k+ = t MU k + F k+ Step 2: DM D T Φ k+ = k+ tdũ (6.7) Step 3: U k+ = Ũ k+ tm D T Φ k+ Step 4: P k+ =Φ k+.