Convergence of the Fully Discrete Incremental Projection Scheme for Incompressible Flows

The present paper addresses the convergence of a first-order in time incremental projection scheme for the time-dependent incompressible Navier–Stokes equations to a weak solution. We prove the convergence of the approximate solutions obtained by a semi-discrete scheme and a fully discrete scheme using a staggered finite volume scheme on non uniform rectangular meshes. Some first a priori estimates on the approximate solutions yield their existence. Compactness arguments, relying on these estimates, together with some estimates on the translates of the discrete time derivatives, are then developed to obtain convergence (up to the extraction of a subsequence), when the time step tends to zero in the semi-discrete scheme and when the space and time steps tend to zero in the fully discrete scheme; the approximate solutions are thus shown to converge to a limit function which is then shown to be a weak solution to the continuous problem by passing to the limit in these schemes.


Introduction
The incompressible Navier-Stokes equations for a homogeneous fluid read: where the density and the viscosity are set to one for the sake of simplicity, and where (2) T > 0, and Ω is a connected, open and bounded subset of R 3 , with a Lipschitz boundary ∂Ω.
Note that we only consider the three dimensional setting in this work, but the analysis may be carried out in a similar (and often somewhat simpler) manner in the one or two dimensional setting.The variables u and p are respectively the velocity and the pressure in the flow, and Eqns.(1a) and (1b) respectively enforce the momentum conservation and the mass conservation and incompressibility of the flow.This system is supplemented with the boundary condition (3) u = 0 on (0, T ) × ∂Ω, and the initial condition (4) u(0) = u 0 in Ω.
The function u 0 is the initial datum for the velocity and the function f is the source term.Throughout the paper, we shall assume that (5) f ∈ L 2 ((0, T ) × Ω) 3 and u 0 ∈ E(Ω), where E(Ω) is the subset of H 1 0 (Ω) 3 of divergence-free functions, defined by E(Ω) = {u ∈ H 1 0 (Ω) 3 such that divu = 0}.Note that in fact, the initial condition is assumed to be in E(Ω) for the sake of simplicity.It could be considered in L 2 (Ω) 3 only, see Remark 2.3.
Let us define the weak solutions of Problem ( 1)-( 4) in the sense of Leray [17].
Definition 1.1 (Weak solution).Under the assumptions (2) and (5), a function u ∈ L 2 (0, T ; E(Ω)) ∩ L ∞ (0, T ; L 2 (Ω) 3 ) is a weak solution of the problem (1)-( 4) if 3 , divw = 0 a.e. in Ω × (0, T ) .The first projection method to solve the system (1) was designed over 50 years ago, and is known as the Chorin-Temam algorithm [4,21,22].It consists in a prediction step based on a linearized momentum equation without the pressure gradient, and a pressure correction step that enforces the divergence-free constraint.This method and its variants are now often referred to (following [13]) as non incremental projection schemes, in opposition to the incremental projection schemes that were obtained by adding the old pressure gradient in the prediction step (see [12] for a first-order time scheme and [23] of a second order time scheme).These latter schemes are indeed incremental in the sense that the correction step may now be seen as solving an equation on the time increment of the pressure.They seem to be much more efficient from a computational point of view [13] and have been the object of several error analysis, under some regularity assumptions on the solution of the continuous problem, in the semi discrete setting, see [13] and references therein.
The non incremental schemes have been the object of some analyses in the fully discrete setting.In [1] some error estimates are derived for a non incremental scheme with a discretization by the finite element, under some regularity assumptions on the exact solution, In a recent paper the approximate solutions of a fully discrete non incremental scheme with a uniform staggered discretization [15] are shown to converge to a weak solution (and so without any regularity assumption on the solution of (1)) under the condition that h ≤ δt 3−α where h and δt are respectively the mesh size and the time step and with 0 < α ≤ 2.
However, to our knowledge, up to now, no proof of convergence exists for the fully discrete incremental projection schemes, even though they are the most used in practice.The purpose of the present work is therefore to fill this gap and to show the convergence of the incremental projection method with a discretization by a staggered finite volume scheme based on a (non uniform) MAC grid, without any regularity assumption on the exact solution.
The Marker-And-Cell (MAC) scheme, introduced in the middle of the sixties (see [14]), is one of the most popular methods (see e.g.[18] and [24]) for the approximation of the Navier -Stokes equations in the engineering framework, because of its simplicity, its efficiency and its remarkable mathematical properties.Although originally presented as a finite difference scheme on uniform meshes, the MAC scheme is in fact a finite volume scheme and as such can be used on non uniform meshes.The convergence analysis of the staggered finite volume scheme on the MAC mesh using a fully implicit time scheme may be found in [11], and we shall use several of the tools developed therein.We also refer to this latter paper for some more references on studies of the MAC scheme.
The paper is organized as follows.Section 2 deals with the convergence analysis for the semi-discrete projection algorithm.The fully discrete scheme is analysed in Section 3; we only give the main ingredients of the staggered space discretization that we use, and which is often referred to as the MAC scheme.To avoid a lengthy description, the precise definitions of the now classical discrete MAC operators are to be found in [11].
Before starting the analysis of the semi-discrete and fully discrete schemes, we wish to recall, for the sake of clarity, that: , where 1 ≤ p < +∞ and Ω is an open set of R 3 , the space E ′ is identified to L q (Ω), q = p/(p − 1).• For T > 0 and E = L 1 ((0, T ), L 2 (Ω)), the space L ∞ ((0, T ), L 2 (Ω)) is identified with E ′ .
In the appendix, we give some useful technical lemmas.

Analysis of the time semi-discrete incremental projection scheme
We consider a partition of the time interval [0, T ], which we suppose uniform to alleviate the notations, so that the assumptions read:

2.1.
The time semi-discrete scheme.Under the assumptions (7), the usual first order time semi-discrete incremental projection scheme (see [20]) reads: where n stands for the outward normal unit vector to the boundary ∂Ω and Let us briefly account for the existence of a solution at each step of this algorithm.

Find ũn+1
The existence of the predicted velocity is then a consequence of Lemma A.1.

Correction step -A weak form of Eqns. (8d)-(8e) reads
Find • ∇ϕ dx = 0 for any ϕ ∈ H 1 (Ω), so that u n+1 N belongs to the space V (Ω) of "L 2 -divergence-free functions" defined by (11) V The existence of (u n+1 N , p n+1 N ) ∈ V (Ω) × H 1 (Ω) satisfying ( 10) is a consequence of the decomposition result of Lemma A.2 given in the appendix.Indeed, this correction step is the decomposition stated in Lemma A.2 applied to the predicted velocity ũn+1 .Note that p n+1 N is uniquely defined thanks to (8f).
Remark 2.1.Summing (8b) at step n and (8d) We may thus state the following existence result and define the approximate solutions obtained by the projection scheme (8).
Remark 2.2 (On the boundary conditions).The original homogeneous Dirichlet boundary conditions (3) of the strong formulation (1) is imposed on the weak solution through the functional space H 1 0 (Ω) 3 .Note that this condition is only imposed on the predicted velocity in the algorithm (8).Indeed, the corrected velocity does not satisfy the full Dirichlet condition (3) but only the no slip condition imposed by (8e).The compactness of the sequence of predicted velocities ũ together with the convergence of u − ũ towards zero in L 2 as the time step tends to zero will be the mean to prove that the Dirichlet boundary condition is finally satisfied on the limit of the numerical approximations.Note also that there is no need for a boundary condition on the pressure in the correction step.In fact, it can be inferred from the correction step (10) that the incremental pressure ψ n+1 = p n+1 − p n satisfies a Poisson equation on Ω with a Neumann boundary condition on the boundary, but this is a redundant information that does not need to be implemented.We refer to [19] for an interesting discussion on these boundary conditions.

Remark 2.3 (On the initial condition). In fact, the existence of a solution (see Lemma A.1) only requires the initial velocity u 0
N to be in V (Ω), so that we could relax the assumption on the initial condition u 0 ∈ E(Ω) to u 0 ∈ L 2 (Ω) 3 and take u 0 = P V (Ω) u 0 as the orthogonal projection of u 0 onto the closed subspace V (Ω) of L 2 (Ω) 3 , also known as the Leray projection.In this case, u 0 N can be computed as u 0 = u 0 − ∇ψ where ψ ∈ H 1 (Ω) is a solution (unique, up to a constant) of the following problem (see Lemma A.2) Theorem 2.1 (Convergence of the semi-discrete in time projection algorithm).Under the assumptions (2) and (5), consider for N ≥ 1, the time discretization defined by (7), and the approximate solutions u N and ũN of the projection algorithm (8) as given in Definition 2.1.Then there exists ū ∈ L 2 (0, T ; E(Ω)) ∩ L ∞ (0, T ; L 2 (Ω) 3 ) such that up to a subsequence, • the sequence (ũ N ) N ≥1 converges to ū in L 2 (0, T ; L 2 (Ω) 3 ) and weakly in L 2 (0, T ; H 1 0 (Ω) 3 ), • the sequence (u N ) N ≥1 converges to ū in L 2 (0, T ; L 2 (Ω) 3 ) and ⋆-weakly in L ∞ (0, T ; L 2 (Ω) 3 ).Moreover the function ū is a weak solution to (1) in the sense of Definition 1.1.
Proof.Here are the main steps of the proof; each step is detailed in one of the following paragraphs.
• Step 2: compactness and convergence in L 2 (detailed in section 2.3) This is the tricky part of the proof.Since the sequence (ũ N ) N ≥1 converges weakly in L 2 (0, T ; H 1 0 (Ω) 3 ), some estimate on the discrete time derivative would be sufficient to obtain the convergence in L 2 (0, T ; H 1 0 (Ω) 3 ) by a Kolmogorovlike theorem.A difficulty to obtain this estimate arises from the presence of the pressure gradient in Equation (8b), which needs to be "killed" by multiplying this latter equation by a divergence-free function.This function ϕ should also be regular enough so that the nonlinear divergence term makes sense: hence we choose ϕ ∈ L 2 (0, T ; W 1,3 0 (Ω) 3 ) such that divϕ = 0, and define the following semi-norm on (L 2 (Ω)) 3 : ) semi-norm of the time translates of the predicted velocity u N are then obtained from the semi-discrete momentum equation (8b): see Lemma 2.3.Note that this is only an intermediate result; indeed, in order to gain compactness, we need an estimate on the time translates of the predicted velocity in the L 2 (L 2 ) norm.The idea is then to first introduce the following semi-norm on (L 2 (Ω)) 3 .(18) |v| * ,0 = sup{ where V (Ω) is the space of L 2 divergence-free functions defined by (11).Note that where P V (Ω) is the orthogonal projection operator onto V (Ω).Then, thanks to a Lions-like lemma (Lemma 2.4 below), we get that for any ε > 0, there exists By (15), we have an L 2 (0, T ; H 1 0 (Ω) 3 ) bound on the predicted velocities; we have also seen that the time translates of u N for the L 2 (| • | * ,1 ) semi-norm tend to 0 as N → +∞ (Lemma 2.3 below).Therefore, by (20), the time translates of u N for the L 2 (| • | * ,0 ) semi-norm also tend to 0 as N → +∞.In order to show that the L 2 (L 2 ) norm of the time translates of ũN tend to 0, we remark that if v ∈ V (Ω), then |v| * ,0 = ||v|| L 2 (Ω) and conclude thanks to (16), see Lemma 2.5).

• Step 3: convergence towards the weak solution (detailed in section 2.4)
Owing to a Kolmogorov-type theorem (see e.g.[9, Corollary 4.41]), the estimates of steps 1 and 2 yield that there exist subsequences, still denoted (u N ) N ≥1 and (ũ N ) N ≥1 , that converge to ū in L 2 (0, T ; L 2 (Ω) 3 ).In section 2.4, we pass to the limit in the scheme to obtain that ū satisfies (6); therefore ū is a weak solution to (1) in the sense of Definition 1.1.
Remark 2.4 (Uniqueness and convergence of the whole sequence).In the case where uniqueness of the solution is known, then the whole sequence converges ; this is for instance the case in the 2D case [17], see e.g.[3, Chapter 5, Section 1.3] for more on this subject.
2.4.Proof of step 3: convergence to a weak solution.By Lemma 2.5, up to a subsequence, the sequence of predicted velocities (ũ N ) N ≥1 converges to some limit ū ∈ (L 2 (0, T ; L 2 (Ω) 3 ), and owing to (16), so does the sequence (u N ) N ≥1 .There remains to check that ū is a weak solution to (1) in the sense of Definition 1.1.This is a result that we call "Lax-Wendroff consistency", following the famous paper [16] see e.g.[7]: assuming that the approximate solutions converge boundedly to a limit, this limit is a weak solution to the continuous problem.
x), for any x ∈ Ω, and let ϕ N : (0, T ) → E(Ω) and f N : (0, T ) → L 2 (Ω) 3 be defined by The regularity of f and ϕ implies that: Multiplying (12) by δt N ϕ n N , integrating over Ω and summing over n ∈ 1, N − 1 yields Using the fact that ϕ N N = 0 in Ω the first term of the left hand side reads By the triangle inequality, Since the sequence (ũ N ) N ≥1 converges to ū in L 2 (0, T ; L 2 (Ω) 3 ), we obtain (28) lim The second term in the left hand-side reads The convergence of the sequence (ũ N ) N ≥1 in L 2 (0, T ; L 2 (Ω) 3 ), the weak convergence of the sequence (u N ) N ≥1 in L 2 (0, T ; L 2 (Ω) 3 ), the convergence of the sequence The third term in the left hand-side may be written The weak convergence of the sequence (∇ũ N ) N ≥1 in L 2 (0, T ; L 2 (Ω) 3 ) and the convergence of the sequence (∇ϕ N ) N ≥1 in L 2 (0, T ; L 2 (Ω) 3 ) implies The right hand-side satisfies The convergence of the sequence (f N ) N ≥1 in L 2 (0, T ; L 2 (Ω) 3 ) and the convergence of the sequence (ϕ Using (28)-(31) and passing to the limit in (27) gives the expected result.

Analysis of the fully discrete projection scheme
Our purpose is now to adapt the proof of convergence of the semi-discrete case to the fully discrete case.We choose as an example of space discretization a staggered discretization on a (possibly non uniform) rectangular grid of R 3 .The resulting scheme, often referred to as a MAC scheme, was analysed in [11] for an timeimplicit scheme.The idea here is to prove its convergence for the incremental projection scheme.We consider the following assumptions on Ω and on the timespace discretization, indexed by N (in the convergence analysis, the time and space steps will tend to 0 as N tends to +∞).
T > 0, Ω is an open rectangular subset of R 3 , with boundary faces that are orthogonal to one of the vectors of the canonical basis of (32a) Note that at this step, we are only considering one time step δt N = T N and one discretization mesh D N , which is also indexed by N .This might seem strange, but it is in view of the convergence analysis for which a sequence (D N , δt N ) N ≥1 will be considered, with h N , δt N → 0 as N → +∞.
The regularity of the mesh is measured by the following parameter: (33) with | • | the Lebesgue measure (this notation is used in the following for either the R 3 or the R 2 measure).We refer to [11] for the precise definition ot the discrete spaces and operators.The approximate pressure belongs to the set L N (Ω) of functions that are piecewise constant on the so called primal cells K of the (primal) mesh M N : p = K∈MN p K 1 K .The i-th component of the approximate velocities belongs to the set H (i) N (Ω) of functions that are piecewise constant on the dual cells D σ ∈ E (i) , where E (i) denotes the set of faces of the mesh that are orthogonal to e i .Denoting by E(K) the set of faces of a given cell K ∈ M N , and by σ = K|L an interface between two neighbouring cells K and L, a dual cell D σ ∈ E ∩ E(K) is defined by where x K denotes the mass center of K and x K,∂Ω the orthogonal projection of x K on ∂Ω We thus define three dual meshes of Ω.
Theorem 3.1 (Convergence of the fully discrete projection algorithm).Under the assumption (5), let (δt N , D N ) be a sequence of time space discretizations satisfying (32), such that h N → 0 as N → +∞ and such that the mesh regularity parameter θ N defined by (33) remains bounded.Let u N : (0, T ) → E N (Ω) and ũN : (0, T ) → H N,0 (Ω) be the approximate predicted and corrected velocities defined by the scheme (34) and Definition 3.1.Then there exists ū ∈ L 2 (0, T ; Moreover the function ū is a weak solution to (1) in the sense of Definition 1.1.
Proof.We give here the main steps of the proof, which follows that of the semidiscrete case; these steps are detailed in the following paragraphs.
• Step 1: first estimates and weak convergence (detailed in Section 3.2).Let us define, for q ∈ N * , a discrete W 1,q 0 (Ω) 3 -norm for the discrete velocity fields.
There remains to show that ū is a weak solution in the sense of Definition 1.1 and in particular that ū satisfies (6).The weak convergence is not sufficient to pass to the limit in the scheme, because of the nonlinear convection term, so that we first need to get some compactness on one of the subsequences (ũ N ) N ∈N or (u N ) N ∈N (since, by (16), their difference tends to 0 in the L 2 norm).
• Step 2: compactness and convergence in L 2 (detailed in section 3.3) We adapt Step 2 of the convergence proof of the semi-discrete case.Using the bound (40) on the sequence (ũ N ) N ≥1 , some estimate on the discrete time derivative would be sufficient to obtain the convergence in L 2 (0, T ; H 1 0 (Ω) 3 ) by a Kolmogorov-like theorem.As in the semi-discrete case, a difficulty arises from the presence of the (discrete) pressure gradient in Equation ( 12); we get rid of it by multiplying this latter equation by a discrete divergence-free function, chosen as the interpolate of a regular function ϕ ∈ L 2 (0, T ; (W 1,3 0 (Ω)) 3 ) such that divϕ = 0. Let us then define the discrete equivalent of the semi-norm ( 17) on H N,0 (Ω) by: (45 Note that we have the following identity, which is the discrete equivalent of (19).
(46) |w| * ,0,N = P EN (Ω) w L 2 (Ω) 3 , for any w ∈ H N,0 (Ω), where P EN (Ω) is the orthogonal projection operator onto E N (Ω).Then, thanks to a discrete equivalent of the Lions-like 2.4 lemma (Lemma 3.4 below), we get that for any ε > 0, there exists From this latter inequality, using Lemma 3.3 on the time translates of u N for the L 2 (| • | * ,1 ) semi-norm and the bound (40), we get that the time translates of u N for the L 2 (| • | * ,0,N ) semi-norm also tend to 0 as N → +∞.
• Step 3: convergence towards the weak solution (detailed in Section 3.4) Owing to a discrete Aubin-Simon-type theorem [9, Theoreme 4.53], the estimates of steps 1 and 2 yield that there exist subsequences, still denoted (u N ) N ≥1 and (ũ N ) N ≥1 , that converge to ū in L 2 (0, T ; L 2 (Ω) 3 ).Passing to the limit in the scheme (34) then yields that ū satisfies (6) and in particular that ū is a weak solution to (1).
Remark 3.4 (Uniqueness and convergence of the whole sequence).If the solution of the continuous problem is unique, then again the whole sequence converges.
Proof.By Lemma B.1 with α = 1 δtN , we have for n ∈ 0, N − 1 Squaring the relation (34d), integrating over Ω, multiplying by δtN 2 and owing to (34e) and to the discrete duality property of the MAC scheme [11, Lemma 2.4], we get Summing this latter relation with the previous relation yields for n ∈ 0, N − 1 We then get the relation (48) using the Cauchy-Schwarz inequality and the discrete Poincaré estimate [8, Lemma 9.1] after summing over the time steps.

3.3.
Estimates on the time translates and compactness.Lemma 3.3 (A first estimate on the time translates).Under the assumptions of Theorem 3.1, there exists C 4 > 0 only depending on |Ω|, the L 2 -norm of u 0 and the L 2 -norm of f such that for any N ≥ 1 and for any τ ∈ (0, T ) defined by (22).Using (38), we thus get that where ∇ N is the gradient operator of the velocity defined on each dual rectangular grid, see [11,Section 2]), we get that Let us reproduce at the fully discrete level the computations done for each of these terms in the proof of Lemma 2.3.
By a technique similar to that of [11,Lemma 3.5], we get that (49) 3 , we define P N v as the vector function with piecewise constant components: the i-th component of P N v is constant on each dual cell D σ , σ ∈ E, and equal to the mean value of v i on the face σ.By [11,Lemma 3.7], P N is a Fortin operator in the sense that it preserves the divergence; in particular, Lemma 3.4 (Lions-like, fully discrete version).Consider a rectangular domain Ω of R 3 and a sequence of MAC grids (D N ) N ≥1 of Ω satisfying (32c) such that h N → 0 as N → +∞ and such that the mesh regularity parameter θ N defined by (33) remains bounded.Then, for any ε > 0, there exists C ε > 0 and N ε ≥ 1 depending on ε such that for any N ≥ N ε and for any w ∈ H N,0 (Ω), ( 47) is satisfied.
Proof.Let ε > 0; let us show by contradiction that there exists C ε > 0 and N ε ≥ 1 depending on ε such that for any N ≥ N ε and for any w ∈ H N,0 (Ω) Suppose that this is not so, then there exist ε > 0 and a subsequence of MAC grids of Ω still denoted by (D N ) N ≥1 and a sequence (w N ) N ≥1 of functions such that w N ∈ H N,0 (Ω) for any N ≥ 1 and, thanks to (46), By a homogeneity argument, we may choose P E N (Ω) w N L 2 (Ω) 3 = 1; it then follows from the latter inequality that the sequence ( w N 1,2,N ) N ≥1 is bounded and that |w N | * ,1,N → 0 as N → +∞.Hence there exists a subsequence still denoted by (w N ) N ≥1 that converges in L 2 (Ω) 3 to a function w ∈ H 1 0 (Ω) 3 , see e.g.[6, Theorem 3.1].Lemma 3.5 given below then yields that P E N (Ω) w N → P V (Ω) w in L 2 (Ω) 3 and in particular P V (Ω) w L 2 (Ω) 3 = 1.(Recall that P V (Ω) : L 2 (Ω) 3 → L 2 (Ω) 3 is the orthogonal projection in L 2 (Ω) 3 onto the space V (Ω).) For any ϕ ∈ W (Ω), we have P N (ϕ) ∈ E N (Ω).Since w N − P EN (Ω) w N ⊥ E N and by definition of |w N | * ,1,N , it follows that By the W 1,q stability of the operator P N stated in [10, Theorem 1], there exists C 8 only depending on |Ω| and on θ N in a nondecreasing way, such that 3 , for any ϕ ∈ W 1,3 0 (Ω) 3 .Passing to the limit in this inequality yields that This in turn implies that there exists ξ ∈ H 1 (Ω) such that P V (Ω) w = ∇ξ.Using the fact that P V (Ω) w ∈ V (Ω) we have Lemma 3.5.Let N ≥ 1 and let D N = (M N , E N ) be a MAC grid of Ω in the sense of (32c), such that (h N ) N ≥1 converges to zero and such that (θ N ) N ≥1 is bounded, with θ N defined by (33).Let (v N ) N ≥1 be a sequence of functions such that v N ∈ H N,0 (Ω) for any N ≥ 1 and (v N ) N ≥1 converges to v in L 2 (Ω) 3 .Then the sequence Proof.Using the fact that (v N ) N ≥1 is bounded in L 2 (Ω) 3 we obtain that the sequence (P EN (Ω) v N ) N ≥1 is bounded in L 2 (Ω) 3 .Hence there exists a subsequence still denoted by (P EN (Ω) v N ) N ≥1 that converges to a function ṽ weakly in L 2 (Ω) 3 .Thanks to the discrete duality property stated in [11, Lemma 2.4], we have, for any where Π N ϕ is the piecewise constant function defined by Π Passing to the limit in the previous identity gives Ω ṽ • ∇ϕ dx = 0, for any ϕ ∈ C ∞ c (R 3 ).
We then obtain that ṽ ∈ V (Ω).Since P N preserves the divergence [11, Lemma 3.7], the following identity holds for any ϕ ∈ Passing to the limit in the previous identity gives We then obtain that ṽ = P V (Ω) v and the sequence (P EN (Ω) v N ) N ≥1 converges to P V (Ω) v weakly in L 2 (Ω) 3 .We can write Using the convergence of the sequence (v N ) N ≥1 to v in L 2 (Ω) 3 and the weak convergence of the sequence (P The weak convergence of the sequence (P EN (Ω) v N ) N ≥1 to P V (Ω) in L 2 (Ω) 3 and convergence of the sequence ( 3 gives the expected result. Since the predicted velocities are bounded in the • 1,2,N norm (see Lemma 3.2), their | • | * ,0,N semi-norm is controlled by their | • | * ,1,N semi-norm thanks to Lemma 3.4.As in Lemma 2.5, we can therefore obtain an estimate on the time translates for the L 2 (0, T ; L 2 (Ω) 3 ) norm, and, as a consequence, the L 2 (0, T ; L 2 (Ω) 3 ) convergence of the predicted velocities.Lemma 3.6.Under the assumptions of Theorem 3.1 the sequence (ũ N ) N ≥1 satisfies uniformly with respect to N , and is therefore relatively compact in L 2 (0, T ; L 2 (Ω) 3 ).
Proof.We follow the proof of Lemma 2.5.By the triangle inequality, For any N , A N (τ ) → 0 as τ → 0, but owing to (43), we get that A N (τ ) → 0 as τ → 0, uniformly with respect to N .Let us prove that this is also the case for B N (τ ) → 0. Since u N (t) ∈ E N (Ω) for any t ∈ (0, T ) we have for any t ∈ (0, T − τ ) Now thanks to Lemma 3.4, for any ε > 0, there exists C ε > 0 and N ε ≥ 1 such that for any N ≥ N ε and for any t ∈ (0, T − τ ) In particular for any N ≥ N ε and for any τ ∈ (0, T ) we have Therefore, owing to lemmas 3.2 and 3.3, for any N ≥ N ε and for any τ ∈ (0, T ) Proof.Using Lemma A.4 there exists a sequence (u n ) n≥0 of functions of V (Ω) ∩ C 1 c (Ω) 3 converging to u in L 2 (Ω) 3 .Consider the following problem: By the Lax-Milgram theorem, there exists a unique ũn ∈ H 1 0 (Ω) 3 to this problem; indeed, the left hand-side of (59) is a bilinear continuous and coercive form on Let us now give the decomposition result which was used for the proof of existence of a solution to the correction step (10).Then w = v + ∇ψ with Ω v • ∇ξ dx = 0 for any ξ ∈ H 1 (Ω).
The following lemma gives a characterisation of the gradient which is used in the proof of Lemma 2.4.Its proof is a simple consequence of a result of M. E. Bogovskii [2] and refer to the very clear presentation of [5] for more on this subject.Proof.We recall that L 2 0 (Ω) = {q ∈ L 2 (Ω) such that Ω q(x) dx = 0}.A classical result [2] gives the existence of an linear continuous operator B : L 2 0 (Ω) → H 1 0 (Ω) 3 such that div(B(q)) = q a.e. in Ω. Furthermore B(ϕ) ∈ C ∞ c (Ω) 3 for any ϕ ∈ C ∞ c (Ω) ∩ L 2 0 (Ω).For q ∈ L 2 0 (Ω) we set T (q) = Ω f •B(q) dx.The mapping T is a linear continuous form on L 2 0 (Ω).There exists ξ ∈ L 2 0 (Ω) such that A consequence of this lemma is the following interesting per se density result.Proof.Equipped with the L 2 (Ω) 3 -norm, the space V (Ω) is a Hilbert space.In order to prove this density result, we prove that, in this Hibert space, V(Ω) ⊥ = {0}.
discretization in the sense of [11, Definition 2.1], with M N (resp.E N ) the set of cells (resp.faces), (32c) h N = max K∈MN diamK is the space step.
of the time translates of the predicted velocity u N are then obtained from the discrete momentum equation(12): see Lemma 3.3.Again, this is only an intermediate result since we seek an estimate on the time translates of the predicted velocity in the L 2 (L 2 ) norm.So next, as in the semi-discrete case, we introduce the discrete equivalent of the semi-norm | • | * ,0,N .
11, Lemme 2.3] and therefore there exists C 9 ∈ R + depending only on Ω and on θ N in a nondecreasing way, such that