An extension of the MAC scheme to locally refined meshes : convergence analysis for the full tensor time dependent Navier Stokes equations

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1 An extension of the MAC scheme to ocay refined meshes : convergence anaysis for the fu tensor time dependent Navier Stokes equations Eric Chénier, Robert Eymard, Thierry Gaouët and Raphaèe Herbin Apri 0, 0 Abstract A variationa formuation of the standard MAC scheme for the approximation of the Navier-Stokes probem yieds an extension of the scheme to genera D and 3D domains and more genera meshes. An origina discretization of the triinear form of the noninear convection term is proposed; it is designed so as to vanish for discrete divergence free functions. This property aows us to give a mathematica proof of the convergence of the resuting approximate soutions, for the noninear Navier-Stokes equations in both steady state and time dependent regimes, without any sma data condition. Numerica exampes (anaytica steady and time dependent ones, incined driven cavity) confirm the robustness and the accuracy of this method. Keywords: MAC scheme, incompressibe steady and time dependent Navier-Stokes equations, non conforming grids. MSC00: 65N08,76D05 Introduction The Marker-And-Ce (MAC) scheme, introduced in [4] is one of the most popuar methods [0, 6] for the approximation of the Navier-Stokes equations in the engineering framework, because of its simpicity and of its remarkabe mathematica properties. The discrete unknowns are the components of the veocity and the pressure on staggered grids; the mass conservation equation and momentum conservation equations are discretized in such a way that the kinetic energy remains controed. The first error anaysis seems to be that of [] in the case of the time dependent Stokes equations on uniform square grids. The mathematica anaysis of the scheme was aso performed for the Stokes equations in [7] for uniform rectanguar meshes, and generaized to non uniform rectanguar meshes and irreguar source terms in []. Error estimates may be obtained by viewing the MAC scheme as a mixed finite eement method of the vorticity formuation [], or by a mixed method in primitive variabes, with the pressures approximated by Q finite eements [3]. Aong the same ines, it is proven in [5] that a divergence conforming DG scheme based on the owest order Raviart-Thomas space on rectanguar meshes is agebraicay equivaent to the MAC scheme. Error estimates for rectanguar meshes were aso obtained for the reated covoume method, see [5] and references therein. Mathematica studies of the MAC scheme for the non inear Navier-Stokes equations are scarcer. To our knowedge, the ony convergence study is that of [8] for the steady state Navier-Stokes equations and for uniform rectanguar grids. Extensions of the MAC scheme on genera unstructured grids are not easy to derive, and most of the mathematica anayses that we described are restricted to rectanguar ces. The covoume approach [9] may be seen as one way to generaize the MAC scheme on unstructured grids. MAC schemes on trianguar Université Paris-Est, eric.chenier@univ-mv.fr Université Paris-Est, robert.eymard@univ-mv.fr Aix-Marseie Université, thierry.gaouet@univ-amu.fr Aix-Marseie Université, raphaee.herbin@univ-amu.fr

2 meshes are proposed and tested in [5]. A variationa MAC scheme based on acute trianges is proposed in [7], and the convergence anaysis is competed. The aim of this paper is to provide the compete mathematica anaysis of an extension of the MAC scheme on possiby non conforming meshes, aowing oca refinement. This extended MAC scheme is a sight modification of a MAC ike scheme which was presented in [4]. The modification concerns the discretization of the momentum equation, which was performed on dua Voronoï ces in [4], whie it is performed by a inear finite eement method on a Deaunay trianguation buit from the centers of the set of edges (in D), or faces (in 3D) where each component of the veocity is defined. This modification was found necessary in the mathematica anaysis of the scheme for the steady state and time dependent noninear Navier-Stokes equations, in order to prove the convergence of the scheme. It aso aows to easiy hande the case of the fu tensor viscosity. The convergence of a coocated finite voume scheme was proven in [9] for the steady state and time dependent Navier Stokes equations, and we sha use some of the toos therein in the present anaysis. The outine of the paper is the foowing. In Section, we give the weak form of the steady state and time dependent Navier Stokes equations. In Section 3 we first write a discrete variationa formuation of the standard MAC scheme on the inear Stokes probem, and use this variationa formuation to extend the MAC scheme to more compex geometries. The extension to the steady state Navier Stokes equation is presented in Section 4 where we aso prove the convergence of the approximate soutions to a weak soution. The proof of convergence reies on a proper choice of the convection term, which takes the mass conservation into account. We then consider the case of the time dependent probem in Section 5, and again prove the convergence of the method in this case, thanks to a discrete Aubin Simon type resut. In Section 6, the efficiency of the extended scheme is iustrated by numerica exampes on a non rectanguar domain, using a oca refinement aong the boundary of the domain. The first exampe is a steady state probem on a circuar domain, for which the exact soution is known so that we may assess the numerica order of convergence. We then consider the incined driven cavity probem, which impies the discretization of a non rectanguar domain. The comparison with the iterature shows a very good accuracy. We finay consider the time dependent Green Tayor vortex probem, that we approximate on a circuar domain. Weak formuation of the Navier Stokes equations Let be an open bounded set of R d, where d denotes the space dimension. In the remainder of this paper, we assume the domain to be poygona (d = ) or poyhedra (d = 3), in the sense that is a finite union of subsets of hyperpanes of R d. Let β [0, + ) denote the Reynods number. We first consider the steady state case with a time-independent forcing term f L () in the momentum equation. Then, a weak soution to the steady state Navier-Stokes equations with homogeneous Dirichet boundary conditions is a vector function u with components (u (i) ) i=,...,d, such that u E(), p L () with p(x)dx = 0, S(u, v)(x)dx + β (u(x) )u(x) v(x)dx () p(x)divv(x)dx = f(x) v(x)dx, v H0 () d, where E() = {v = (v (i) ) i=,...,d H 0 () d, divv = 0 a.e. in }. We consider two cases for the stress tensor S(u, v), namey: - the usua simpified form for the incompressibe Stokes or Navier-Stokes probem, which reads S(u, v)(x) = u(x) : v(x) = - the fu stress tensor d u (i) (x) v (i) (x), for a.e. x, u, v H0 () d, () i= S(u, v)(x) = λdivu(x)divv(x) + µɛ(u)(x) : ɛ(v)(x), for a.e. x, u, v H 0 () d, (3) where µ > 0 and 3λ + µ 0 (these vaues coud depend on the space variabe through a couped variabe, such as the temperature) and ɛ(u) i,j (x) = ( iu (j) (x) + j u (i) (x)), for i, j =,..., d.

3 3 The coefficient β is stricty positive in the genera (noninear) case and is set to 0 to obtain the inear Stokes probem. We then consider the time dependent case, for which we consider a finite time T of study of the fow, a time-dependent forcing term f L ( (0, T )) d in the momentum equation, and an initia condition u ini L () d. Then, a weak soution to the time dependent Navier-Stokes equations with homogeneous Dirichet boundary conditions is a vector function u with components (u (i) ) i=,...,d, such that u L (0, T ; E()) L (0, T ; L () d ), = T 0 T + 0 T 0 u(x, t) t ϕ(x, t) dx dt S(u, ϕ)(x, t) dx dt + β T 0 u ini (x) ϕ(x, 0) dx (u(x, t) )u(x, t) ϕ(x, t)dx dt f(x, t) ϕ(x, t) dx dt, ϕ L (0, T ; E()) C c ( (, T )) d. Remark. It may be proved that any weak soution u of (4) satisfies t u L 4/d (0, T ; E() ) in the foowing cassica sense T T ϕ(t) t u(t), v E(),E()dt = ϕ (t) u(x, t)v(x)dxdt, 0 0 v E(), ϕ Cc ((0, T )). (4) 3 A variationa MAC scheme Our extension of the MAC scheme to non conforming meshes is based on a discrete variationa formuation. Hence in this section, we begin by considering a variationa formuation of the standard MAC scheme for the approximation of the Stokes probem, that is () with β = 0. We then extend this variationa scheme to non conforming meshes. 3. The standard MAC scheme for conforming meshes For the sake of simpicity, et us first consider a two dimensiona square domain =]x, x[ ]y, y[. The domain is discretized in the MAC way, that is with a staggered arrangement of the rectanguar discretization ces for the pressure and each of the veocity components, as depicted in Figure. Let N and M be two σ E () σ E () y M+ = y σ K V () V σ () y j+ y j y j e () x = x x i x i x i+ x N+ = x y = y e () Figure : Notations for the standard MAC scheme positive integers, and et M be the set of pressure grid ces (with the notations given in Figure : { } M = ]x i, x i+ [ ]y j, y j+ [, i N, j M.

4 4 The above notations may easiy be extended to the case d = 3. For the space dimension d equa to or 3, we denote by E = d i= E (i) the set of the edges or faces of the mesh, where E (i) is the set of edges associated to the i-th component of the veocity. In order to define the norma veocity fux from one ce to a neighbouring one, we introduce, for any pair σ, σ E (k), k = or, the transmissivity τ (k) σ,σ ce V (k) σ and ce V (k) σ : τ (k) σ,σ between (k) V = σ V (k) σ, (5) d(x σ, x σ ) where V σ (k) V (k) (k) σ denotes the ength of the ine segment which is the intersection of V σ and V (k) σ, and d(x σ, x σ ) denotes the distance between the points x σ and x σ. For instance, for a vertica edge σ = {x i+ } ]y j, y j+ [ E (), one has: τ () σ,σ = x i+ 3 y j+ y j x if σ = {x i+ 3 } ]y i+ j, y j+ [, (6) x i+ x i if σ = {x y j+ y i+ } ]y j j+, y j+ 3 [. For any K M, we denote by E K the subset of E int containing a edges (or faces) of K which are interna, and by E (k) K = E K E (k). For an interna edge σ E int separating two ces K and L, we sha write σ K L = L K (at this stage, we coud write σ = K L but in the generaized MAC scheme, interfaces wi be aowed to contain more than one edge or face). We denote by (e (k) ) k=,...,d the canonica orthonorma basis of R d and, for σ K L, with K, L M and by n K,σ the unit norma vector to σ outward to K. We then write σ K L in the case where σ K L E (k) for some k =,..., d and n K,σ e (k) =. We finay represent by D = (M, E) the coection of a the space discretization data. Let us consider here the simpest case of the Stokes equations, i.e. β = 0, with the stress tensor given by (). The standard MAC scheme may then be written: Find (u σ ) σ E R, (p K ) K M R ; K p K = 0, d k= σ E (k) K K M σ u σ e (k) n K,σ = 0, K M, σ E (k) τ (k) σ,σ (u σ u σ) + σ (p L p K ) = V (k) σ f (k) (x)dx, k =,..., d, σ E (k), and K, L M such that σ K L. (7a) (7b) Let us define H M () as the set of piecewise functions constant in K M (pressure unknown), and H (k) E () as the set of piecewise functions which are constant in V σ, for σ E (k) (kth component of the veocity unknown). Let H E () = {v = (v (k) ) k=,...,d ; v (k) H (k) E ()}. The piecewise constant discrete divergence of v H E () is the function div M v H M () defined by: div M v(x) = div K v = K d k= σ E K σ v K,σ, for a.e. x K, K M, v H E (), (8) where v K,σ = v σ n K,σ e (k). The discrete mass conservation equation (7a) can then be simpy written as: div M u(x) = 0, for a.e. x. (9) We now introduce a variationa formuation of the discrete momentum equation (7b) by recaing (see e.g. [8, Chapter 3] that if the points x σ are the nodes of a Deaunay trianguation, then τ () σ,σ = ξ () σ (x) ξ () σ (x)dx, where the functions (ξ () σ ) σ E () are the P finite eement basis functions defined on the Deaunay trianguation, such as the one depicted in Figure (in which the nodes are the points x σ, for a σ E () ). The

5 5 Figure : Trianguar mesh for the P finite eement approximation û () of u (). functions (ξ σ () ) σ E () are defined in a simiar way, and we may then define û (k) H0 () such that û (k) = σ E (k) u σ ξ (k) σ, k =,..., d. (0) We then define an inner product, E on H E () (this hods for both cases () and (3)) by u, v E = S(û, v)(x)dx, u, v H E (), () which we expect to approximate the inner product S(u, v)(x)dx of the continuous probem. We then obtain, mutipying (7) by v σ and summing on k =, and σ E (k), u, v E p(x)div M v(x)dx = f(x) v(x)dx, v H E (), () A discrete variationa formuation of the standard MAC scheme (7) is therefore: Find u H E () and p H M () s. t. K p K = 0 and (9) and () hod. (3) K M 3. The extended MAC scheme for non conforming meshes Let us now turn to an extension of the above depicted standard MAC scheme to more genera meshes (incuding oca refinement as in Figure 6; note that these oca refinements may be used to foow the contours of a genera non rectanguar domain, as in Figures 3 and 4. We consider D or 3D meshes of, which are such that a interna edges (D) or faces (3D) (from now on, we ony use the word face, in D or 3D) have their norma vector parae to one of the basis vector e (k) of the space R d, for some k =,..., d. In other words, a interna edges must be aigned with one of the reference axes. Note that on the other hand, the externa faces, that is the faces of the mesh ying on the boundary need not be aigned with the axes: they are ony assumed to be panar. In fact a tited cavity such as the one depicted in Figure 3 is easiy meshed with such a grid. Curved boundaries may aso be meshed with such grids, by using oca refinement cose to the boundaries, such as in Figure 5. We denote by M the set of pressure ces. Exampes of resuting pressure grids are depicted in the eft part of Figures 3 and 5. For a K M, we again denote by E K the set of a interna faces of K (therefore, the faces of K which are on are not eements of E K ), and we define the set E as the union over K M of a the sets E K. It is assumed that a given σ E int is entirey incuded in an interface between two ces, say K and L; we sha write σ K L. Note that an interface K L is aowed to contain severa faces of the mesh; this may for instance happen in adaptive mesh refinement (and de-refinement) procedures. We then introduce the set E (k) as the subset of E which contains a the interna faces whose norma is parae to the basis vector e (k). For any σ E, x σ denotes the center of gravity of σ.

6 6 In order to get a discrete variationa MAC-ike scheme, we consider, for any k =,..., d, the set of interna points V (k) int = (x σ ) σ E (k) and a given famiy of externa points V (k) ext containing at east a the vertices of. The ony additiona difficuty on the non conforming grid is the discretization of the diffusion term, which we simpy discretize by a inear finite eement approximation. To this purpose, we introduce a Deaunay trianguation T (k) of whose vertices are V (k) int V(k) ext. Such a trianguation of is a set of simpices: trianges in D, tetrahedra in 3D. Each simpex has d + vertices which beong to V (k) int V(k) ext. The trianguation is assumed to satisfy the Deaunay property, which means that the interior of the circumcirce (in D) or of the circumsphere (in 3D) of any simpex T T (k) does not contain any eement of V (k) int V(k) ext. Exampes of Deaunay trianguations constructed from the edge mid-points are iustrated in Figures 3 and 5 (midde and right parts). We then denote, for any σ E (k), the function ξ σ (k), which is continuous, piecewise P on any T T (k), and whose vaue is at the point x σ and 0 at points x σ for any σ E (k) \ {σ}. Let {V (k) σ as foows:, σ E (k) } be the Voronoï mesh associated to the famiy (x σ ) σ E (k), defining the Voronoï ces V (k) σ = {x, d(x, x σ ) < d(x, x σ ), σ E (k) \ {σ}}, σ E (k). See Figures 4 and 6 for exampes of the superposition of the Deaunay trianguations and the Voronoï mesh construction. We finay denote by D the coection of a the space discretization data. Remark 3. Note that in the case of a uniform rectanguar mesh, the Voronoï ces thus defined are equa to the veocity ces defined in the previous section. However, this is no onger true if a non uniform mesh is used, even in the conforming case; indeed, in this atter case, the Voronoï ces V σ (k) are again rectanges, but they are not equa to the (rectanguar) veocity ces of the cassica MAC scheme. In the case of hanging nodes, they are no onger rectanguar, as can be seen in Figure 6, where we depict the Deaunay grid and the Voronoï ces for the horizonta and vertica veocities. Figure 3: The pressure (eft) and veocity grids (midde: horizonta veocity, right: vertica veocity). Figure 4: The Deaunay trianguation and the Voronoï ces (eft: horizonta veocity, right: vertica veocity). We may again define H M () as the set of piecewise functions constant on the pressure ces K M, the set H (k) E () of piecewise constant functions on the dua grid ces V σ, for σ E (k) ; this discrete set is the space of functions meant to approximate the k-th component of the veocity. We then denote by H E () the set of a v = (v (k) ) k=,...,d with v (k) H (k) E (). We then define: v (k) = v σ ξ σ (k) H0 (), (4) σ E (k) we denote by v = ( v (k) ) k=,...,d, and we define the norm v E = v L () d d, v H E(). (5) The extended MAC scheme for the Stokes equations (β = 0) is again (8)-(3), appying Definition () for, E.

7 7 Figure 5: The pressure (eft) and veocity grids (midde: horizonta veocity, right: vertica veocity). Figure 6: The Deaunay trianguation and the Voronoï ces (eft: horizonta veocity, right: vertica veocity). 4 The extended MAC scheme for the steady Navier-Stokes equations 4. Discretization of the noninear convection term In order to write this generaized scheme for the Navier-Stokes equations, we ony need to give a discretization of the noninear term (u(x) )u(x) v(x)dx. To this purpose, we introduce a discrete triinear form b E which aims at discretizing the triinear form b defined over (H0 ()) 3 by b(u, v, w) = (u v)w dx. We begin by defining some interpoation operators between H E () and (H M ()) d. Definition 4. (From H E () to H M ()... and back) Let D be a possiby non conforming mesh as depicted in Section 4. For v H E (), we define Π K v by its components (Π K v) (k) : (Π K v) (k) = σ v σ, k =,..., d, (6) σ E (k) K σ σ E (k) K and Π M v H M () = (H M ()) d as the piecewise constant function equa to (Π K v) on ce K. We then define Π E v H E () as the foowing piecewise constant function on the Voronoï ces: Π E w = σ E int σ K L (Π Kw + Π L w) Vσ, where Vσ denotes the characteristic function of V σ, that is Vσ (x) = if x V σ and 0 otherwise, Definition 4. (Discrete gradient and convection term ) For v H M (), we define its discrete gradient E v H E () by: ( E v = () E v,..., (d) E ) t v (i), with E v = σ v Vσ σ E (i) and σ v = { (v L v K ) σ V, σ if σ E int, σ K L, 0, if σ E ext, (7)

8 8 where σ K L means that σ K L E (k) for some k =,..., d and n K,σ e (k) =. We then define the foowing triinear form b E on (H E ()) 3 by: b E (u, v, w) = (u E )Π M v Π E w dx, for u, v, w H E () (8) where, for any ṽ H M (), d (u E )ṽ(x) = u σ σ ṽ Vσ (x). i= σ E (i) It is we known that the continuous triinear form satisfies b(u, v, v) = v divu dx for a u, v H0 (), and therefore, b(u, v, v) = 0 if divu = 0. Simiary, we have the foowing resut for the discrete triinear form. Lemma 4. (Properties of the triinear form) Let u, v, w H E () and et b E be defined by (8). Then: b E (u, v, w) = K M σ E K σ K L where u K,σ = u σ n K,σ e (k). Moreover, σ u K,σ Π L v Π K v b E (u, v, v) = Π K w, u, v, w H E (), (9) Π M v div M u, (0) where the discrete divergence operator div M is defined by (8); therefore, if div M u = 0, b E (u, v, v) = 0. Proof. From (8) we get that b E (u, v, w) = σ E σ K L u σ (Π L v Π K v) σ (Π Kw + Π L w). = u σ (Π L v Π K v) σ Π K w + u σ (Π K v Π L v) σ Π K w. σ E σ K L σ E σ L K Reordering the summation over the edges of each ce K yieds (9). Taking w = v in (9) then yieds that b E (u, v, v) = K M σ E K σ K L σ u K,σ ( ΠL v Π K v Π K v ), which yieds (0) by conservativity. The extended MAC scheme for the Navier-Stokes equation then reads: Find u H E () and p H M () s. t. K p K = 0, K M div M u(x) = 0, for a.e. x. u, v E p(x)div M v(x)dx + β b E (u, u, v) = f(x) v(x)dx, v H E (). (a) (b) (c) Remark 4. (Link with the cassica MAC scheme) In the case of the Stokes probem on a conforming rectanguar grid, the scheme presented here is equivaent to the cassica MAC scheme. In the case of the noninear Navier Stokes equations, the scheme is not quite the same, since the centred MAC scheme has a 3 point stenci in each direction whereas the scheme presented here has a 5 point stenci. This is due to the fact that we chose to take Π E w rather than w in the expression of the triinear form (8), in order to obtain the property (0), which is quite usefu to obtain the estimates on u.

9 9 4. Mathematica anaysis Let D be a possiby non conforming mesh, such as introduced in Section 3.. We define h D as the maximum vaue of the diameters of a T T (k) and a V σ (k), σ E (k), k =,..., d. The reguarity θ D of D is defined as the minimum vaue of:. a ratios diam(t )/diam(t ), for a neighbouring simpices T, T T (k), k =,..., d,. a ratios T /diam(t ) d, T T (k), k =,..., d, 3. a ratios V σ (k) vertex x (k) σ. / T, T / V (k) σ, diam(v (k) σ 4. a ratios σ diam(k)/ V σ, σ E int, K M σ. Let us then introduce the foowing interpoation operators. )/diam(t ), diam(t )/diam(v σ (k) ), for any T T (k) with Definition 4.3 (Interpoation operators) For v = (v (),..., v (d) ) H0 () d, et P E v H E () be defined by the components (P E v) (k) of its piecewise constant vaues on the dua ces V σ (k), for k =,..., d. (P σ v) (k) = v (k) ds(x), σ E (k). () σ σ For ϕ H 0 (), et P M ϕ H M () be defined by its piecewise constant vaues on the ces K M: (P M ϕ) K = ϕ(x K ), K M. (3) Foowing the ideas of [] appied to the piecewise P interpoation P E v of v, one may check that the interpoation operator P E satisfies the foowing property. Lemma 4. Under the assumptions of Definition 4.3, there exists C, ony depending on θ and, such that P E v E C v H () d. (4) Lemma 4.3 (Properties of the discrete divergence) Let v (H0 ()) d, et P E be the interpoation operator defined by ().and et p H M. Then p(x)div M P E v(x)dx = p(x)divv(x)dx. (5) Moreover, for any u H E, and for any q H M (), one has: q(x)div M udx = E q(x) u(x)dx. (6) Proof. The first resut is an obvious consequence of the reation 8 defining the discrete divergence operator and of the fact that p is piecewise constant on the ces K M. The second resut foows from the fact that q(x)div M udx = K M = σ K L = q K d k= σ E (k) K (q K q L ) σ u σ E q(x) u(x)dx. σ u σ e (k) n K,σ The foowing emma gives some estimates on the triinear form which are used to obtain some estimates on the soutions of the schemes, for both the steady state treated in this section and the time dependent case in Section 5.

10 0 Lemma 4.4 (Estimates on the triinear form) Let D be a discretization in the sense given in Section 4. and et b E be the triinear form defined by (9). Then there exist C and C 3, ony depending on any θ θ D and on, such that b E (u, v, w) C u (L 4 ()) d v E w (L 4 ()) d C 3 u E v E w E, u, v, w H E (). (7) Moreover, there exists C 4, ony depending on and θ, such that b E (u, u, v) C 4 u / L () u 3/ E v E, u H E (); div M u = 0, v H E (). (8) Proof. From the definition (8) of the triinear form b E denoting by ṽ (j) (resp. w (j) ) the j-th component of Π M v (resp. Π E w), we get thanks to the Cauchy Schwarz inequaity, we get that: b E (u, v, w) d d k= j= u (k) L 4 ()) (k) E ṽ(j) L ()) w (j) L 4 ()) d u (L 4 ()) d E(Π M v) (L ()) d d Π Ew (L 4 ()) d. Let us then remark that the components of Π Kw+Π L w are barycentric combinations of neighbouring terms w σ ; hence there exists C 5, ony depending on and θ, such that Π E w (L4 ()) d C 5 w (L4 ()) d. In order to bound the term E (Π M v) (L ()) d d, we first define the functions Π(k) v and Π (k) + v by and Π (k) v(x) = Π K v, σ E (k), for a.e. x V σ (k), and K, L such that σ K L, + v(x) = Π L v, σ E (k), for a.e. x V σ (k), and K, L such that σ K L. Π (k) We finay define δ (k) (x) = V (k) σ σ for a x V σ (k). We then get that (k) E Π Mv (L ()) = Π (k) d δ (k) (x) + v(x) Π (k) v(x) dx d ( = Π (k) δ (k) (x) + v () (x) Π (k) v (x)) () dx. = In order to appy Lemma 7., we first remark that, thanks to the reguarity hypotheses of the mesh, there exists some C 6 > 0, ony depending on θ, such that δ (k) (x) C 6 δ(x), with δ(x) defined in Lemma 7.. Since the functions Π (k) v () (x) ony depend on the discrete unknowns vσ, we may introduce the functions ψ k,,σ such that Π (k) v () (x) =,σ(x), which are the piecewise constant functions defined by: σ E (k), with K, L such that σ K L, ψ k,,σ (x) = σ E () v σψ k, σ σ E () K σ if σ E () \ E K, for a.e. x V (k) σ, 0 if σ E () \ E K or if x / V (k) The functions ψ k,,σ satisfy the hypothesis (49) of Lemma 7.; furthermore, thanks to the reguarity of the mesh, the mesh dependent bound C 8 defined by (50) remains bounded by a function of θ. We may thus appy Lemma 7., which yieds (5). This impies ( ) Π (k) δ (k) (x) v () (x) v () C 9 (x) dx (C 6 ) v() L () d. σ.

11 Simiary, we have the same inequaity with Π (k) + v () instead of Π (k) v (), which shows the existence of C 7 > 0, ony depending on θ, such that E (Π M v) (L ()) C 7 v d d L () d d, thus concuding the eft inequaity of (7). Now, thanks to the equivaence of the norms proved in Lemma 7. in the appendix (inequaity (55)), there exists C 8 R +, ony depending on any θ θ D, such that u (k) L 4 () C 8 û (k) L 4 (), Appying the standard Soboev inequaity û (k) L 4 () C sob û (k) L () d, where C sob ony depends on and d, we concude that there exists C 9, ony depending on and on any θ θ D, such that u (k) L 4 () C 9 û L () d d. Therefore, we concude the right inequaity of (7). Let us then prove (8); since div M u = 0, we have which proves, using (7), that Using the Cauchy Schwarz inequaity, we have: b D (u, u, v) = b D (u, v, u), b D (u, u, v) C u L 4 () v E. u L 4 () u / L () u 3/ L 6 (), We again appy Lemma 7.: there exists C 0 R +, ony depending on θ and, such that u L 6 () C 0 û L 6 (). Appying the standard Soboev inequaity, we get that û L 6 () C sob û L () d d, where C sob R + depends ony on and d; therefore, by the definition (5) of the norm E, we get: b D (u, u, v) C u / L () u 3/ E v E. which concudes the proof of Lemma 4.4. Lemma 4.5 (Estimates on the veocity and the pressure for the Navier-Stokes probem) Let D be a discretization in the sense given in Section 4. and et θ θ D. Let (u, p) H E () H M () be a soution to (). Then there exists C, ony depending on (and on λ and µ in the case (3), this dependency is no onger mentioned in this paper), such that u satisfies the foowing estimates: Moreover, there exists C ony depending on β, θ and such that u E C f (L ()) d. (9) p L () C f (L ()) d. (30) As a direct consequence, there exists one and ony one soution (if β = 0) and at east one soution (if β > 0) to the scheme (). Proof. We et v = u in (). Let us first remark that we have p(x)div Mu(x)dx = 0 and that by the definition (9) of b E, we have b E (u, u, u) = K M Π Ku d k= σ E (k) K σ u σ e (k) n K,σ = 0,

12 since div M u(x) = 0. Hence we obtain C 3 u E S(u, u) L () = f(x) u(x)dx, where we denote by C 3 > 0 the constant invoved in the Korn inequaity, which hods for both cases () and (3). We then have, thanks to the Cauchy Schwarz inequaity, u E f L () d u L () d. Let û be defined by (0). Appying Lemma 7., we get that there exists C 4, ony depending on θ, such that Thanks to the Poincaré inequaity, we may write u L () d C 4 û L () d. û L () d diam() u E. (3) Therefore we get (9). Let us now turn to the proof of (30). We use the foowing property, first due to Nečas [6], see aso [3]: there exists v H 0 () d such that p(x) = divv(x) for a.e. x and there exists C 5, ony depending on, such that v H () d C 5 p L (). (3) Let P E be the interpoation operator defined by (). By Lemma 4.3, we have: p(x)div M P E v(x)dx = p(x)divv(x)dx = p(x) dx. We then get, using P E v H E () as test function in Scheme (), u, P E v E p(x) dx + β b E (u, u, P E v) = f(x) P E v(x)dx. (33) Thanks to the Cauchy Schwarz inequaity, we get that there exists C 6 (ony depending on λ and µ in the case (3), this dependency is no onger recaed), such that u, P E v E C 6 u E P E v E. Using Inequaity (4), we get that there exists C 7, ony depending on θ and, such that u, P E v E C 7 u E v H () C 7C d f (L ()) dc 5 p L (). We may aso write, thanks to the Poincaré inequaity and to (3), f(x) P E v(x)dx diam()c 5 f (L ()) d p L (), Therefore, using (9) and (7), we get the existence of C 8 such that ) p L () C 8 ( f (L ()) + d f (L ()) p d L (). which concudes the proof of (30). In the case where β = 0, the inequaities (9) and (30) suffice to prove that the square inear system issued from Scheme () (repacing one equation of (9) by p(x)dx = 0) is invertibe. If β > 0, using the topoogica degree argument as done in [9], we concude to the existence of at east one soution to Scheme (). The next emma shows a property of weak convergence of the discrete gradient which is used in the passage to the imit in the scheme.

13 3 Lemma 4.6 (Weak convergence of the gradient) Let F be a famiy of discretizations in the sense given in Section 4.. For any D F, et v D H M (), such that: the famiy ( E v D ) D F is bounded in (L ()) d, (34) the famiy (v D ) D F converges weaky in L () to v L () as h D 0. (35) Then v H 0 (), and the famiy ( E v D ) D F converges to v weaky in L () as h D tends to 0. Proof. Let ψ = (ψ,..., ψ d ) (C (R d )) d ; then E v D (x) ψ(x)dx = E v D (x) P E ψ(x)ds(x) + R E where P E is the interpoation operator defined by (), and R E C ψ h D E v D (L ()) 0 as h D 0, thanks to Assumption (34). Now, by (6) and (5) of Lemma 4.3, E v D (x) P E ψ(x)ds(x) = v D (x)divψ(x)dx. Therefore, by Assumption (35), E v D (x) P E ψ(x)ds(x) v(x)divψ(x)dx, which shows that E v D tends to v in the distribution sense. Since the sequence ( E ) D F is bounded in L (), we have that v L (), and proonging v D by 0 outside of, we get that v = 0 in R d \. Therefore, E v D tends to v H0 () weaky in (L ()) d. Theorem 4. (Convergence of the scheme) Let F be a famiy of discretizations in the sense given in Section 4., such that there exists θ > 0 with θ θ D for a D F. For any D = (M, E) F, et (u E, p M ) H E ()s H D () denote a soution to Scheme (). Then there exists a weak soution (u, p) of () such that, up to a subsequence, u E converges in L () d to u and p M converges in L () d to p as h D 0. Moreover, if β = 0, the whoe famiy converges to the unique weak soution of () as h D 0. Proof. Let (D () ) N be a sequence of eements of F, such that h D () tends to 0 as and such that there exists θ > 0 with θ D () θ, for a m N. Let (u E (), p M ()) denote a soution of () for the discretization D = (M, E, δt ). From estimate (9), using Reich s theorem, we deduce that there exists u H0 () d (with u = (u (k) ) k=,...,d ) and a subsequence of (D () ) m N, again denoted by (D () ) N, such that û (k) H M () 0 () weaky converges in H0 () d (therefore strongy in L () d ) to u (k) as for k =,..., d. Using Lemma 7. and the estimate (9), we get that u E () aso converges in L () d to u as. Thanks to the estimate (30) on the pressure, we may then consider a subsequence of (D () ) N, again denoted by (D () ) N, such that p M () weaky converges in L () to some function p L () as, with p(x)dx = 0. We now have to prove that (u, p) is a weak soution of (), which we do by passing to the imit on the weak form of the scheme. Let us first show that u E(), i.e. divu(x) = 0 for a.e. x. Let ϕ Cc (); mutipying (b) by P M ϕ and integrating over yieds, thanks to Lemma 4.3: P M ϕ(x) div M u E (x)dx = E P M ϕ(x) u E (x)dx = 0. Thanks to the reguarity of the mesh, there exists C 9 depending ony on ϕ and θ such that E P M ϕ (L ()) C 9. Therefore, by Lemma 4.6, d E P M ϕ tends to ϕ weaky in (L ()) d as h D tends to 0. Hence, passing to the imit in the above equation, we get that ϕ(x) u(x)dx = 0 for any ϕ Cc (),which shows that divu(x) = 0 for a.e. x. Let ϕ Cc () d. Let us now show () with v = ϕ. We take (again omitting some indices (), thus denoting D = D () ) v E = P E ϕ H E (), as defined by (), as test function in Scheme (). We get that u D, P E ϕ E p D (x)div M P E ϕ(x)dx + β b E (u D, u D, P E ϕ) = f(x) P E ϕ(x)dx.

14 4 Using Lemma 4.3 and the fact that p D () weaky converges in L () to p L (), we obtain that p D ()(x)div D ()P E ()ϕ(x)dx = p(x)divϕ(x)dx. im Thanks to the weak and strong convergence properties of the different sequences, we get that im u D (), P E ()ϕ E = im S(û () D (), P E ()ϕ)(x)dx = S(u, ϕ)(x)dx. We aso have, thanks to the definition () of P E ϕ, f(x) P E ()ϕ(x)dx = im f(x) ϕ(x)dx. Let us now turn to the study of the imit of b E ()(u D (), u D (), P E ()ϕ). By the definition (9) of b E, again dropping some indices (), we have: b E (u D, u D, P E ϕ) = (u M (x) E )(Π D u E )(x) Π D (P E ϕ(x))dx, Thanks to the reguarity of the mesh, the famiy E (Π D u E ) is bounded in (L ()) d. Moreover, using Lemma 7. and the estimate (9), we get that there exists C 0, ony depending on θ, such that Π D u D u D L () d h DC 0 u D E, which proves that Π D ()u D () converges to u in L () d. Therefore, thanks to Lemma 4.6, that E ()u D () converges weaky in (L ()) d to u. Since we have that Π D ()ϕ converges in (L ()) d to ϕ, we obtain the convergence of b E ()(u D (), u D (), P E ()ϕ) to (u(x) )u(x) ϕ(x)dx as, which competes the proof that () hods for a v H0 () d by density. 5 The extended MAC scheme for the time dependent Navier-Stokes equations 5. Definition of the scheme Let D be the coection of a the space discretization data as given in Section 4.. The discrete approximation of the time dependent Navier Stokes equations (4) is then given by the foowing reations. The initia condition is approximated by u (0) H E () defined by u (0) σ = V (k) σ V (k) σ u (k) ini (x)dx, σ E (k), (36) and, for a given time step δt > 0, et us denote by t (n) = nδt for n 0. The scheme is defined by δ (n+ ) D,δt u(x) = u(n+) (x) u (n) (x), for a.e. x, (37) δt and, for a given α [, ], div M u (n+ ) (x) = 0, for a.e. x, (38) for a n N, (u (n+), p (n+ ) ) H E () H M (), p (n+ ) (x)dx = 0, u (n+ ) = ( α)u (n) + αu (n+), δ (n+ ) D,δt u(x) v(x)dx + u (n+ ), v E +β b D (u (n+ ), u (n+ ), v) = δt t (n+) t (n) p (n+ ) (x)div M v(x)dx f(x, t) v(x)dxdt, v H E (). (39)

15 5 We then denote û D,δt (x, t) = û (n+ ) (x), u D,δt (x, t) = u (n+ ) (x), and p D,δt (x, t) = p (n+ ) (x), and we define the discrete time derivative for a.e. (x, t) (t (n), t (n+) ), n N, (40) δ D,δt u(x, t) = δ (n+ ) D,δt u(x) for a.e. (x, t) (t (n), t (n+) ), n N. Remark 5. (Time discretization) We consider a constant time step ony for the sake of carity of notations. The mathematica anaysis is sti vaid with a variabe time step. Note that the above scheme corresponds to a Crank Nicoson-ike scheme for α =, and to an impicit scheme for α =. 5. Mathematica anaysis Lemma 5. (Estimates on the veocity and the pressure) Let δt = T/N T with N T N, et D be a discretization as defined in Section 4.. Let the initia condition u (0) be defined by (36), and, for a given α [, ], et (u(n+), u (n+ ), p (n+ ) ) H E () H E () H M () such that (38)-(39) hods for n =,..., N T. Then there exists C, ony depending on f, u ini, T and, such that the foowing discrete L (H ) and L (L ) estimates hod: N T n=0 δt u (n+ ) E C, (4) and u (n) L () d C, n = 0,..., N T. (4) Moreover we have the foowing L (L ) estimate on the pressure: p D,δt L (0,T ;L ()) C, n = 0,..., N, (43) where C ony depends on D, β, δt, f, u ini and. Therefore there exists at east one soution to (38)-(39) if β > 0 and exacty one soution if β = 0. Proof. Let us first remark that (4), for n = 0, is due to the Cauchy Schwarz inequaity appied to (36). We et u = u (n+ ) in (39). The noninear term again vanishes, which eads, using the Young, Poincaré and Korn inequaities, to: u(n+) L () d u(n) L () d + (α ) u(n+) u (n) L () d + δt C 3 u (n+ ) E diam() f (L ( (t (n),t (n+) )) d + δt u(n+ ) E. Summing for n = 0,..., N ead to both (4) and (4). Foowing the same ideas as in the proof of Lemma 4.5, we get the existence of C 3, ony depending on θ, β and such that ( δt p (n+ ) L () C 3 δt u (n+ ) E + δt u (n+ ) E+ u (n+) u (n) L () d + δt f (L ( (t (n) t (n+) ) d ). Summing the above inequaity on n = 0,..., N T, using the Cauchy Schwarz inequaity for the terms N T δt u (n+ ) E and N T N T δt f (L ( (t (n),t (n+) )) d, inequaity (4) for the term δt u (n+ ) E n=0 and inequaity (4) for the term n=0 N T n=0 u (n+) u (n) L () d ead to (43). The existence of at east one soution to the scheme is then again deduced from the use of the topoogica degree resuts, as in [9], as we as the uniqueness of the soution if β = 0. Since our aim is to appy Theorem 7., we define a second norm on H E () in the next definition. n=0

16 6 Definition 5. (Dua norm) Let us define the foowing dua norm on H E (): w H E (), w,e = sup { w(x) v(x)dx, v H E(), v E = and div M v = 0 }. (44) Let us denote by Y E the space H E () equipped with the norm,e. We define a continuous embedding of Y E in E() (reca that E() = {v (H0 ()) ; divv = 0}) by the reation w, v E(),E() = w(x) P E v(x)dx, w H E (), v E(). (45) where P E v is defined by () in Lemma.??. Note that, thanks to (4) we get that where C ony depends on θ and. w E() C w,e, w H E (), (46) In view of appying Theorem 7., et us study the dua norm of the discrete time derivative. Lemma 5. (Estimate on the dua norm of the discrete time derivative) Let δt = T/N T with N T N, et D be a discretization as defined in Section 4. and et θ < θ D. Let (u (0) ) be defined by (36), and, for a given α [, ], et (u(n+), u (n+ ), p (n+ ) ) H E () H E () H M () such that (38)-(39) hods for n =,..., N T. Then there exists C 4, ony depending on f, u ini, T,, β and θ, such that δ D,δt u L 4/3 (0,T ;Y E ) C 4. (47) (Note that, thanks to (46), a simiar inequaity hods on δ D,δt u L 4/3 (0,T ;E() ).) Proof. Let us take v H E (), with div M v = 0, as test function in Scheme (39). Using (45) eads to δ (n+ ) D,δt u(x) v(x)dx + u (n+ ), v E + β b D (u (n+ ), u (n+ ), v) = t (n+) f(x, t) v(x)dxdt. δt t (n) Since u (n+ ) satisfies (38) we get thanks to the estimate (8) of Lemma 4.4 and to the estimate (4) that: b D (u (n+ ), u (n+ ), v) C 5 u (n+ ) 3/ E v E, where C 5, ony depends on f, u ini, T, and θ. We then get that there exists C 6, ony depending on f, u ini, T, β, and θ, such that δ D,δt u(t),e C 6 u (n+ ) E + u (n+ ) 3/ t (n+) E + f(, t)dt. δt Remarking that u (n+ ) 4 3 E + u (n+ ) E and that 4 ( t (n+) 3 ) t (n+) f(, t)dt + f(, t) δt t δt L () ddt (n) L () d t (n) + δt t (n+) t (n) t (n) f(, t) L () ddt, L () d we get that there exists C 7, ony depending on f, u ini, T, β, and θ, such that ( δ D,δt u(t) 4 3,E C 7 + u (n+ ) E + ) t (n+) f(, t) L δt () ddt. Integrating for t (0, T ) and using the discrete L (H ) estimate on the veocity (4), we get that there exists C 4, ony depending on f, u ini, T,, β and θ, such that (47) hods, which concudes the proof. t (n)

17 7 Theorem 5. (Convergence of the scheme) Let F be a famiy of time space discretizations (D, δt), where D is a discretization in the sense given in Section 4. and there exists N T N such that δt = T/N T. We assume that there exists θ > 0 with θ θ D for a D F (see Section 4.). For any D = (M, E, δt) F, et (u E,δt, p mesh,δt ) denote a soution to Scheme (36)-(9)-(39) for a given α [, ]. Then there exists a weak soution u of (4) such that, up to a subsequence, u E,δt converges in L ( (0, T )) d to u as h D 0 and δt 0, which shows that t u L 4/3 (0,T ;E() ) C 4. (48) Moreover, if β = 0, the whoe famiy converges to the unique weak soution of (4) as h D 0 and δt 0. Proof. Let D = (M, E, δt ) N be a sequence of eements of F, such that h D and δt tend to 0 as. From thereon, we denote u M,δt by u for short. Step Proof that hypotheses (h-h4) of Theorem 7. hod, and consequences. In our setting, the space B of Theorem 7. is L () d. We take B = {w H E ; div D w = 0}. The norm X is the norm E, and the norm Y is defined in Definition 5.. Let (w ) N be a sequence of functions of B such that w E C for some C R +. Then by definition of the norm w E and thanks to Reich s theorem, the sequence (ŵ ) E converges in L () d to some w L () d ; therefore, by the inequaity (5) of 7. given in the Appendix, the sequence (w ) E aso converges to w in L () d. Thus, assumption (h) of Theorem 7. is satisfied. Let us then show that assumption (h) is aso satisfied. Let (w ) N be a sequence of functions of L () d such that w B and w E C for some C R +, and such that there exists w B with w w in B and w,e 0 as. By definition of the norm,e, we have w (x) w (x)dx w,e w E 0 as, which shows that w = 0. From estimates (4) and (47), we get that hypotheses (h3) and (h4) of Theorem 7. are satisfied. Therefore, we deduce that there exists u L (0, T ; L () d ) and a subsequence of (D, δt ) N, again denoted by (D, δt ) N, such that u converges in L (0, T ; L () d ) to u as. Step Proof that u is a weak soution of (4). As in [9], we easiy get that u L (0, T ; H 0 () d ) and we prove the convergence in L ( (0, T )) d to u of û. Lemma 7. impies the same convergence property for u. We get that divu(x, t) = 0 in the same manner as in the steady case. The proof that u satisfies (4) foows the same steps as the Navier- Stokes steady case, using a divergence free test function. The convergence of the additiona time term to the corresponding continuous imit is then cassica. Step 3 Proof that (48) hods. Thanks to (47), extracting a subsequence such that δ D,δt u weaky converges in the Soboev space L 4/3 (0, T ; E() ), we may pass to the imit in (45). We thus get T ϕ(t) im δ D,δt u (t), v E() 0,E()dt = v E(), ϕ Cc ((0, T )). The preceding reation aows to concude that T t u = im δ D,δt u (t), 0 ϕ (t) u(x, t)v(x)dxdt, and therefore that (48) hods.

18 8 6 Numerica exampes 6. An anaytica steady probem We consider a probem where the continuous soution of the Navier Stokes equations () in the case () with β = is given by: ū (x, x ) = π sin (πx ) cos(πx ) sin(πx ) ū (x, x ) = π cos(πx ) sin(πx ) sin (πx ) p(x, x ) = sin (πx ) sin (πx ) in a circe with centre (0.5, 0.5) and radius We consider four meshes for the mass conservation M j, j = 0,..., 3, defined in the foowing way:. a structured square 0 0 is given on the square [0, ] [0, ],. for i = 0,..., 3, et us spit in 4 contro voumes each grid bock whose centre (x, x ) satisfies (x 0.5) + (x 0.5) / i, 3. get rid of a the contro voumes K with centre (x, x ) such that (x 0.5) + (x 0.5) > Let us denote card(m j ) the number of contro voumes of the mesh M j. We get that card(m 0 ) = 604, card(m ) = 646, card(m ) = 559 and card(m 3 ) = 034. The L errors of unknowns u, u, p, respectivey denoted by e (u ), e (u ), e (p), are respectivey computed in the Voronoï grids associated to the veocity components and in M j. Left part of Figure (7) shows the errors og 0 (e (u )) and og 0 (e (p)) with respect to og 0 (/ card(m j )) for j = 0,..., 3. On right part of Figure 7 are potted the stream ines for the finest mesh. The veocity u (x) p.09x x Figure 7: Left: The L error with respect to the number of contro voumes. Right: Stream ines. components and the pressure are shown in Figure (8) for two meshes. Athough the veocity fieds are accuratey computed on the coarsest mesh, the pressure fieds show osciations where neighbouring contro voumes have contrasted sizes. However, these osciations are decreasing whie refining the mesh. 6. An incined driven cavity We consider a 30 o incined driven cavity, which corresponds to non homogeneous Dirichet boundary conditions for the veocity on the horizonta upper boundary of a paraeogram (see Figure 9) in the case (). We consider the case where the Reynods number is equa to 000, and we discretize the domain using refinement strategies. We thus show the fexibiity of this extended MAC scheme in this case. We obtain the resut shown in Figure 0. The resuts are compared to the iterature [6] in Tabe. The resuts show the expected precision.

19 9 Figure 8: Horizonta component of the veocity (eft), vertica component of the veocity (midde), pressure (right) for j = 0 (top) and j = (bottom). ξ ξ 0 0 Figure 9: An exampe of mesh, and the two media axes ξ and ξ. Figure 0: Veocity components, Dm u() (eft) u() (right) #M min u() (ξ ) ξ min u() (ξ ) ξ max u() (ξ ) ξ Generaized MAC scheme [6] 0400 (= 30 ) Tabe : Maximum and minimum of the veocity components aong the centerines ξ and ξ. 6.3 A time dependent case We consider the Green-Tayor case with β = 0, in the same domain Ω as the one considered in Section 6., which is a circe, with center (0.5, 0.5) and radius 0.45, on the time domain [0, 0.]. Nonhomogeneous

20 0 Dirichet boundary conditions are imposed. The soution is given, in the case (), by u () (x, x, t) = cos(πx ) sin(πx ) exp( π t), u () (x, x, t) = sin(πx ) cos(πx ) exp( π t), p(x, x, t) = β cos(πx ) + cos(πx ) exp( 4π t). 4 We pot in Figure the L errors evauated at time t = 0. as a function of the space step ( 0.5 og 0 (#M)). For each computation, the time step is set such that the tempora error is negigibe with respect to the spatia one. The sequence of time and space steps are given in Tab. : the ratio of two successive time steps is 4 whereas it is ony for the space step. We observe that the approximate pressure does not seem to converge in the case where α = /, which does not occur in the case α =. If we pot the vaues (p(n ) + p (n+ ) ) instead of p (n+ ), we observe that this post-processed pressure numericay converges. A the converging curves show an order. In order to check the behaviour of the pressure in the case α = /, we show, in Figures and 3, the pressures fieds, for two meshes, the eft and right figures showing two consecutive times steps, and the midde one showing the average vaue between these ones. This shows that, athough the scheme obtained in the case α = / may ead to osciations in pressures, it is simpe to get back converging pressures. δt #M Tabe : Time and space steps used for spatia order of convergence α = 5.5 α = /, average pressure α = / og 0(#M) Figure : L errors (+: veocity : pressure) as a function of the size of the mesh. Figure : An exampe of pressure soution at two consecutive time steps. p (n ) (eft) (p(n ) + p (n+ ) ) (midde) and p (n+ ) with #M = 604 and δt = In order to check the convergence order with respect to the time step, we have chosen the sequence described by Tabe 3. The errors are potted in Figure 4 (for the case α = /, we have compared the exact pressure with the average pressure (p(n ) + p (n+ ) ) instead of p (n+ ) ). This shows an order for α = /, and for α =.

21 Figure 3: An exampe of pressure soution at two consecutive time steps. p(n ) (eft) (p(n ) + p(n+ ) ) (midde) and p(n+ ) with #M = 559 and δt = δt #M Tabe 3: Time and space steps used for time order of convergence α= α = / og0(δt) Figure 4: L errors (+: veocity : pressure) as a function of the time step. 7 Concusion The extension of the MAC scheme presented here seems to show interesting properties for practica computations. The grids are easy to construct, and compex domains may easiy be dicretized with the possibiity of oca refinement. This coud ead to the use of this scheme in industria and arge scae probems. Work is in progress to design a discrete noninear convection term which woud resume to the cassica MAC scheme in the case of a uniform rectanguar grid, whie retaining the same properties in order to obtain the mathematica convergence of the discrete soutions. Appendix Interpoation resuts The first emma that we give is a technica resut which aows to bound, under adequate geometrica conditions, the difference between various interpoates by the gradient of a P reconstruction. Lemma 7. (Comparison between interpoates) Let Ω Rd, with d N be an open poygona connected open set, such that there exists a H conforming simpicia mesh T over Ω. We denote by T the eements of M, by V the set of a vertices, and by VT the set of a vertices of T and by TT T the set of the simpices sharing a face with T. For s V, we denote by ξs the P finite eement basis function associated to the node xs. We assume that are given some non-negative functions ψs, for a s V, such that X ψs (x) =, for a.e. x Ω. (49) s V

22 We denote by V the set of a pairs of vertices (s, s ) V such that ψ s (x)ξ s(x)dx > 0, and we assume that a set T (s, s ) T is defined such that it contains a the eements T T with s V T, it contains at east one T T such that s V T and for any pair (T, T ) T (s, s ), there exists a sequence of simpices T,..., T M such that T i+ T Ti for i =,..., M, T = T and T = T M. Let C 8 > be defined by C 8 = max{ diam(t )d, T T } { diam(t ) T diam(t ), T T, T T T } {#T (s, s ), (s, s ) V}. (50) For (u s ) s V R V, et us denote û = s V u sξ s and u = s V u sψ s, and, for any T T and x T, et us denote δ(x) = diam(t ). Then there exists C 9 > 0 ony depending on C 8, such that so that in particuar, δ (u û) L () C 9 û d L () d, (5) u û L () d C 9 max diam(t ) û L T T () d. (5) Proof. By definition of û and u, thanks to the fact that s V ξ s(x) = s V ψ s(x) =, and appying the Cauchy Schwarz inequaity, we get: δ (u û) L () = d δ(x) ( u s ξ s (x) u s ψ s (x)) dx s V s V = δ(x) ( (u s u s )ξ s (x)ψ s (x)) dx, s V s V δ(x) (u s u s ) ξ s (x)ψ s (x)dx. (s,s ) V We remark that the cosure of T (s, s ) contains s, s and an interior connected path from s to s, whose intersection with any T T (s, s ) is either a nonzero vector w(t, s, s ) R d or empty (then we set w(t, s, s ) = 0, and therefore satisfies w(t, s, s ) = s s. T T (s,s ) We may then write, since û is affine on each T T, u s u s = T T (s,s ) T ûẇ(t, s, s ), denoting by T û = s V T u s ξ s (x), for a x T, the constant gradient of û in the simpex T. We get (u s u s ) T T (s,s ) w(t, s, s ) which provides, denoting by (L(s, s )) = T T (s,s ) diam(t ), (u s u s ) (L(s, s )) T T (s,s ) T T (s,s ) T û, T û, and therefore δ (u û) L () d (s,s ) V = T T (L(s, s )) δ(x) T û T T (s,s ) T û ξ s (x)ψ s (x)dx (L(s, s )) δ(x) ξ s (x)ψ s (x)dx, (s,s ) V T