Convergence of the MAC scheme for the incompressible Navier-Stokes equations

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Navier-Stokes equations

Thierry Gallouët, Raphaele Herbin, J.-C Latché, K. Mallem

To cite this version:

Thierry Gallouët, Raphaele Herbin, J.-C Latché, K. Mallem. Convergence of the MAC scheme for

the incompressible Navier-Stokes equations. Foundations of Computational Mathematics, Springer

Verlag, 2018, 18 (1), pp.249-289. �10.1007/s10208-016-9338-4�. �hal-01189014v4�

INCOMPRESSIBLE NAVIER-STOKES EQUATIONS ON NON-UNIFORM GRIDS

T. Gallou¨ et

, R. Herbin

, J.-C. Latch´ e

and K. Mallem

Abstract. We prove in this paper the convergence of the Marker And Cell (MAC) scheme for the discretization of the steady-state and time-dependent incompressible Navier-Stokes equations in prim- itive variables, on non-uniform Cartesian grids, without any regularity assumption on the solution. A priori estimates on solutions to the scheme are proven; they yield the existence of discrete solutions and the compactness of sequences of solutions obtained with family of meshes the space step and, for the time-dependent case, the time step of which tend to zero. We then establish that the limit is a weak solution to the continuous problem.

2010 AMS Subject Classification. Primary 65M08, 76N15 ; Secondary 65M12, 76N19.

The dates will be set by the publisher.

KeywordsFinite-volume methods, MAC scheme, incompressible Navier-Stokes.

Communicated by Douglas N. Arnold.

Contents

1. Introduction 1

2. Space discretization 3

3. The steady case 10

3.1. The scheme 10

3.2. Stability and existence of a solution 10

3.3. Convergence analysis 16

4. The time-dependent case 20

4.1. Time discretization 20

4.2. Estimates on discrete solutions and existence 21

4.3. Convergence analysis 22

4.4. Case of the unsteady Stokes equations 25

5. Appendix: Discrete functional analysis 27

References 28

1. Introduction

Let Ω be an open bounded domain ofR^d withd= 2 ord= 3. The steady-state incompressible Navier-Stokes equations read:

div ¯u= 0 in Ω, (1a)

−∆¯u+ (¯u·∇)¯u+∇p¯= ¯f in Ω, (1b)

¯

u= 0 on ∂Ω. (1c)

Keywords and phrases:Finite-volume methods, MAC scheme, incompressible Navier-Stokes.

1 I2M UMR 7373, Aix-Marseille Universit´e, CNRS, Ecole Centrale de Marseille. 39 rue Joliot Curie. 13453 Marseille, France.

(raphaele.herbin@univ-amu.fr)

2 I2M UMR 7373, Aix-Marseille Universit´e, CNRS, Ecole Centrale de Marseille. 39 rue Joliot Curie. 13453 Marseille, France.

(raphaele.herbin@univ-amu.fr)

3 IRSN, BP 13115, St-Paul-lez-Durance Cedex, France (jean-claude.latche@irsn.fr)

4 I2M UMR 7373, Aix-Marseille Universit´e, CNRS, Ecole Centrale de Marseille. 39 rue Joliot Curie. 13453 Marseille, France.

(khadidja.mallem@univ-amu.fr)

where ¯ustands for the (vector-valued) velocity of the flow, ¯pfor the pressure and ¯f is a given field of L²(Ω)^d, and where, for two given vector fields v= (v1, . . . , vd) andw= (w1, . . . , wd), the quantity (v·∇)w is a vector field whose components are ((v·∇)w)i=Pd

k=1vk∂kwi,i∈

|1, d|

. A weak formulation of Problem (1) reads:

Find (¯u,p)¯ ∈H₀¹(Ω)^d×L²₀(Ω) such that,∀(v, q)∈H₀¹(Ω)^d×L²₀(Ω), Z

Ω

∇u¯ :∇v dx+ Z

Ω

((¯u·∇)¯u)·v dx− Z

Ω

¯

pdivv dx= Z

Ω

f¯·v dx, (2a) Z

Ω

qdiv ¯u dx= 0, (2b)

whereL²₀(Ω) stands for the subspace ofL²(Ω) of zero mean-valued functions.

The time-dependent Navier-Stokes equations are also considered:

div ¯u= 0 in Ω×(0, T), (3a)

∂tu¯−∆¯u+ (¯u·∇)¯u+∇p¯= ¯f in Ω×(0, T), (3b)

¯

u= 0 on ∂Ω×(0, T), (3c)

¯

u(x,0) =u0 in Ω. (3d)

This problem is posed for (x, t) in Ω×(0, T) whereT ∈R^∗

+; the right-hand side ¯f is now a given vector field ofL²(Ω×(0, T))^d and the initial datumu0belongs to the spaceE(Ω) of divergence-free functions, defined by:

E(Ω) =

u∈H₀¹(Ω)^d ; divu= 0 a.e. in Ω . A weak formulation of the transient problem (3) reads (see e.g. [3]):

Findu∈L²(0, T;E(Ω))∩L^∞(0, T;L²(Ω)^d) such that,∀v∈L²(0, T;E(Ω))∩C_c^∞(Ω×[0, T)),

− Z T

0

Z

Ω

u(x, t)¯ ·∂tv(x, t) dxdt− Z

Ω

u0(x)·v(x,0) dx+ Z T

0

Z

Ω

∇u(x, t) :¯ ∇v(x, t) dxdt +

Z T 0

Z

Ω

((¯u·∇)¯u)(x, t)·v(x, t) dxdt= Z T

0

Z

Ω

f¯(x, t)·v(x, t) dxdt.

(4)

The Marker-And-Cell (MAC) scheme, introduced in the middle of the sixties [21], is one of the most popular methods [28, 33] for the approximation of the Navier-Stokes equations in the engineering framework, because of its simplicity, its efficiency and its remarkable mathematical properties. The aim of this paper is to show, under minimal regularity assumptions on the solution, that sequences of approximate solutions obtained by the discretization of problem (1)(resp. (3)) by the MAC scheme converge to a solution of (2)(resp. (4)) as the mesh size (resp. the mesh size and the time step) tends (resp. tend) to 0.

For the linear problems, the first error analysis seems to be that of [29] in the case of the time-dependent Stokes equations on uniform square grids. The mathematical analysis of the scheme was performed for the steady- state Stokes equations in [26] for uniform rectangular meshes with H²-regularity assumption on the pressure.

Error estimates for the MAC scheme applied to the Stokes equations have been obtained by viewing the MAC scheme as a mixed finite element method [19,20] or a divergence conforming DG method [22]. Error estimates for rectangular meshes were also obtained for the related covolume method, see [6] and references therein. Using the tools that were developed for the finite volume theory [11,12], an order 1 error estimate for non-uniform meshes was obtained in [1], with order 2 convergence for uniform meshes, under the usual regularity assumptions (H² for the velocities, H¹ for the pressure). It was recently shown in [24] that under higher regularity assumptions (C⁴ for the velocities and C³ for the pressure) and an additional convergence assumption on the pressure, superconvergence is obtained for non uniform meshes. Note also that the convergence of the MAC scheme for the Stokes equations with a right-hand side inH⁻¹(Ω) was proven in [2].

Mathematical studies of the MAC scheme for the nonlinear Navier-Stokes equations are scarcer. A pioneering work was that of [27] for the steady-state Navier-Stokes equations and for uniform rectangular grids. More recently, a variant of the MAC scheme was defined on locally refined grids and the convergence proof was performed for both the steady-state and time dependent cases in two or three space dimensions [4]. A MAC-like

scheme was also studied for the stationary Stokes and Navier-Stokes equations on two-dimensional Delaunay- Vorono¨ı grids [9]. For the Stokes equations on uniform grids, the scheme given in [4] coincides with the usual MAC scheme that is classically used in CFD codes. However, for the Navier-Stokes equations, the nonlinear convection term is discretized in [4] and [9] in a manner reminiscent of what is sometimes done in the finite element framework (see e.g. [32]), which no longer coincides with the usual MAC scheme, even on uniform rectantular grids; this discretization entails a larger stencil, and numerical experiments [5] seem to show that it is not as efficient as the classical MAC scheme.

Our purpose here is to analyse the genuine MAC scheme for the steady-state and transient Navier-Stokes equations in primitive variables on a non-uniform rectangular mesh in two or three dimensions, and, as in [4], without any assumption on the data nor on the regularity of the the solutions. The convergence of a subsequence of approximate solutions to a weak solution of the Navier-Stokes equations is proved for both the steady and unsteady case, which yields as a by product the existence of a weak solution, well known since the work of J.

Leray [23]. In the case where uniqueness of the solution is known, the whole sequence of approximate solutions can be shown to converge, see remarks 3.14 and 4.4.

This paper is organized as follows. In Section 2, the MAC space grid and the discrete operators are introduced.

In particular, the velocity convection operator is approximated so as to be compatible with a discrete continuity equation on the dual cells ; this discretization coincides with the usual discretization on uniform meshes [28], contrary to the scheme of [4]. The MAC scheme for the steady state Navier-Stokes equations and its weak formulation are introduced in Section 3. Velocity and pressure estimates are then obtained, which lead to the compactness of sequences of approximate solutions. Any prospective limit is shown to be a weak solution of the continuous problem. Section 4 is devoted to the unsteady Navier-Stokes equations. An essential feature of the studied scheme is that the (discrete) kinetic energy remains controlled. We show the compactness of approximate sequences of solutions thanks to a discrete Aubin-Simon argument, and again conclude that any limit of the approximate velocities is a weak solution of the Navier-Stokes equations, thanks to a passage to the limit in the scheme. In the case of the unsteady Stokes equations, some additional estimates yield the compactness of sequences of approximate pressures; this entails that the approximate pressure converges to a weak solution of the Stokes equations as the mesh size and time steps tend to 0.

2. Space discretization

Let Ω be a connected subset ofR^d consisting in a union of rectangles (d= 2) or orthogonal parallelepipeds (d= 3); without loss of generality, the edges (or faces) of these rectangles (or parallelepipeds) are assumed to be orthogonal to the canonical basis vectors, denoted by (e⁽¹⁾, . . . ,e^(d)).

Definition 2.1 (MAC grid). A discretization of Ω with a MAC grid, denoted byD, is defined byD= (M,E), where:

– M stands for the primal grid, and consists in a conforming structured partition of Ω in possibly non uniform rectangles (d= 2) or rectangular parallelepipeds (d= 3). A generic cell of this grid is denoted by K, and its mass center by xK. The pressure is associated to this mesh, and M is also sometimes referred to as ”the pressure mesh”.

– The set of all faces of the mesh is denoted byE; we haveE = E_int∪E_ext, where E_int (resp. E_ext) are the edges ofEthat lie in the interior (resp. on the boundary) of the domain. The set of faces that are orthogonal toe⁽ⁱ⁾ is denoted byE⁽ⁱ⁾, fori ∈

|1, d|

. We then have E⁽ⁱ⁾ =E⁽ⁱ⁾

int ∪E⁽ⁱ⁾

ext, where E⁽ⁱ⁾

int (resp.

E⁽ⁱ⁾_ext) are the edges ofE⁽ⁱ⁾ that lie in the interior (resp. on the boundary) of the domain.

Forσ∈E_int, we writeσ=K|Lifσ=∂K∩∂L. A dual cellDσ associated to a faceσ∈Eis defined as follows:

- ifσ=K|L∈E_int thenDσ =DK,σ∪DL,σ, whereDK,σ (resp. DL,σ) is the half-part ofK (resp.

L) adjacent toσ(see Fig. 1 for the two-dimensional case);

- ifσ∈E_ext is adjacent to the cell K, thenDσ=DK,σ. We obtaindpartitions of the computational domain Ω as follows:

Ω =∪σ∈E(i)Dσ, i∈

|1, d|

,

and thei^th of these partitions is calledi^thdual mesh, and is associated to thei^thvelocity component, in a sense which is clarified below. The set of the faces of the i^th dual mesh is denoted byEe⁽ⁱ⁾ (note that these faces may be orthogonal to any vector of the basis ofR^d and not onlye⁽ⁱ⁾) and is decomposed into

the internal and boundary edges: eE⁽ⁱ⁾=eE⁽ⁱ⁾

int∪eE⁽ⁱ⁾_ext. The dual face separating two duals cellsDσandDσ^′

is denoted byǫ=σ|σ^′.

To define the scheme, we need some additional notations. The set of faces of a primal cellKand a dual cell Dσare denoted byE(K) andEe(Dσ) respectively. Forσ∈E, we denote byxσthe mass center ofσ. The vector nK,σ stands for the unit normal vector toσoutward K. In some case, we need to specify the orientation of a geometrical quantity with respect to the axis:

- a primal cell K will be denoted K = [−→

σσ^′] if σ, σ^′ ∈ E⁽ⁱ⁾ ∩E(K) for some i ∈

|1, d|

are such that (xσ^′−xσ)·e⁽ⁱ⁾>0;

- we writeσ=−−→

K|Lifσ∈E⁽ⁱ⁾and −−−−→xKxL·e⁽ⁱ⁾>0 for somei∈

|1, d|

; - the dual faceǫseparatingDσ andDσ^′ is written ǫ=−−→

σ|σ^′ if−−−→xσxσ^′·e⁽ⁱ⁾>0 for somei∈

|1, d|

.

For the definition of the discrete momentum diffusion operator, we associate to any dual faceǫa distance dǫ as sketched on Figure 1. For a dual face ǫ∈eE(Dσ),σ∈E⁽ⁱ⁾,i∈

|1, d|

, the distancedǫis defined by:

dǫ=





d(xσ,xσ^′) if ǫ=σ|σ^′ ∈eE⁽ⁱ⁾

int, d(xσ, ǫ) ifǫ∈Ee⁽ⁱ⁾

ext, (5)

whered(·,·) denotes the Euclidean distance inR^d.

Dσ

K

L

σ=K|L σ^′′

×

×

× xσ^′

xσ xσ^′′

ǫ2 ǫ3

σ^′

ǫ1=σ|σ^′

∂Ω

dǫ3

dǫ2

dǫ1

Figure 1. Notations for control volumes and dual cells (in two space dimensions, for the second component of the velocity).

The sizeh^M and the regularityη^M of the mesh are defined by:

h^M= max

diam(K), K∈M , (6)

η^M= maxn|σ|

|σ^′|, σ∈E⁽ⁱ⁾, σ^′ ∈E^(j), i, j∈

|1, d|

, i6=jo

, (7)

where| · |stands for the (d−1)-dimensional measure of a subset ofR^d−1(in the sequel, it is also used to denote thed-dimensional measure of a subset ofR^d).

The discrete velocity unknowns are associated to the velocity cells and are denoted by (uσ)_σ∈E⁽ⁱ⁾,i∈

|1, d|

, while the discrete pressure unknowns are associated to the primal cells and are denoted by (pK)K∈M. The discrete pressure spaceL^M is defined as the set of piecewise constant functions over each of the grid cellsK of M, and the discretei^thvelocity spaceHE⁽ⁱ⁾ as the set of piecewise constant functions over each of the grid cells Dσ, σ∈E⁽ⁱ⁾. The set of functions ofL^M with zero mean value is denoted byL^M,0. As in the continuous case,

the Dirichlet boundary conditions are (partly) incorporated into the definition of the velocity spaces, by means of the introduction of the spacesHE(i),0⊂HE(i), i∈

|1, d|

, defined as follows:

HE(i),0=n

u∈HE(i), u(x) = 0 ∀x∈Dσ, σ∈E⁽ⁱ⁾_exto .

We then setH^E,0 =Qd

i=1HE⁽ⁱ⁾,0. Defining the characteristic function 11A of any subsetA⊂Ω by 11A(x) = 1 ifx∈Aand 11A(x) = 0 otherwise, thedcomponents of a functionu∈H^E,0 and a functionp∈L^M may then be written:

ui= X

σ∈E(i)

uσ11Dσ, i∈

|1, d|

and p= X

K∈M

pK11K.

Let us now introduce the discrete operators which are used to write the numerical scheme.

Discrete Laplace operator– For i∈

|1, d|

, thei^th component of the discrete Laplace operator is defined by:

−∆E⁽ⁱ⁾ : HE⁽ⁱ⁾,0−→HE⁽ⁱ⁾,0

ui7−→ −∆E⁽ⁱ⁾ui=− X

σ∈E⁽ⁱ⁾

(∆u)σ11Dσ, with −(∆u)σ= 1

|Dσ| X

ǫ∈eE(Dσ)

φσ,ǫ(ui),

φσ,ǫ(ui) =









|ǫ|

dǫ

(uσ−uσ^′), if ǫ=σ|σ^′ ∈Ee⁽ⁱ⁾

int,

|ǫ|

dǫuσ, ifǫ∈Ee⁽ⁱ⁾

ext∩eE(Dσ),

(8)

wheredǫ is defined by (5). The numerical diffusion flux is conservative:

φσ,ǫ(ui) =−φσ^′,ǫ(ui), ∀ǫ=σ|σ^′ ∈eE⁽ⁱ⁾

int. (9)

The discrete Laplace operator of the full velocity vector is defined by

−∆^E: H^E,0−→H^E,0

u7→ −∆^Eu= (−∆E⁽¹⁾u1, . . . ,−∆E^(d)ud). (10) Let us now recall the definition of the discreteH₀¹-inner product [11]: theH₀¹-inner product betweenu∈HE,0

and v ∈ HE,0 is obtained by taking, for each dual cell, the inner product of the discrete Laplace operator applied touby the test functionvand integrating over the computational domain. A simple reordering of the sums (which may be seen as a discrete integration by parts) yields, thanks to the conservativity of the diffusion flux (9):

∀(u,v)∈HE,0×HE,0, Z

Ω

−∆^Eu·v dx= [u,v]1,E,0= Xd i=1

[ui, vi]_1,E(i),0, with, fori∈

|1, d|

, [ui, vi]_1,E(i),0= X

ǫ∈eE⁽ⁱ⁾

int

ǫ=σ|σ^′

|ǫ|

dǫ

(uσ−uσ^′) (vσ−vσ^′) + X

ǫ∈eE⁽ⁱ⁾_ext ǫ∈eE(Dσ)

|ǫ|

dǫ

uσ vσ. (11)

The bilinear forms

HE(i),0×HE(i),0→R (ui, vi)7→[ui, vi]1,E(i),0

and

H^E,0×H^E,0→R (u,v)7→[u,v]1,E,0

are inner products onHE(i),0 and H^E,0 respectively, which induce the following scalar and vector discreteH₀¹ norms:

kuik²_1,E⁽ⁱ⁾,0= [ui, ui]_1,E(i),0= X

ǫ∈eE⁽ⁱ⁾

int

ǫ=σ|σ^′

|ǫ|

dǫ

(uσ−uσ^′)²+ X

ǫ∈eE⁽ⁱ⁾_ext ǫ∈Ee(Dσ)

|ǫ|

dǫ

u²_σ, fori∈

|1, d|

, (12a)

kuk²_1,E,0= [u,u]1,E,0= Xd i=1

kuik²_1,E(i),0. (12b)

Dǫ

uσ uσ^′

ǫ

(ð₁u1)Dǫ =uσ^′−uσ

dǫ

Note that this definition is still valid if σ or σ^′ are external faces (in which case the corresponding ve- locity is equal to zero). The volumes (Dǫ, forǫ ∈ e

E⁽¹⁾ andǫorthogonal toe⁽¹⁾) thus form a partion of Ω and the definition is complete.

Note also that, in the present case, Dǫ is also a primal cell.

Dǫ

uσ

uσ^′

ǫ

(ð₂u1)Dǫ= uσ^′−uσ

dǫ

Dǫuσ

ǫ

(ð₂u1)Dǫ =−uσ

dǫ

Dǫ

uσ

ǫ (ð₂u1)Dǫ= uσ

dǫ

Figure 2. Notations for the definition of the partial space derivatives of the first component of the velocity, in two space dimensions.

This inner product may also be formulated as theL²-inner product of discrete gradients. To this purpose, we introduced×dnew partitions of the domain Ω, where the (i, j)^th partition consists in an union of rectangles (d= 2) or orthogonal parallelepipeds (d= 3) associated to the dual faces orthogonal toe^(j) of the dual mesh e

E⁽ⁱ⁾ for thei^th component of the velocity. This (i, j)^th partition reads:

(Dǫ)_ǫ∈E_e(i),ǫ⊥e^(j), with

Dǫ=ǫ×[xσxσ^′] ifǫlies inside Ω, ǫ=σ|σ^′, Dǫ=ǫ×[xσxσ,ǫ] ifǫlies on∂Ω, ǫ∈eE(Dσ),

wherexσ,ǫis defined as the orthogonal projection ofxσ onǫ(which is also, in two space dimensions, the vertex ofσlying onǫ). The discrete derivativeð_jui is defined on the (i, j)^th partition and reads:

ifǫlies inside Ω, ǫ=−−→

σ|σ^′, (ð_jui)Dǫ =uσ^′−uσ

dǫ

, ifǫlies on∂Ω, ǫ∈eE(Dσ), (ð_jui)Dǫ =−uσ

dǫ

−−−−→

xσxσ,ǫ·e^(j), (13)

with dǫ defined by (5). These definitions are illustrated on Figure 2. Note that some of these partitions are the same: the (i, j)^th partition coincide with the (j, i)^th and (i, i)^th partitions are the same for i ∈

|1, d|

. In addition, these latters also coincide with the primal mesh: for any sub-volume Dǫ of such a partition, there is K ∈Msuch thatDǫ =K, and we may thus write equivalently (ð_iui)Dǫ or (ð_iui)K. We choose this latter notation in the definition of the discrete divergence below for the sake of consistency, since, if we adopt a variational point of view for the description of the scheme, the discrete velocity divergence has to belong (and indeed does belong) to the space of discrete pressures (see Sections 3 and 4 below for a varitional form of the scheme, in the steady and time-dependent case, respectively). The discrete discrete gradient of each velocity

component ui may now be defined as:

∇_Ee(i)ui= (ð₁ui, . . . ,ð_dui) withð_jui= X

ǫ∈eE⁽ⁱ⁾ ǫ⊥e^(j)

(ð_jui)Dǫ 11Dǫ. (14)

With this definition, it is easily seen that Z

Ω

∇_e

E⁽ⁱ⁾ψ·∇_e

E⁽ⁱ⁾χdx= [ψ, χ]1,E⁽ⁱ⁾,0, forψ, χ∈HE⁽ⁱ⁾,0, andi∈

|1, d|

. (15)

If we extend this definition to the velocity vector by

∇_eEu= (∇_Ee(1)u1, . . . ,∇_eE(d)ud),

we get Z

Ω

∇_eEu:∇_Eev dx= [u,v]1,E,0. This operator satisfies the following consistency result.

Lemma 2.2(Consistency of the discrete partial derivatives of the velocity). LetΠ^Ebe an interpolation operator from C_c^∞(Ω)^d toH^E,0 such that, for any ϕ= (ϕ1,· · ·, ϕd)∈C_c^∞(Ω)^d, there exists Cϕ≥0depending only on ϕ such that

Π^Eϕ= (ΠE(1)ϕ1,· · ·,ΠE(d)ϕd)∈HE(1),0× · · · ×HE(d),0, where

(ΠE(i)ϕi)σ−ϕi(xσ)≤Cϕ h²M, for σ∈E⁽ⁱ⁾, i∈

|1, d|

. (16) Letη^M be the parameter measuring the regularity of the mesh defined by(6). Then there existsCϕ,ηM ≥0, only depending in a non-decreasing way on η^M, such that

|ð_jΠE(i)ϕi(x)−∂jϕi(x)| ≤Cϕ,η h^M for a.e. x∈Ω and for i, j∈

|1, d|

.

As a consequence, if(M_m,E_m)m∈Nis a sequence of MAC grids whose regularity is bounded and whose size tends to 0 asm tends to+∞, then∇_Ee

m(Π^Emϕ)→∇ϕ uniformly asm→+∞.

Discrete divergence and gradient operators– The discrete divergence operator div^M is defined by:

div^M: HE,0−→L^M,0

u7−→div^Mu= X

K∈M

1

|K|

X

σ∈E(K)

|σ|uK,σ 11K, (17) withuK,σ=uσnK,σ·e⁽ⁱ⁾ forσ∈E⁽ⁱ⁾∩E(K), i∈

|1, d|

. (18)

Note that the numerical flux is conservative,i.e.

uK,σ=−uL,σ, ∀σ=K|L∈E_int. (19)

We can now define the discrete divergence-free velocity space:

EE(Ω) =

u∈HE,0 ; div^Mu= 0 . The discrete divergence ofu= (u1, . . . , ud)∈H^E,0 may also be written as

div^Mu= Xd i=1

(ð_iui)K11K, where the discrete derivative (ð_iui)K is defined by Relation (13).

Dσ K uK,σ uK,σ^′

ǫ

uσ,ǫ=1

2(uK,σ^′−uK,σ)

Dσ

K L

σ ǫ

uK,τ uL,τ^′

|ǫ|uσ,ǫ=1

2 (|τ|uK,τ+|τ^′|uL,τ^′) Figure 3. Mass fluxes in the definition of the convection operator for the primal component of the velocity, in two space dimensions.

The gradient (which applies to the pressure) in the discrete momentum balance equation is built as the dual operator of the discrete divergence, and reads:

∇E: L^M−→H^E,0

p7−→∇Ep= (ð₁p, . . . ,ð_dp), (20) whereð_ip∈HE(i),0 is the discrete derivative ofpin thei-th direction, defined by:

ð_ip(x) = |σ|

|Dσ| (pL−pK) ∀x∈Dσ, forσ=−−→

K|L∈E⁽ⁱ⁾

int, i∈

|1, d|

. (21)

Note that, in fact, the discrete gradient of a function of L^M should only be defined on the internal faces, and does not need to be defined on the external faces; it is chosen to be inH^E,0 (that is zero on the external faces) for the sake of simplicity. Again, the definition of the discrete derivatives of the pressure on the MAC grid is consistent in the sense made precise in the following lemma.

Lemma 2.3 (Discrete gradient consistency). Let Π^M be an interpolation operator from C_c^∞(Ω) to L^M such that, for any ψ∈C_c^∞(Ω), there existsCψ≥0 depending only onψ such that

|(Π^Mψ)K−ψ(xK)| ≤Cψ h²M, for K∈M. (22) then there exists Cψ,ηM ≥0 depending only onψ and, in a non-decreasing way, onη^M, such that

|ð_iΠ^Mψ(x)−∂iψ(x)| ≤Cψ,η h^M, for a.e. x∈Ω and fori∈

|1, d|

.

Lemma 2.4 (Discrete div−∇duality). Let q∈L^M andv∈H^E,0 then:

Z

Ω

q div^Mv dx+ Z

Ω

∇Eq·v dx= 0. (23) Proof. Let q∈L^Mandv∈H^E,0. By the definition (17) of the discrete divergence operator and thanks to the conservativity (19) of the flux:

Z

Ω

q div^Mv dx= X

K∈M

qK

X

σ∈E(K)

|σ|vK,σ= X

σ∈E_int,σ=K|L

|σ| (qK−qL)vK,σ.

Therefore, by the definition (21) of the discrete derivative ofq, Z

Ω

qdiv^Mv dx=− Xd

i=1

X

σ∈E(i)

|Dσ|vσð_iq=− Z

Ω

∇Eq·v dx,

which concludes the proof.

Discrete convection operator – Let us consider the momentum equation (1b) for thei^th component of the velocity, and integrate it on a dual cell Dσ, σ ∈ E⁽ⁱ⁾. By the Stokes formula, we then need to discretize P

ǫ∈Ee(Dσ)

R

ǫuiu·nσ,ǫ dγ(x), wherenσ,ǫ denotes the unit normal vector toǫ outwardDσ and dγ(x) denotes thed−1-dimensional Lebesgue measure. For ǫ=σ|σ^′, the convection fluxR

ǫuiu·nσ,ǫ dγ(x) is approximated by |ǫ|uσ,ǫu^∗_ǫ; usually, u^∗_ǫ is chosen as the mean value of the two unknowns uσ and uσ^′. In some situations (high Reynolds number for instance), an upwind choice may be preferred. The two possible choices that will be considered foruǫ are thus:

u^∗_ǫ =u^c_ǫ= uσ+uσ^′

2 (centred choice,∗=c) oru^∗_ǫ =u^up_ǫ =

(uσ ifuσ,ǫ≥0,

uσ^′ otherwise, (upwind choice,∗=up). (24) The quantity |ǫ|uσ,ǫ is the numerical mass flux throughǫ outward Dσ; it must be chosen carefully to obtain the L²-stability of the scheme. More precisely, a discrete counterpart of divu= 0 should be satisfied also on the dual cells. To define uσ,ǫ on internal dual edges, we distinguish two cases (see Figure 3):

- First case – The vectore⁽ⁱ⁾ is normal to ǫ, and ǫis included in a primal cellK, withE⁽ⁱ⁾(K) ={σ, σ^′}.

Then the mass flux throughǫ=σ|σ^′ is given by:

|ǫ|uσ,ǫ= 1

2 (−|σ|uK,σ+|σ^′|uK,σ^′). (25)

Note that, in this relation, all the measures of the face are the same, so this definition equivalently reads uσ,ǫ= (−uK,σ+uK,σ^′)/2.

- Second case – The vectore⁽ⁱ⁾ is tangent toǫ, andǫis the union of the halves of two primal faces τ and τ^′ such thatσ=K|L,τ ∈E(K) andτ^′∈E(L). The mass flux throughǫis then given by:

|ǫ|uσ,ǫ=1

2 (|τ|uK,τ+|τ^′|uL,τ^′). (26)

Again, the numerical flux on a dual face is conservative:

uσ,ǫ=−uσ^′,ǫ, for any dual faceǫ=σ|σ^′. (27) Moreover, if div^Mu= 0, the following discrete free divergence condition holds on the dual cells:

X

ǫ∈Ee(Dσ)

|ǫ|uσ,ǫ= 1 2

X

σ∈E(K)

|σ|uK,σ+1 2

X

σ∈E(L)

|σ|uL,σ= 0. (28)

On the external dual faces associated to free degrees of freedom (which means that we are in the second of the above cases), this definition yields uσ,ǫ= 0, which is consistent with the boundary condition (1c).

Thei-th componentCE⁽ⁱ⁾(u) of the non linear convection operator is defined by:

CE⁽ⁱ⁾(u) : HE⁽ⁱ⁾,0−→HE⁽ⁱ⁾,0

v7−→CE⁽ⁱ⁾(u)v= X

σ∈E⁽ⁱ⁾

int

1

|Dσ| X

ǫ∈eE(Dσ)

|ǫ|uσ,ǫv_ǫ^∗11Dσ, (29)

wherev^∗_ǫ is chosen centred or upwind, as defined in (24). The full discrete convection operatorC^E(u), H^E,0−→

H^E,0is defined by

C^E(u)v= CE⁽¹⁾(u)v1, . . . , CE^(d)(u)vd .

3. The steady case

3.1. The scheme

With the notations introduced in the previous sections, the MAC scheme for the discretization of the steady Navier-Stokes equations (1) on a MAC grid (M,E) reads:

u∈HE,0, p∈LM,0, (30a)

−∆^Eu+C^E(u)u+∇Ep=f, (30b)

div^Mu= 0. (30c)

The discrete right-hand side of the momentum balance equation readsf =P_Ef¯, whereP_Eis the cell mean-value operator defined byP_Ev= (P_E(1)v1,· · ·,P_E(d)vd)∈HE(1),0× · · · ×HE(d),0and, fori∈

|1, d|

, P_E(i): L¹(Ω)−→HE⁽ⁱ⁾,0

vi 7−→P_E(i)vi= X

σ∈E⁽ⁱ⁾

int

vσ11Dσ with, forσ∈E⁽ⁱ⁾

int, vσ= 1

|Dσ| Z

Dσ

vi(x) dx. (31)

Let us define the weak formb^E of the nonlinear convection term:

for (u,v,w)∈H^E,0×H^E,0×H^E,0, b^E(u,v,w) = Xd i=1

bE⁽ⁱ⁾(u, vi, wi), where fori∈

|1, d|

, bE(i)(u, vi, wi) = Z

Ω

CE(i)(u)vi wi dx. (32) We can now introduce a weak formulation of the scheme, which reads:

Find (u, p)∈H^E,0×L^M,0 such that, for any (v, q)∈H^E,0×L^M, Z

Ω

∇_eEu:∇_Eev dx+b^E(u,u,v)− Z

Ω

pdiv^Mv dx= Z

Ω

f·v dx, (33a) Z

Ω

div^Muq dx= 0. (33b)

This formulation is equivalent to the strong form (30).

Remark 3.1 (Convergence of the MAC scheme for the Stokes problem and the gradient schemes theory). Omit- ting the convection terms in (33), we obtain a weak formulation of the MAC scheme for the linear Stokes problem. Moreover, formulating the discreteH¹-inner product as the integral over Ω of dot products of discrete gradients, the MAC scheme can be interpreted as a gradient scheme in the sense introduced in [13] (see [15]

and [8] for more details on the generalization of this formulation to other schemes). Thanks to this result, the (strong) convergence of the velocity and of its discrete gradient to the exact velocity and its gradient can be shown, and thus also the strong convergence of the pressure.

3.2. Stability and existence of a solution

To prove the scheme stability, it is convenient to first reformulate the trilinear form associated to the velocity convection term. To this purpose, we introduce a reconstruction of the velocity components on the partitions which where used for the definition of the discrete velocity gradient. This leads to define d×d class of recon- struction operators, denoted byR^(i,j)

e

E , withR^(i,j)

e

E acting on the i^th component of the velocity and providing a reconstruction of this field on the partition of Ω associated to its j^th partial derivative.

Definition 3.2 (Velocity reconstructions). Let (M,E) be a given MAC mesh, and let i, j ∈

|1, d|

. Let R^(i,j)

e E

be a reconstruction operator defined as follows:

R^(i,j)

e

E : HE(i),0 → L²(Ω) v 7→ R^(i,j)

e

E v= X

ǫ∈eE⁽ⁱ⁾, ǫ⊥e^(j)

(R^(i,j)

e

E v)Dǫ 11Dǫ,

Dǫ

uσ uσ^′

ǫ

(R^(1,1)

e

E u1)Dǫ=αǫuσ+ (1−αǫ)uσ^′

Dǫ

uσ

uσ^′

ǫ

(R^(1,2)

e

E u1)Dǫ=αǫuσ+ (1−αǫ)uσ^′

Dǫuσ

ǫ

(R^(1,2)

e

E u1)Dǫ =αǫuσ

Figure 4. Reconstruction of the first component of the velocity, in two space dimensions.

First line: R^(1,1)

e

E . Second line: R^(1,2)

e

E , inner dual face (left) and dual face lying on the boundary (right). The real numberαǫ is only supposed to satisfyαǫ∈[0,1].

where (R^(i,j)

e

E v)Dǫis a convex combination of the (one of two) discrete values of thei^thcomponent of the velocity lying on faces ofDǫ(see Figure 4).

Such a reconstruction operator satisfies the following stability result.

Lemma 3.3(Stability of the velocity reconstruction operators). Let(M,E)be a given MAC mesh,i, j∈

|1, d|

, and R^(i,j)

e

E be a reconstruction operator, in the sense of Definition 3.2. Then, for p ∈ [1,+∞), there exists CηM ≥ 0, depending only on p and on the parameter η^M characterizing the regularity of the mesh defined by (7), and non-decreasing with respect toη^M, such that, for anyv∈HE(i),0,

kR^(i,j)

e

E vk_L^p_(Ω)≤CηM kvk_L^p_(Ω). Proof. Letp∈[1,+∞),i, j∈

|1, d|

andv∈HE⁽ⁱ⁾,0. We have:

kR^(i,j)

e

E vk^p_Lp(Ω)= X

ǫ∈eE(i), ǫ⊥e^(j)

|Dǫ| |(R^(i,j)

e

E v)Dǫ|^p

= X

ǫ∈eE⁽ⁱ⁾

int

ǫ=−−→ σσ^′, ǫ⊥e^(j)

|Dǫ|αǫvσ+ (1−αǫ)vσ^′

^p+ X

ǫ∈Ee⁽ⁱ⁾

ext

ǫ∈eE(Dσ), ǫ⊥e^(j)

|Dǫ| |αǫvσ|^p.

Since|αǫ| ≤1 and (a+b)^p≤2^p−1(a^p+b^p), fora, b∈[0,+∞), we get:

kR^(i,j)

e

E vk^p_Lp(Ω)≤2^p−1 X

ǫ∈eE⁽ⁱ⁾

int

ǫ=σσ^′, ǫ⊥e^(j)

|Dǫ| |vσ|^p+|vσ^′|^p

+ X

ǫ∈Ee⁽ⁱ⁾

ext

ǫ∈Ee(Dσ), ǫ⊥e^(j)

|Dǫ| |vσ|^p.

Reordering the sums, we obtain that:

kR^(i,j)

e

E vk^p_Lp(Ω)≤2^p−1 X

σ∈E⁽ⁱ⁾

|Vσ| |vσ|^p,

where the volume Vσ is the sum of the two volumes Dǫ such that ǫ is a face of Dσ. It may now be easily checked that there existsCηM depending only on the the parameterη^M and non-decreasing with respect to this

parameter such that|Vσ| ≤CηM|Dσ|, which concludes the proof.

The discretization of the velocity convection term in the i^th momentum balance equation may be seen as a discrete counterpart of div(viu), where vi is the convected component of the velocity field (in the scheme, vi =ui). Multipling this expression by wi and inegrating over Ω yields a continuous counterpartb⁽ⁱ⁾(u, vi, wi) ofbE(i)(u, vi, wi) which reads:

b⁽ⁱ⁾(u, vi, wi) = Z

Ω

div(viu)wi dx.

An integration by parts (supposing that wi vanishes on the boundary) yields:

b⁽ⁱ⁾(u, vi, wi) =− Z

Ω

viu·∇wi dx=− Xd j=1

Z

Ω

viuj∂jwi dx.

The following lemma states a discrete equivalent of this relation.

Lemma 3.4 (Reformulation of b^E). Let (M,E) be a given MAC mesh, i ∈

|1, d|

, and (u,v,w) ∈ H^E,0× H^E,0×H^E,0. LetbE(i)(u, vi, wi) be given by (32). Then there exists two reconstruction operators in the sense of Definition 3.2, denoted by(R^(i,j)

e

E_n )^u and(R^(j,i)

e

E_n )^v, such that:

bE(i)(u, vi, wi) =− Xd j=1

Z

Ω

(R^(i,j)

e

E )^vvi (R^(j,i)

e

E )^uuj ð_jwi dx.

Proof. Let (u,v,w)∈E^E×H²E,0. By definition, bE(i)(u, vi, wi) = X

σ∈E(i)

wσ

X

ǫ∈eE(Dσ)

|ǫ|v^∗_ǫuσ,ǫ.

Reordering the sums, we get by conservativity:

bE⁽ⁱ⁾(u, vi, wi) = X

ǫ=σ|σ^′∈eE⁽ⁱ⁾

int

|ǫ|v^∗_ǫ uσ,ǫ (wσ−wσ^′) =− X

ǫ=σ|σ^′∈eE⁽ⁱ⁾

int

|Dǫ|v_ǫ^∗ uσ,ǫ

wσ^′−wσ

dǫ

.

The sum is over the whole set of dual faceseE⁽ⁱ⁾, so over thedpartitions involved in the definition of the discrete gradient ofwi. In addition, without loss of generality, we may suppose that we have chosen forǫthe orientation such thatǫ=−−→

σ|σ^′. Hence, we get, by definition (13), wσ^′−wσ

dǫ = (ð_jwi)Dǫ,

where j is the index such that ǫ is normal no e^(j). For the centered version of the convection operators, v_ǫ^∗ = (vσ+vσ^′)/2; in the upwind case, it is equal to either vσ or vσ^′ (Relation (24)). In both cases, it is a convex combination of the two discrete values ofvi lying on the faces ofDǫ; there exists thus an operatorR^(i,j)

e E

(still with the same meaning for j), in the sense of Definition 3.2 such that v^∗_ǫ = (R^(i,j)

e

E vi)Dǫ. Finally, from the definition of the convection operator and with the chosen orientation for ǫ, uσ,ǫ is a convex combination of the two values of uj lying on the faces on Dǫ: either the mean value given by (25), if j = i, either the convex combination of (26), if j 6= i. In addition, Dǫ is a volume used in the definition of the j^th discrete partial derivative of thei^th component, and thus also a volume used in the definition of the i^th discrete partial derivative of the j^th component (both partitions are the same). So there exists one reconstruction operator R^(j,i)

e

E such thatuσ,ǫ = (R^(j,i)

e

E uj)Dǫ, which concludes the proof.

Lemma 3.5(Estimates onb^E). Let(M,E)be a MAC grid and letb^Ebe defined by (32). There existsCηM >0, depending only and in non-decreasing way on the regularity parameterη^M of the mesh defined by(7), such that:

∀ (u,v,w)∈H³E,0, |b^E(u,v,w)| ≤CηM kukL⁴(Ω)^d kvkL⁴(Ω)^d kwk1,E,0 (34) and

∀ (u,v,w)∈H³E,0, |b^E(u,v,w)| ≤CηM kuk1,E,0 kvk1,E,0kwk1,E,0. (35) Proof. Leti∈

|1, d|

. Thanks to Lemma 3.4, there exists two reconstruction operators in the sense of Definition 3.2, denoted by (R^(i,j)

e

E )^v and (R^(j,i)

e

E )^u such that:

bE(i)(u, vi, wi) =− Xd j=1

Z

Ω

(R^(i,j)

e

E )^vvi (R^(j,i)

e

E )^uuj ð_jwi dx.

Thanks to H¨older’s inequality, we get, forj∈

|1, d|

:

Z

Ω

(R^(i,j)

e

E )^vvi (R^(j,i)

e

E )^uuj ð_jwi dx≤ k(R^(i,j)

e

E )^vvikL⁴(Ω) k(R^(i,j)

e

E )^uujkL⁴(Ω) kð_jwikL²(Ω),

which, in view of Lemma 3.3 and the identity (15), concludes the proof of Estimate (34). We then deduce (35) by the discrete Sobolev inequality [11, Lemma 3.5] which allows to control the L⁴-norm by the discrete

H¹-norm.

Let us now prove thatb^Eis skew-symmetrical with respect to the last two variables. At the continuous level, this result is obtained as follows. For i∈

|1, d|

, on one side, we have by integration by parts:

Z

Ω

wi div(viu) dx=− Z

Ω

viu·∇wi dx. (36)

On the other side, since divu= 0, div(viu) =u·∇vi, so:

Z

Ω

wi div(viu) dx= Z

Ω

wiu·∇vi dx,

which yields the conclusion. The following lemma states a discrete analogue of this property.

Lemma 3.6 (b^E is skew-symmetrical). Let (u,v,w) ∈ E^E×H^E,0×H^E,0, and let i ∈

|1, d|

. Assume the centred choice for vǫ in the expression of bE(i); then

bE(i)(u, vi, wi) =−bE(i)(u, wi, vi), (37) and therefore,

bE(i)(u, vi, vi) = 0. (38)

Assume now the upwind choice for vǫ in the expression of bE(i); then,

bE(i)(u, vi, vi)≥0. (39)

Proof. We mimick the computation performed in the continuous case. At the discrete level and for the centred formulation of the convection term, we have, by a simple reordering of the sum:

bE(i)(u, vi, wi) = X

σ∈E⁽ⁱ⁾

int

wσ

X

ǫ=σ|σ^′∈eE(Dσ)

|ǫ| vσ+vσ^′

2 uσ,ǫ= X

ǫ=σ|σ^′∈eE⁽ⁱ⁾

int

|ǫ| vσ+vσ^′

2 (wσ−wσ^′)uσ,ǫ. This relation is just obtained by conservativity of the mass flux, by a process which may be seen as a discrete integration by parts, and we have seen that it may be written as a discrete analogue of (36) (Lemma 3.4). On the other hand, thanks to (28) (i.e.the discrete analogue of divu= 0), we have, for any faceσ∈E⁽ⁱ⁾

int: X

ǫ∈Ee(Dσ)

|ǫ|(−vσ)uσ,ǫ=−vσ

X

ǫ∈Ee(Dσ)

|ǫ|uσ,ǫ= 0.