Convergence analysis of a colocated finite volume scheme for the incompressible Navier-Stokes equations on general 2 or 3D meshes

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Convergence analysis of a colocated finite volume scheme for the incompressible Navier-Stokes equations

on general 2 or 3D meshes

Robert Eymard, Raphaele Herbin, Jean-Claude Latché

To cite this version:

Robert Eymard, Raphaele Herbin, Jean-Claude Latché. Convergence analysis of a colocated finite

volume scheme for the incompressible Navier-Stokes equations on general 2 or 3D meshes. SIAM

Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2007, 45 (1), pp.1-

36. �10.1137/040613081�. �hal-00004841v2�

hal-00004841, version 2 - 30 Apr 2008

CONVERGENCE ANALYSIS OF A COLOCATED FINITE VOLUME SCHEME FOR THE INCOMPRESSIBLE NAVIER-STOKES

EQUATIONS ON GENERAL 2 OR 3D MESHES

R. EYMARD

^∗

, R. HERBIN

^†

, AND J.C. LATCH´ E

^‡

Abstract. We study a colocated cell centered finite volume method for the approximation of the incompressible Navier-Stokes equations posed on a 2D or 3D finite domain. The discrete unknowns are the components of the velocity and the pressures, all of them colocated at the center of the cells of a unique mesh; hence the need for a stabilization technique, which we choose of the Brezzi-Pitk¨ aranta type. The scheme features two essential properties: the discrete gradient is the transposed of the divergence terms and the discrete trilinear form associated to nonlinear advective terms vanishes on discrete divergence free velocity fields. As a consequence, the scheme is proved to be unconditionally stable and convergent for the Stokes problem, the steady and the transient Navier-Stokes equations. In this latter case, for a given sequence of approximate solutions computed on meshes the size of which tends to zero, we prove, up to a subsequence, the L

²

-convergence of the components of the velocity, and, in the steady case, the weak L

²

-convergence of the pressure.

The proof relies on the study of space and time translates of approximate solutions, which allows the application of Kolmogorov’s theorem. The limit of this subsequence is then shown to be a weak solution of the Navier-Stokes equations. Numerical examples are performed to obtain numerical convergence rates in both the linear and the nonlinear case.

Key words. Finite Volume, cell centered scheme, colocated discretizations, steady state and transient Navier-Stokes equations, convergence analysis.

AMS subject classifications. 15A15, 15A09, 15A23

1. Introduction. We are interested in this paper in finding an approximation of the fields ¯ u = (¯ u ⁽ⁱ⁾ ) i=1,...,d : Ω × [0, T ] → R ^d , and ¯ p : Ω × [0, T ] → R , weak solution to the incompressible Navier-Stokes equations which write:

∂ t u ¯ ⁽ⁱ⁾ − ν∆¯ u ⁽ⁱ⁾ + ∂ i p ¯ +

d

X

j=1

¯

u ^(j) ∂ j u ¯ ⁽ⁱ⁾ = f ⁽ⁱ⁾ in Ω × (0, T ), for i = 1, . . . , d, div¯ u =

d

X

i=1

∂ i u ¯ ⁽ⁱ⁾ = 0 in Ω × (0, T ).

(1.1) nstocontt

with a homogeneous Dirichlet boundary condition for ¯ u and the initial condition

¯

u ⁽ⁱ⁾ ( · , 0) = ¯ u ⁽ⁱ⁾ _ini in Ω for i = 1, . . . , d.

(1.2) nstoconti

In the above equations, ¯ u ⁽ⁱ⁾ , i = 1, . . . , d denote the components of the velocity of a fluid which flows in a domain Ω during the time (0, T ), ¯ p denotes the pressure, ν > 0 stands for the viscosity of the fluid. We make the following assumptions:

Ω is a polygonal open bounded connected subset of R ^d , d = 2 or 3,

(1.3) ^hypomegat

T > 0 is the finite duration of the flow,

(1.4) hyptimet

ν ∈ (0, + ∞ ),

(1.5) ^hypnut

¯

u ini ∈ L ² (Ω) ^d ,

(1.6) hypuini

f ⁽ⁱ⁾ ∈ L ² (Ω × (0, T )), for i = 1, . . . , d.

(1.7) hypfgt

∗

Universit´ e de Marne-la-Vall´ ee, France, (eymard@math.univ-mlv.fr)

†

Universit´ e de Provence, France (herbin@cmi.univ-mrs.fr)

‡

DPAM, Institut de Radioprotection et Suret´ e Nucl´ eaire, (jean-claude.latche@irsn.fr)

1

We denote by x = (x ⁽ⁱ⁾ ) i=1,...,d any point of Ω, by | . | the Euclidean norm in R ^d , i.e.: | x | ² =

d

X

i=1

(x ⁽ⁱ⁾ ) ² and by dx the d-dimensional Lebesgue measure dx = dx ⁽¹⁾ . . . dx ^(d) .

The weak sense that we consider for the Navier-Stokes equations is the following.

Definition 1.1 (Weak solution for the transient Navier-Stokes equations).

weaksolt

Under hypotheses ( 1.3)-( ^hypomegat 1.7), let the function space ^hypfgt E(Ω) be defined by:

E(Ω) := { ¯ v = (¯ v ⁽ⁱ⁾ ) i=1,...,d ∈ H ₀ ¹ (Ω) ^d , div¯ v = 0 a.e. in Ω } .

(1.8) e0

Then u ¯ is called a weak solution of ( ^nstocontt 1.1)-( 1.2) if ^nstoconti u ¯ ∈ L ² (0, T ; E(Ω)) ∩ L ^∞ (0, T ; L ² (Ω) ^d ) and:



 

 

 

 

 



 

 

 

 

 



∀ ϕ ∈ L ² (0, T ; E(Ω)) ∩ C _c ^∞ (Ω × ( −∞ , T )) ^d ,

− Z T

0

Z

Ω

¯

u(x, t) · ∂ t ϕ(x, t) dx dt − Z

Ω

¯

u ini (x) · ϕ(x, 0) dx +ν

Z T 0

Z

Ω ∇ u(x, t) : ¯ ∇ ϕ(x, t) dx dt + Z T

0

b(¯ u( · , t), u( ¯ · , t), ϕ( · , t)) dt

= Z T

0

Z

Ω

f (x) · ϕ(x, t) dx dt

(1.9) nstocontft

where, for all u, ¯ v ¯ ∈ H ₀ ¹ (Ω) ^d and for a.e. x ∈ Ω, we use the following notation:

∇ u(x) : ¯ ∇ ¯ v(x) =

d

X

i=1

∇ u ¯ ⁽ⁱ⁾ (x) · ∇ ¯ v ⁽ⁱ⁾ (x)

and where the trilinear form b(., ., .) is defined, for all u, ¯ ¯ v, w ¯ ∈ (H ₀ ¹ (Ω)) ^d , by b(¯ u, v, ¯ w) = ¯

d

X

k=1 d

X

i=1

Z

Ω

¯

u ⁽ⁱ⁾ (x)∂ i ¯ v ^(k) (x) ¯ w ^(k) (x) dx.

(1.10) ^deftricont

Remark 1.1. From ( 1.9), we get that a weak solution ^nstocontft u of ( 1.1)-( ^{nstocontt nstoconti} 1.2) in the sense of Definition ^weaksolt 1.1 satisfies ∂ t u ¯ ∈ L ^4/d (0, T ; E(Ω) ^′ ), and is therefore a weak solution in the classical sense, such that u( ¯ · , 0) is the orthogonal L ² -projection of u ¯ ini on { ¯ v ∈ L ² (Ω) ^d , div¯ v = 0, trace(¯ v · n ∂Ω , ∂Ω) = 0 } (see for example [36] or ^temam [7]). ^bf

Numerical schemes for the Stokes equations and the Navier-Stokes equations have been extensively studied: see giraultraviart, patankar,peyret-taylor,pironneau, gunzburger,glo

[23, 33, 34, 35, 25, 24] and references therein. Among different schemes, finite element schemes and finite volume schemes are frequently used for mathematical or engineering studies. An advantage of finite volume schemes is that the unknowns are approximated by piecewise constant functions: this makes it easy to take into account additional nonlinear phenomena or the coupling with alge- braic or differential equations, for instance in the case of reactive flows; in particular, one can find in [33] the presentation of the classical finite volume scheme on rect- ^patankar angular meshes, which has been the basis of many industrial applications. However, the use of rectangular grids makes an important limitation to the type of domain

2

which can be gridded and more recently, finite volume schemes for the Navier-Stokes equations on triangular grids have been presented: see for example [26] where the ^nico vorticity formulation is used and [6] where primal variables are used with a Chorin ^boivin type projection method to ensure the divergence condition. Proofs of convergence for finite volume type schemes for the Stokes and steady-state Navier-Stokes equations are have recently been given for staggered grids ^chou [9], [26], ^nico [13], ^cras [14], ^stagg [4], following the ^benartzi pioneering work of Nicolaides et al. [31], ^nicol [32]. ^nicwu

In this paper, we propose the mathematical and numerical analysis of a discretiza- tion method which uses the primitive variables, that is the velocity and the pressure, both approximated by piecewise constant functions on the cells of a 2D or 3D mesh.

We emphasize that the approximate velocities and pressures are colocated, and there- fore, no dual grid is needed. The only requirement on the mesh is a geometrical assumption needed for the consistency of the approximate diffusion flux (see [15] and ^book section ( ^secdisc 2) for a precise definition of the admissible discretizations).

As far as we know, this work is a first proof of the convergence, of a finite volume scheme which is of large interest in industry. Indeed, industrial CFD codes (see e.g.

fluent

[28], [1]) use colocated cell centered finite volume schemes; leaving aside implemen- ^neptune tation considerations, the principle of these schemes seems to differ from the present scheme only by the stabilization choice. The main reasons why this scheme is so popular in industry are:

• a colocated arrangement of the unknowns,

• a very cheap assembling step, (no numerical integration to perform)

• an easy coupling with other systems of equations.

The finite volume scheme studied here is based on three basic ingredients. First, a stabilization technique ` a la Brezzi-Pik¨aranta [8] is used to cope with the instability ^brez of colocated velocity/pressure approximation spaces. Second, the discretization of the pressure gradient in the momentum balance equation is performed to ensure, by construction, that it is the transpose of the divergence term of the continuity constraint. Finally, the contribution of the discrete nonlinear advection term to the kinetic energy balance vanishes for discrete divergence free velocity fields, as in the continuous case. These features appear to be essential in the proof of convergence.

We are then able to prove the stability of the scheme and the convergence of dis- crete solutions towards a solution of the continuous problem when the size of the mesh tends to zero, for the steady linear case (generalized Stokes problem), the stationary and the transient Navier-Stokes equations, in 2D and 3D. Our results are valid for general meshes, do not require any assumption on the regularity of the continuous solution nor, in the nonlinear case, any small data condition. We emphasize that the convergence of the fully discrete (time and space) approximation is proven here, using an original estimate on the time translates, which yields, combined with a classical estimate on the space translates, a sufficient relative compactness property.

An error analysis is performed in the steady linear case, under regularity assump- tions on the solution. An error bound of order 0.5 with respect to the step size is obtained in the discrete H ¹ norm and the L ² norm for respectively the velocity and the pressure. Of course, this is probably not a sharp estimate, as can be seen from the numerical results shown in Section 5. Indeed, a better rate of convergence can be ^secnum proved under additional assumptions on the mesh [20]. ^EHL

3

This paper is organized as follows. In section 2, we introduce the discretization ^secdisc tools together with some discrete functional analysis tools. Section 3 is devoted to ^secfvslin the linear steady problem (Stokes problem), for which the finite volume scheme is given and convergence analysis and error estimates are detailed. The complete finite volume scheme for the nonlinear case is presented in section 4, in both the steady and ^secfvsnlin transient cases. We then develop the analysis of its convergence to a weak solution of the continuous problem. We give some numerical results in section 5, and finally ^secnum conclude with some remarks on open problems (section ^seconcrem 6).

secdisc

2. Spatial discretization and discrete functional analysis.

2.1. Admissible discretization of Ω. We first recall the notion of admissible discretization for a finite volume method, which is given in [15]. ^book

Definition 2.1 (Admissible discretization, steady case). Let Ω be an open adisc

bounded polygonal (polyhedral if d = 3) subset of R ^d , and ∂Ω = Ω \ Ω its boundary. An admissible finite volume discretization of Ω, denoted by D , is given by D = ( M , E , P ), where:

- M is a finite family of non empty open polygonal convex disjoint subsets of Ω (the “control volumes”) such that Ω = ∪ ^K ∈M K. For any K ∈ M , let

∂K = K \ K be the boundary of K and m K > 0 denote the area of K.

- E is a finite family of disjoint subsets of Ω (the “edges” of the mesh), such that, for all σ ∈ E , there exists a hyperplane E of R ^d and K ∈ M with σ = ∂K ∩ E and σ is a non empty open subset of E. We then denote by m σ > 0 the (d-1)-dimensional measure of σ. We assume that,for all K ∈ M , there exists a subset E ^K of E such that ∂K = ∪ ^σ ∈E

K

σ. It then results from the previous hypotheses that, for all σ ∈ E , either σ ⊂ ∂Ω or there exists (K, L) ∈ M ² with K 6 = L such that K ∩ L = σ; we denote in the latter case σ = K | L.

- P is a family of points of Ω indexed by M , denoted by P = (x K ) _K∈M . The coordinates of x K are denoted by x ⁽ⁱ⁾ _K , i = 1, . . . , d. The family P is such that, for all K ∈ M , x K ∈ K. Furthermore, for all σ ∈ E such that there exists (K, L) ∈ M ² with σ = K | L, it is assumed that the straight line (x K , x L ) going through x K and x L is orthogonal to K | L. For all K ∈ M and all σ ∈ E ^K , let z σ be the orthogonal projection of x K on σ. We suppose that z σ ∈ σ.

An example of two neighbouring control volumes K and L of M is depicted in Figure 2.1. ^fig_maille

The following notations are used. The size of the discretization is defined by:

size( D ) = sup { diam(K), K ∈ M} .

For all K ∈ M and σ ∈ E ^K , we denote by n K,σ the unit vector normal to σ outward to K. We denote by d K,σ the Euclidean distance between x K and σ. The set of interior (resp. boundary) edges is denoted by E ^int (resp. E ^ext ), that is E ^int = { σ ∈ E ; σ 6⊂ ∂Ω } (resp. E ^ext = { σ ∈ E ; σ ⊂ ∂Ω } ). For all K ∈ M , we denote by N ^K the subset of M of the neighbouring control volumes. For all K ∈ M and L ∈ N ^K , we set n KL = n _K,K|L , we denote by d _K|L the Euclidean distance between x K and x L .

We shall measure the regularity of the mesh through the function regul( D ) defined by

regul( D ) = inf n _d

K,σ

diam(K) , K ∈ M , σ ∈ E ^K o

∪ n _d

K,K|L

d

_K|L

, K ∈ M , L ∈ N ^K o

∪ n

1

card( E

K

) , K ∈ M o . regul (2.1)

4

d_K,σ x_L

x_K

L K|L

K

mσ

d_K|L

Fig. 2.1. Notations for an admissible mesh fig_maille

2.2. Discrete functional properties. Finite volume schemes are discrete bal- ance equations with an adequate approximation of the fluxes, see e.g. [15]. Recent ^book works dealing with cell centered finite volume methods for elliptic problems [21], ^convpardeg [16], ^cras100

stagg

[14] introduce an equivalent variational formulation in adequate functional spaces.

Here we shall follow this latter path, also introducing discrete analogues of the con- tinuous Laplace, gradient, divergence and transport operators, each of them featuring properties similar to their continuous counterparts.

Definition 2.2. Let Ω be an open bounded polygonal subset of R ^d , with d ∈ N _∗ . espdis

Let D = ( M , E , P ) be an admissible finite volume discretization of Ω in the sense of definition 2.1. We denote by ^adisc H _D (Ω) ⊂ L ² (Ω) the space of functions which are piecewise constant on each control volume K ∈ M . For all w ∈ H _D (Ω) and for all K ∈ M , we denote by w K the constant value of w in K. The space H _D (Ω) is embedded with the following Euclidean structure: For (v, w) ∈ (H _D (Ω)) ² , we first define the following inner product (corresponding to Neumann boundary conditions)

h v, w i D = 1 2

X

K ∈M

X

L ∈N

K

m _K|L d K | L

(v L − v K )(w L − w K ).

(2.2) ^amundis

We then define another inner product (corresponding to Dirichlet boundary conditions) [v, w] _D = h v, w i D + X

K∈M

X

σ∈E

K

∩E

ext

m σ

d K,σ v K w K .

(2.3) ^amudis

Next, we define a seminorm and a norm in H _D (Ω) (thanks to the discrete Poincar´e inequality ( 2.4) given below) by ^poindis

| w | D = ( h w, w i D ) ^1/2 , k w k D = ([w, w] _D ) ^1/2 .

We define the interpolation operator P _D : C(Ω) → H _D (Ω) by (P _D ϕ) K = ϕ(x K ), for all K ∈ M , for all ϕ ∈ C(Ω).

Similarly, for u = (u ⁽ⁱ⁾ ) i=1,...,d ∈ (H _D (Ω)) ^d , v = (v ⁽ⁱ⁾ ) i=1,...,d ∈ (H _D (Ω)) ^d and w = (w ⁽ⁱ⁾ ) i=1,...,d ∈ (H _D (Ω)) ^d , we define:

k u k D =

d

X

i=1

[u ⁽ⁱ⁾ , u ⁽ⁱ⁾ ] _D

! ^1/2

, [v, w] _D =

d

X

i=1

[v ⁽ⁱ⁾ , w ⁽ⁱ⁾ ] _D ,

5

and P _D : C(Ω) ^d → H _D (Ω) ^d by (P _D ϕ) K = ϕ(x K ), for all K ∈ M , for all ϕ ∈ C(Ω) ^d . The discrete Poincar´e inequalities (see [15]) write: ^book

k w k L

²

(Ω) ≤ diam(Ω) k w k D , ∀ w ∈ H _D (Ω),

(2.4) poindis

and there exists C Ω > 0, only depending on Ω, such that k w k ² L

²

(Ω) ≤ C Ω | w | ² _D , ∀ w ∈ H _D (Ω) with

Z

Ω

w(x)dx = 0.

(2.5) poindismoy

We define a discrete divergence operator div _D : (H _D (Ω)) ^d → H _D (Ω), by:

div _D (u)(x) = 1 m K

X

L ∈N

K

A KL · (u K + u L ), for a.e. x ∈ K, ∀ K ∈ M ,

(2.6) ^divdisc

with

A KL = m K | L

d _K|L

x L − x K

2 = 1

2 m K | L n KL , ∀ K ∈ M , ∀ L ∈ N ^K .

(2.7) defAB

We then set E _D (Ω) = { u ∈ (H _D (Ω)) ^d , div _D (u) = 0 } .

Remark 2.1. Any definition of A KL such that A KL = m _K|L a KL n KL with a KL ≥ 0 and a KL + a LK = 1, combined with the definition div _D (u)(x) = _m ¹

K

P

L∈N

K

(A KL · u K − A LK · u L ), produces the same results of convergence as those which are proven in this paper. On particular meshes, one can prove a better error estimate, choosing a KL = d(x L , K | L)/d KL (see ^EHL [20]). Nevertheless, in the general framework of this paper, other choices do not improve the convergence result and the error estimate.

Therefore, we set in this paper a KL = 1/2, which corresponds to ( 2.7). The advantage ^defAB of this choice is that it leads to simpler notations and shorter equations.

The adjoint of this discrete divergence defines a discrete gradient ∇ D : H _D (Ω) → (H _D (Ω)) ^d :

( ∇ D u) K = 1 m K

X

L∈N

K

A KL (u L − u K ), ∀ K ∈ M , ∀ u ∈ H _D (Ω).

(2.8) ^discgrad

This operator ∇ D then satisfies the following property.

Proposition 2.3. Let ( D ^(m) ) m ∈

N

be a sequence of admissible discretizations of weakconvgrad

Ω in the sense of Definition 2.1, such that ^adisc lim

m→∞ size( D ^(m) ) = 0. Let us assume that there exists C > 0 and α ∈ [0, 2) and a sequence (u ^(m) ) m ∈

N

such that u ^(m) ∈ H _D

^(m)

(Ω) and | u ^(m) | ² _D

m

≤ C size( D ^(m) ) ^−α , for all m ∈ N .

Then the following property holds:

m→+∞ lim Z

Ω

P _D

_m

ϕ(x) ∇ D

m

u ^(m) (x) + u ^(m) (x) ∇ ϕ(x)

dx = 0, ∀ ϕ ∈ C _c ^∞ (Ω),

(2.9) wcvgrad

and therefore:

m→+∞ lim Z

Ω ∇ D

m

u ^(m) (x) · P _D

_m

ψ(x)dx = 0, ∀ ψ ∈ C _c ^∞ (Ω) ^d ∩ E(Ω),

(2.10) wcvdiv

where E(Ω) is defined by ( 1.8). ^e0

6

Proof. Let us assume the hypotheses of the above lemma, and let i = 1, . . . , d and ϕ ∈ C _c ^∞ (Ω) be given. Let us study, for m ∈ N , the term

T ₁ ^(m) _wcv1 = Z

Ω

P _D

_m

ϕ(x) ∇ D

m

u ^(m) (x) + u ^(m) (x) ∇ ϕ(x) dx.

From ( 2.7) and ( ^defAB 2.8), we get that ^discgrad T ₁ ^{wcv1 (m)} = X

σ ∈E

int

,σ=K | L

(u ^(m) _L − u ^(m) _K )m K | L R ^(m) _KL , where

R ^(m) _KL = 1

2 (ϕ(x K ) + ϕ(x L )) − 1 m K | L

Z

K | L

ϕ(x)dγ(x)

! n KL . Thanks to the Cauchy-Schwarz inequality,

| T ₁ ^{wcv1 (m)} | ² ≤ | u ^(m) | ² D

m

X

σ ∈E

int

,σ=K | L

R ^(m) _KL

2

m K | L d KL . One has P

σ ∈E

int

,σ=K | L m _K|L d KL ≤ dm(Ω). Thanks to the existence of C ϕ > 0 which only depends on ϕ such that | R _KL ^(m) | ≤ C ϕ size( D ^(m) ) and since α < 2, we then get that

m lim →∞ T ^wcv1 ₁ ^(m) = 0, which yields ( ^wcvgrad 2.9).

Proposition 2.4 (Discrete Rellich theorem). Let ( D ^(m) ) m ∈

N

be a sequence of ad- cpct

missible discretizations of Ω in the sense of definition 2.1, such that ^adisc lim

m →∞ size( D ^(m) ) = 0. Let us assume that there exists C > 0 and a sequence (u ^(m) ) m ∈

N

such that u ^(m) ∈ H _D

^(m)

(Ω) and k u ^(m) k D

m

≤ C for all m ∈ N .

Then, there exists ¯ u ∈ H ₀ ¹ (Ω) and a subsequence of (u ^(m) ) m ∈

N

, again denoted (u ^(m) ) m ∈

N

, such that:

1. the sequence (u ^(m) ) m ∈

N

converges in L ² (Ω) to u ¯ as m → + ∞ , 2. for all ϕ ∈ C _c ^∞ (Ω), we have

m → lim + ∞ [u ^(m) , P _D

_m

ϕ] _D

_m

= Z

Ω ∇ u(x) ¯ · ∇ ϕ(x)dx,

(2.11) cstd

3. ∇ D

m

u ^(m) weakly converges to ∇ u ¯ in L ² (Ω) ^d as m → + ∞ and ( 2.9) holds. ^wcvgrad Proof. The proof of the first two items is given in [15] (see proof of Theorem ^book

91. pp 773–774). Since we have | u ^(m) | D

m

≤ k u ^(m) k D

m

, we can apply proposition weakconvgrad

2.3, which gives the third item.

Remark 2.2. Following ^hcv [12], if we denote

D ^K,σ = { tx K + (1 − t)y, t ∈ (0, 1), y ∈ σ } , ∀ K ∈ M , ∀ σ ∈ E ^K , we may alternatively define a discrete gradient ∇ ˜ D : H _D (Ω) → (L ² (Ω)) ^d , by:

for all K ∈ M ,

∇ ˜ D u(x) = _d ^d

KL

(u L − u K )n KL , for a.e. x ∈ D ^K,K | L ∪ D ^L,K | L , ∀ L ∈ N ^K ,

∇ ˜ D u(x) = d d K,σ

(0 − u K )n K,σ , for a.e. x ∈ D ^K,σ , ∀ σ ∈ E ^K ∩ E ^ext .

7

A result similar to that of Proposition 2.4 holds with this definition of a discrete gradi- ^cpct ent, and in fact, it can be shown that the weak convergence of ∇ ˜ D

m

u ^(m) is equivalent to the weak convergence of ∇ D

m

u ^(m) .

secfvslin

3. Approximation of the linear steady problem.

3.1. The Stokes problem. We first study the following linear steady problem:

find an approximation of ¯ u and ¯ p, weak solution to the generalized Stokes equations, which write:

η¯ u − ν ∆¯ u + ∇ p ¯ = f in Ω div¯ u = 0 in Ω,

(3.1) ^stocont

For this problem, the following assumptions are made:

Ω is a polygonal open bounded connected subset of R ^d , d = 2 or 3

(3.2) ^hypomega

ν ∈ (0, + ∞ ), η ∈ [0, + ∞ ),

(3.3) hypnu

f ∈ L ² (Ω) ^d .

(3.4) hypfg

We then consider the following weak sense for problem ( 3.1). ^stocont Definition 3.1 (Weak solution for the steady Stokes equations).

weaksol

Under hypotheses ( 3.2)-( ^hypomega 3.4), let ^hypfg E(Ω) be defined by ( 1.8). Then ^e0 (¯ u, p) ¯ is called a weak solution of ( 3.1) (see e.g. ^{stocont temam} [36] or [7]) if ^bf



 

 

 

 

¯

u ∈ E(Ω), p ¯ ∈ L ² (Ω) with R

Ω p(x)dx ¯ = 0, η

Z

Ω

¯

u(x) · ¯ v(x)dx + ν Z

Ω ∇ u(x) : ¯ ∇ ¯ v(x)dx − Z

Ω

¯

p(x)div¯ v(x)dx = Z

Ω

f (x) · ¯ v(x)dx, ∀ ¯ v ∈ H ₀ ¹ (Ω) ^d .

(3.5) stocontf

The existence and uniqueness of the weak solution of ( 3.1) in the sense of the ^stocont above definition is a classical result (again, see e.g. [36] or ^temam [7]). ^bf

3.2. The finite volume scheme. Under hypotheses ( 3.2)-( ^{hypomega hypfg} 3.4), let D be an admissible discretization of Ω in the sense of Definition ^adisc 2.1. It is then natural to write an approximate problem to the Stokes problem ( 3.5) in the following way. ^stocontf



 

 

 



 

 

 



u ∈ E _D (Ω), p ∈ H _D (Ω) with Z

Ω

p(x)dx = 0 η

Z

Ω

u(x) · v(x)dx + ν[u, v] _D

− Z

Ω

p(x)div _D (v)(x)dx = Z

Ω

f (x) · v(x)dx ∀ v ∈ H _D (Ω) ^d

(3.6) schvf0

As we use a colocated approximation for the velocity and the pressure fields, the scheme must be stabilized. Using a non-consistant stabilization ` a la Brezzi-Pitk¨aranta

8

[8], we then look for (u, p) such that brez



 

 

 

 

 

 

 

 

 

 

 

 

(u, p) ∈ H _D (Ω) ^d × H _D (Ω) with Z

Ω

p(x)dx = 0 η

Z

Ω

u(x) · v(x)dx + ν [u, v] _D

− Z

Ω

p(x)div _D (v)(x)dx = Z

Ω

f (x) · v(x)dx ∀ v ∈ H _D (Ω) ^d Z

Ω

div _D (u)(x)q(x)dx = − λ size( D ) ^α h p, q i D ∀ q ∈ H _D (Ω)

(3.7) schvf

where λ > 0 and α ∈ (0, 2) are adjustable parameters of the scheme which will have to be tuned in order to make a balance between accuracy and stability.

System ( 3.7) is equivalent to finding the family of vectors (u ^schvf K ) K ∈M ⊂ R ^d , and scalars (p K ) _K∈M ⊂ R solution of the system of equations obtained by writing for each control volume K of M :



 

 

 

 



 

 

 

 



η m K u K − ν X

L ∈N

K

m K | L

d _K|L (u L − u K ) − ν X

σ ∈E

K

∩E

ext

m σ

d K,σ

(0 − u K )

+ X

L∈N

K

A KL (p L − p K ) = Z

K

f (x)dx X

L ∈N

K

A KL · (u K + u L ) − λ size( D ) ^α X

L ∈N

K

m K | L

d K | L

(p L − p K ) = 0

(3.8) schvfS

supplemented by the relation

X

K∈M

m K p K = 0

(3.9) moypnulle

Defining p σ = (p K + p L )/2 if σ = K | L, and p σ = p K if σ ∈ E ^ext ∩ E ^K , and using the fact that P

σ∈E

K

m σ n K,σ = 0, one notices that: P

L∈N

K

A KL (p L − p K ) is in fact equal to P

σ∈E

K

m σ p σ n K,σ , thus yielding a conservative form, which shows that ( 3.8) ^schvfS is indeed a finite volume scheme.

The existence of a solution to ( 3.7) will be proven below. ^schvf seccvglin

3.3. Study of the scheme in the linear case. We first prove a stability estimate for the velocity.

Proposition 3.2 (Discrete H ¹ estimate on velocities). Under hypotheses ( 3.2)- ^hypomega estl1l2

( 3.4), let ^hypfg D be an admissible discretization of Ω in the sense of definition 2.1. Let ^adisc λ ∈ (0, + ∞ ) and α ∈ (0, 2) be given. Let (u, p) ∈ H _D (Ω) ^d × H _D (Ω) be a solution to ( 3.7). Then the following inequalities hold: ^schvf

ν k u k D ≤ diam(Ω) k f k (L

²

(Ω))

^d

,

(3.10) estimu

and

ν λ size( D ) ^α | p | ² D ≤ diam(Ω) ² k f k ² (L

²

(Ω))

^d

.

(3.11) estimp

9

Proof. We apply ( 3.7) setting ^schvf v = u. We get η

Z

Ω

u(x) ² dx + ν k u k ² D − Z

Ω

p(x)div _D (u)(x)dx = Z

Ω

f (x) · v(x)dx.

Since η ≥ 0, the second equation of ( ^schvf 3.7) with q = p and Young’s inequality yield that:

η Z

Ω

u(x) ² dx + ν k u k ² D + λ size( D ) ^α | p | ² D ≤ diam(Ω) ²

2ν k f k ² (L

²

(Ω))

^d

+ ν

2diam(Ω) ² k u k ² (L

²

(Ω))

^d

. Using the Poincar´e inequality ( 2.4) gives ^poindis

ν k u k ² D + λ size( D ) ^α | p | ² D ≤ diam(Ω) ²

2ν k f k ² (L

²

(Ω))

^d

+ ν 2 k u k ² D , which leads to ( 3.10) and ( ^estimu 3.11). ^estimp

We can now state the existence and the uniqueness of a discrete solution to ( 3.7). ^schvf Corollary 3.3. [Existence and uniqueness of a solution to the finite exunpos

volume scheme] Under hypotheses ( 3.2)-( ^{hypomega hypfg} 3.4), let D be an admissible discretization of Ω in the sense of Definition ^adisc 2.1. Let λ ∈ (0, + ∞ ) and α ∈ (0, 2) be given. Then there exists a unique solution to ( 3.7). ^schvf

Proof. System ( 3.7) is a linear system. Assume that ^schvf f = 0. From propositions

estl1l2

3.2 and using ( 2.5), we get that ^poindismoy u = 0 and p = 0. This proves that the linear system ( ^schvf 3.7) is invertible.

We then prove the following strong estimate on the pressures.

Proposition 3.4 (L ² estimate on pressures). Under hypotheses ( 3.2)-( ^hypomega 3.4), let ^hypfg estp

D be an admissible discretization of Ω in the sense of definition ^adisc 2.1 and let θ > 0 be such that regul( D ) > θ. Let λ ∈ (0, + ∞ ) and α ∈ (0, 2) be given. Let (u, p) ∈ H _D (Ω) ^d × H _D (Ω) be a solution to ( 3.7). Then there exists ^schvf C 1 estisp , only depending on d, Ω, η, ν, λ, α and θ, and not on size( D ), such that the following inequality holds:

k p k ^L

²

^(Ω) ≤ C 1 ^estisp k f k (L

²

(Ω))

^d

.

(3.12) estimsp

Proof. We first apply a result by Neˇcas ^necas [29]: thanks to R

Ω p(x)dx = 0, there exists C 2 derham > 0, which only depends on d and Ω, and ¯ v ∈ H ₀ ¹ (Ω) ^d such that div¯ v(x) = p(x) for a.e. x ∈ Ω and

k ¯ v k H

¹₀

(Ω)

^d

≤ C 2 ^derham k p k ^L

²

^(Ω) .

(3.13) ineqderham

We then set

v _σ ⁽ⁱ⁾ = 1 m σ

Z

σ

¯

v ⁽ⁱ⁾ (x)dγ(x), ∀ σ ∈ E , ∀ i = 1, . . . , d.

(note that v ⁽ⁱ⁾ σ = 0 for all σ ∈ E ^ext and i = 1, . . . , d) and we define v ∈ H _D (Ω) ^d by v _K ⁽ⁱ⁾ = 1

m K

Z

K

¯

v ⁽ⁱ⁾ (x)dx, ∀ K ∈ M , ∀ i = 1, . . . , d.

10

Applying the results given p 777 in [15], we get that there exists ^book C 3 dirnh > 0, only depending on d and θ, such that

(v ⁽ⁱ⁾ _K − v ⁽ⁱ⁾ _σ ) ² ≤ C 3 ^dirnh diam(K) m σ

Z

K

( ∇ v ⁽ⁱ⁾ (x)) ² dx,

(3.14) inebook1

and

k v k D ≤ C 3 ^dirnh k ¯ v k ^H

¹0

(Ω)

^d

.

(3.15) continjh1d

We then have Z

Ω

p(x)div _D v(x)dx = X

K ∈M

p K

X

L ∈N

K

A KL · (v K + v L ) = T 2 esp1 + T 3 esp2 , where

T 2 ^esp1 = X

K ∈M

p K

X

L ∈N

K

2A KL · v _K|L

= X

K ∈M

p K

X

L ∈N

K

Z

K|L

¯

v(x) · n KL dγ(x)

= Z

Ω

p(x)div¯ v(x)dx = k p k ² L

²

(Ω) , and

T 3 ^esp2 = X

K ∈M

p K

X

L ∈N

K

m K | L

1

2 (v K + v L ) − v K | L

· n KL

= X

σ=K|L∈E

int

m K | L (p K − p L ) 1

2 (v K + v L ) − v K | L

· n KL .

We then have, thanks to the Cauchy-Schwarz inequality T ₃ ^{esp2 2} ≤ | p | ² D

X

σ=K | L ∈E

int

m K | L d KL

1

2 (v K + v L ) − v K | L

2

.

Applying Inequality ( 3.14) and thanks to ( ^{inebook1 1} ₂ (v K + v L ) − v _K|L ) ² ≤ ¹ 2 ((v K − v _K|L ) ² + (v L − v _K|L ) ² ), we get that

T ^esp2 ₃ ² ≤ | p | ² D

X

σ=K | L ∈E

int

d KL C 3 ^dirnh size( D ) Z

K∪L d

X

i=1

( ∇ v ⁽ⁱ⁾ (x)) ² dx.

This in turn implies the existence of C 4 estisp1 > 0, only depending on d and θ, such that T ₃ ^{esp2 2} ≤ C 4 ^estisp1 size( D ) ² | p | ² D k ¯ v k ² H

₀¹

(Ω)

^d

.

Thanks to ( ^ineqderham 3.13), we then get, gathering the previous results Z

Ω

p(x)div _D v(x)dx ≥ k p k ² L

²

(Ω) − C 4 ^estisp1 size( D ) | p | D C ^derham 2 k p k ^L

²

^(Ω) .

(3.16) latche

11

We then introduce v as a test function in ( ^schvf 3.7). We get Z

Ω

p(x)div _D (v)(x)dx = η Z

Ω

u(x) · v(x)dx + ν [u, v] _D − Z

Ω

f (x) · v(x)dx.

(3.17) ^vdansschvf

Applying the discrete Poincar´e inequality, ( 3.15) and ( ^continjh1d 3.16), we get the existence of ^latche C 5 estisp3 , only depending on d, Ω, f , η, ν , λ and θ, such that

k p k ² L

²

(Ω) − C ^estisp1 4 size( D ) | p | D C ^derham 2 k p k ^L

²

^(Ω) ≤ C ^estisp3 5 k u k D + k f k ^L

²

^(Ω)

^d

k p k ^L

²

^(Ω) . We now apply ( 3.10) and ( ^estimu 3.11). Since size( ^estimp D ) ² ≤ size( D ) ^α diam(Ω) ^{2 − α} , the condi- tion α ≤ 2 suffices to produce ( 3.12) from the above inequality, a factor 1/λ ^estimsp being introduced in the expression of C ^estisp 1 (it is therefore not possible to let λ tend to 0 in ( ^estimsp 3.12)).

We then have the following result, which states the convergence of the scheme ( ^schvf 3.7).

Proposition 3.5 (Convergence in the linear case). Under hypotheses ( 3.2)- ^hypomega cvgce

( 3.4), let ^hypfg (¯ u, p) ¯ be the unique weak solution of the Stokes problem ( 3.1) in the sense ^stocont of definition 3.1. Let ^weaksol λ ∈ (0, + ∞ ), α ∈ (0, 2) and θ > 0 be given and let D be an admissible discretization of Ω in the sense of definition 2.1 such that ^adisc regul( D ) ≥ θ.

Let (u, p) ∈ H _D (Ω) ^d × H _D (Ω) be the unique solution to ( 3.7). ^schvf

Then u converges to u ¯ in (L ² (Ω)) ^d and p weakly converges to p ¯ in L ² (Ω) as size( D ) tends to 0.

Proof. Under the hypotheses of the above proposition, let ( D ^(m) ) _m∈

^N

be a se- quence of admissible discretizations of Ω in the sense of definition 2.1, such that ^adisc lim _m→∞ size( D ^(m) ) = 0 and such that regul( D ^(m) ) ≥ θ, for all m ∈ N .

Let (u ^(m) , p ^(m) ) ∈ H _D

^(m)

(Ω) ^d × H _D

^(m)

(Ω) be given by ( 3.7) for all ^schvf m ∈ N . Let us prove the existence of a subsequence of ( D ^(m) ) m ∈

N

such that the corresponding sequence (u ^(m) ) m ∈

N

converges in (L ² (Ω)) ² to ¯ u and the sequence (p ^(m) ) m ∈

N

weakly converges in (L ² (Ω)) ² to ¯ p, as m → ∞ . Then the proof is complete thanks to the uniqueness of (¯ u, p). ¯

Using ( 3.10), we obtain (see ^estimu [18], ^cvnl [15]) an estimate on the translates of ^book u ^(m) : for all m ∈ N , there exists C 6 cc2 > 0, only depending on Ω, ν , f and g such that

R

Ω (u ^(m,k) (x + ξ) − u ^(m,k) (x)) ² dx ≤ C ^cc2 6 | ξ | ( | ξ | + 4size( D ^(m) )), for k = 1, . . . , d, ∀ ξ ∈ R ^d ,

(3.18) transx

where u ^(m,k) denotes the k-th component of u ^(m) . We may then apply Kolmogorov’s theorem, and obtain the existence of a subsequence of ( D ^(m) ) m ∈

N

and of ¯ u ∈ H ₀ ¹ (Ω) ² such that (u ^(m) ) m ∈

N

converges to ¯ u in L ² (Ω) ² . Thanks to proposition 3.4, we extract ^estp from this subsequence another one (still denoted u ^(m) ) such that (p ^(m) ) m ∈

N

weakly converges to some function ¯ p in L ² (Ω). In order to conclude the proof of the conver- gence of the scheme, there only remains to prove that (¯ u, p) is the solution of ( ¯ 3.5), ^stocontf thanks to the uniqueness of this solution.

Let ϕ ∈ (C _c ^∞ (Ω)) ^d . Let m ∈ N such that D ^(m) belongs to the above extracted subsequence and let (u ^(m) , p ^(m) ) be the solution to ( ^schvf 3.7) with D = D ^(m) . We suppose that m is large enough and thus size( D ^(m) ) is small enough to ensure for all K ∈ M such that K ∩ support(ϕ) 6 = ∅ , then ∂K ∩ ∂Ω = ∅ holds. Let us take v = P _D

^(m)

ϕ in ( ^schvf 3.7). Applying proposition 2.4, we get ^cpct

n→∞ lim [u ^(m) , P _D

^(m)

ϕ] _D

^(m)

= Z

Ω ∇ ¯ u(x) : ∇ ϕ(x)dx.

12

Moreover, it is clear that

n lim →∞

Z

Ω

f (x) · P _D

^(m)

ϕ(x)dx = Z

Ω

f (x) · ϕ(x)dx, and

n lim →∞ η Z

Ω

u ^(m) (x) · P _D

^(m)

ϕ(x)dx = η Z

Ω

¯

u(x) · ϕ(x)dx.

Thanks to the weak convergence of the sequence of approximate pressures, to ( 3.11) ^estimp and to the hypothesis α < 2, we now apply proposition weakconvgrad

2.3, which gives

n lim →∞

Z

Ω

p ^(m) (x)div _D

^(m)

(P _D

^(m)

ϕ)(x)dx = Z

Ω

¯

p(x)divϕ(x)dx.

(3.19) convp

The last step is to prove that div(¯ u) = 0 a.e. in Ω. Let ϕ ∈ C _c ^∞ (Ω) and let m ∈ N be given. Let us take q = P _D

^(m)

ϕ in ( ^schvf 3.7). We get T ₄ ^(m) _X = − T ₅ ^(m) _Y , where

T ₄ ^{X (m)} = Z

Ω

div _D

^(m)

(x)(u ^(m) )P _D

^(m)

ϕ(x)dx.

and

T ^Y ₅ ^(m) = λ size( D ^(m) ) ^α h p ^(m) , P _D

^(m)

ϕ i D . On the one hand, the third item of proposition 2.4 produces ^cpct

n lim →∞ T ₄ ^{X (m)} =

d

X

i=1

Z

Ω

ϕ(x)∂ i u ¯ ⁽ⁱ⁾ dx.

On the other hand, using the Cauchy-Schwarz inequality, we get:

T ^Y ₅ ^(m) ≤ λsize( D ^(m) ) ^α | p ^(m) | D | P _D

^(m)

ϕ | D

Therefore, thanks to ( 3.11) and to the regularity of ^estimp ϕ (that implies that | P _D

^(m)

ϕ | D

remains bounded independently on size( D ^(m) )) we obtain lim n →∞ T ^Y ₅ ^(m) = 0. This in turn implies that:

d

X

i=1

Z

Ω

ϕ(x)∂ i ¯ u ⁽ⁱ⁾ (x)dx = 0, for all ϕ ∈ C _c ^∞ (Ω),

(3.20) vandiv

which proves that ¯ u ∈ E(Ω).

Remark 3.1 (Strong convergence of the pressure). Note that the proof of the strong convergence of p to p ¯ is a straightforward consequence of the error estimate stated in Proposition ^ester 3.6 below, which holds under additional regularity hypotheses.

3.4. An error estimate. We then have the following result, which states an error estimate for the scheme ( 3.7). ^schvf

Proposition 3.6 (Error estimate in the linear case). Under hypotheses ( 3.2)- ^hypomega ester

( 3.4), we assume that the weak solution ^hypfg (¯ u, p) ¯ of the Stokes problem ( 3.1) in the ^stocont sense of definition ( ^weaksol 3.1) is such that (¯ u, p) ¯ ∈ H ² (Ω) ^d × H ¹ (Ω). Let λ ∈ (0, + ∞ ) and α ∈ (0, 2) be given, let D be an admissible discretization of Ω in the sense of definition

13

adisc

2.1 and let θ > 0 such that regul( D ^(m) ) ≥ θ. Let (u, p) ∈ H _D (Ω) ^d × H _D (Ω) be the solution to ( 3.7). Then there exists ^schvf C 7 ester1 , which only depends on d, Ω, ν , η and θ such that

k u − u ¯ k ² L

²

(Ω) ≤ C ^ester1 7 ε(λ, size( D ), p, ¯ u), ¯

(3.21) eqester1

λ size( D ) ^α | p | ² D ≤ C 7 ^ester1 ε(λ, size( D ), p, ¯ u) ¯

(3.22) eqester2

k p − p ¯ k ² L

²

(Ω) ≤ C ^ester1 7 ε(λ, size( D ), p, ¯ u). ¯

(3.23) eqester3

where

ε(λ, size( D ), p, ¯ ¯ u) = min λsize( D ) ^α , _λ ¹ size( D ) ^{2 − α}

×

k p ¯ k ² H

¹

(Ω) + k u ¯ k ² H

²

(Ω)

.

(3.24) epsilon

Proof. We define (ˆ u, p) ˆ ∈ H _D (Ω) ^d × H _D (Ω) by ˆ u = P _D u, which means ˆ ¯ u K = ¯ u(x k ) for all K ∈ M , and ˆ p K = _m ¹

K

R

K p(x)dx ¯ for all K ∈ M . Integrating the first equation of ( 3.1) on ^stocont K ∈ M gives

η Z

K

¯

u(x)dx + X

σ∈E

K

− ν R

σ ∇ ¯ u(x) : n K,σ dγ(x)+

R

σ p(x)n ¯ K,σ dγ(x)

= Z

K

f (x)dx.

(3.25) stocontK

We introduce, for K ∈ M , ε ^u _K = ˆ u K − m ¹

K

R

K u(x)dx, and, for ¯ L ∈ N ^K : R K,L = _d ¹

K|L

(ˆ u L − u ˆ K ) − m

K|L

¹

R

σ ∇ u(x) : ¯ n K,σ dγ(x), and for σ ∈ E ^K ∩ E ^ext , R K,σ = _d

_K,σ

¹ (0 − u ˆ K ) − m ¹

σ

R

σ ∇ u(x) : ¯ n K,σ dγ(x);

moreover, we define for L ∈ N ^K : ε ^p _{K | L} = ¹ ₂ (ˆ p K + ˆ p L ) − m

K|L

¹

R

K | L p(x)dγ(x), ¯ and for σ ∈ E ^K ∩ E ^ext , ε ^p _σ = ˆ p K − m ¹

σ

R

σ p(x)dγ(x). Using these notations and the relation ¯ P

σ∈E

K

m σ n K,σ = 0, we get from ( 3.25) ^stocontK η m K u ˆ K − ν X

L∈N

K

m _K|L d K | L

(ˆ u L − u ˆ K ) + X

σ∈E

K

∩E

ext

m σ

d K,σ (0 − u ˆ K )

! + X

L ∈N

K

A KL (ˆ p L − p ˆ K ) = Z

K

f (x)dx + R K , with

R K = η m K ε ^u _K − ν X

L∈N

K

m K | L R K,L + X

σ∈E

K

∩E

ext

m σ R K,σ

!

+ X

σ∈E

K

m σ ε ^p _σ n K,σ . We then set δu = ˆ u − u and δp = ˆ p − p. We then get, substracting the first relation of the scheme ( ^schvfS 3.8) to the above equation,

η R

Ω δu(x)v(x)dx + ν [δu, v] _D − Z

Ω

δp(x)div _D (v)(x)dx = R

Ω R(x)vdx, ∀ v ∈ H _D (Ω) ^d ,

(3.26) eqester4

and, setting v = δu in ( 3.26), ^eqester4 η

Z

Ω

δu(x) ² dx + ν k δu k ² D − Z

Ω

δp(x)div _D (δu)(x)dx = Z

Ω

R(x)δu(x)dx.

14

We now integrate the second equation of ( 3.1) on ^stocont K ∈ M . This gives X

σ ∈E

K

Z

σ

¯

u(x) · n K,σ dγ(x) = 0, ∀ K ∈ M . Using ¯ u ∈ H ₀ ¹ (Ω), we then obtain

X

L∈N

K

A KL · (ˆ u K + ˆ u L ) = X

L∈N

K

m _K|L ε ^u _{K | L} , ∀ K ∈ M

with

ε ^u _K|L = 1

2 (ˆ u K + ˆ u L ) − 1 m K | L

Z

K | L

¯

u(x)dγ(x)

!

· n KL , ∀ K ∈ M , ∀ L ∈ N ^K . We then give, substracting the second relation of the scheme ( ^schvfS 3.8) to the above equa- tion,

Z

Ω

div _D (δu)(x)δp(x)dx = λ size( D ) ^α h p, p ˆ − p i D + T 6 esterm1 , with

T 6 ^esterm1 = X

K | L ∈E

int

m K | L ε ^u _K|L (δp K − δp L ), Gathering the above results, we get

η R

Ω δu(x) ² dx + ν k δu k ² D + λ size( D ) ^α | p | ² D = λ size( D ) ^α h p, p ˆ i D + R

Ω R(x) · δu(x)dx + T 6 ^esterm1 . eqester5 (3.27)

Let us study the terms at the right hand side of the above equation. We have, using the Young inequality,

h p, p ˆ i D ≤ 1

4 | p | ² D + | p ˆ | ² D ≤ 1

4 | p | ² D + C 8 k p ¯ k ² H

¹

(Ω) .

(3.28) eqester6

We then study R

Ω R(x) · δu(x)dx = T 7 esterm2 + T 8 esterm3 + T 9 esterm4 , with T 7 ^esterm2 = η

Z

Ω

ε ^u (x) · δu(x)dx,

T ^esterm3 8 = ν X

K ∈M

X

L ∈N

K

m K | L R K,L + X

σ ∈E

K

∩E

ext

m σ R K,σ

!

· δu K ,

and

T 9 ^esterm4 = X

K ∈M

X

σ ∈E

K

m σ ε ^p _σ n K,σ · δu K .

Thanks to interpolation results proven in [15] and to ( ^{book poindis} 2.4), we obtain T 7 ^esterm2 ≤ C 9 size( D ) ² k u ¯ k ² H

²

(Ω) + ν

4 k δu k ² D ,

(3.29) eqester7

15

T 8 ^esterm3 ≤ C 10 size( D ) ² k u ¯ k ² H

²

(Ω) + ν 4 k δu k ² D ,

(3.30) eqester8

and

T 9 ^esterm4 ≤ C 11 size( D ) ² k p ¯ k ² H

¹

(Ω) + ν 4 k δu k ² D .

(3.31) eqester9

We then study T 6 ^esterm1 . We have T 6 ^esterm1 = T 10 esterm1a − T 11 esterm1b with T 10 ^esterm1a = X

K | L ∈E

int

m K | L ε ^u _{K | L} (ˆ p K − p ˆ L ), which verifies

T 10 ^esterm1a ≤ C 12 size( D )

k p ¯ k ² H

¹

(Ω) + k u ¯ k ² H

²

(Ω)

,

(3.32) ^eqester10

and

T 11 ^esterm1b = X

K | L ∈E

int

m K | L ε ^u _{K | L} (p K − p L ), which verifies

T 11 ^esterm1b ≤ 1

4 λ size( D ) ^α | p | ² D + C 13 1

λ size( D ) ^{2 − α} k u ¯ k ² H

²

(Ω) .

(3.33) eqester10bis

Gathering equations ( 3.27)-( ^eqester5 eqester10bis

3.33) gives

k δu k ² D + λ size( D ) ^α | p | ² D ≤ C 14 estertot ε(λ, size( D ), p, ¯ u), ¯

where ε(λ, size( D ), p, ¯ u) is defined by ( ¯ 3.24). This in turn yields ( ^epsilon 3.21) and ( ^eqester1 3.22). We ^eqester2 then again follow the method used in the proof of Proposition 3.4. Using ^estp R

Ω p(x)dx ˆ = 0 and therefore R

Ω δp(x)dx = 0, let ¯ v ∈ H ₀ ¹ (Ω) ^d be given such that div¯ v(x) = δp(x) for a.e. x ∈ Ω and

k v ¯ k H

₀¹

(Ω)

^d

≤ C 2 ^derham k δp k L

²

(Ω) .

(3.34) ^eqester11

We again set

v _σ ⁽ⁱ⁾ = 1 m σ

Z

σ

¯

v ⁽ⁱ⁾ (x)dγ(x), ∀ σ ∈ E , ∀ i = 1, . . . , d.

and we define v ∈ H _D (Ω) ^d by v _K ⁽ⁱ⁾ = 1

m K

Z

K

¯

v ⁽ⁱ⁾ (x)dx, ∀ K ∈ M , ∀ i = 1, . . . , d.

The same method gives k δp k ² L

²

(Ω) ≤

Z

Ω

δp(x)div _D (v)(x)dx + C 4 ^estisp1 size( D ) | p | D k v ¯ k H

₀¹

(Ω)

^d

≤ Z

Ω

δp(x)div _D (v)(x)dx + C 15 ester15 size( D ) ² | p | ² D + 1

4 k δp k ² L

²

(Ω) . We now use v as test function in ( ^eqester4 3.26). We get

Z

Ω

δp(x)div _D (v)(x)dx = η Z

Ω

δu(x)v(x)dx + ν [δu, v] _D + Z

Ω

R(x)vdx.

16

Gathering the two above inequalities, ( 3.29), ( ^{eqester7 eqester8} 3.30), ( 3.31) and ( ^eqester9 3.34) produces ^eqester11 k δp k ² L

²

(Ω) ≤ 1

2 k δp k ² L

²

(Ω) + C 16 ester17 size( D ) ²

k p ¯ k ² H

¹

(Ω) + k u ¯ k ² H

²

(Ω)

+C 17 ester16 k δu k ² D + C 15 ^ester15 size( D ) ² | p | ² D .

Applying ( ^eqester1 3.21) and ( ^eqester2 3.22) gives ( 3.23). ^eqester3

Remark 3.2. In the above result, it suffices to let α = 1 to obtain the proof of an order 1/2 for the convergence of the scheme. We recall that this result is not sharp, and that the numerical results show a much better order of convergence.

secfvsnlin

4. The finite volume scheme for the Navier-Stokes equations. Before handling the transient nonlinear case, we first address in the following section the steady-state case.

stnl

4.1. The steady-state case. For the following continuous equations, η u ¯ ⁽ⁱ⁾ − ν∆¯ u ⁽ⁱ⁾ + ∂ i p ¯ +

d

X

j=1

¯

u ^(j) ∂ j u ¯ ⁽ⁱ⁾ = f ⁽ⁱ⁾ in Ω, for i = 1, . . . , d, div¯ u =

d

X

i=1

∂ i u ¯ ⁽ⁱ⁾ = 0 in Ω.

(4.1) nstocontss

with a homogeneous Dirichlet boundary condition, we define the following weak sense.

Definition 4.1 (Weak solution for the steady Navier-Stokes equations). Under weaksolss

hypotheses ( 3.2)-( ^{hypomega hypfg} 3.4), let E(Ω) be defined by ( 1.8). Then ^e0 (¯ u, p) ¯ is called a weak solution of ( 4.1) if ^nstocontss



 

 

 



 

 

 



¯

u ∈ E(Ω), p ¯ ∈ L ² (Ω) with Z

Ω

¯

p(x)dx = 0, η

Z

Ω

¯

u(x) · v(x)dx ¯ + ν Z

Ω ∇ u(x) : ¯ ∇ v(x)dx ¯

− Z

Ω

¯

p(x)div¯ v(x)dx + b(¯ u, u, ¯ ¯ v) = Z

Ω

f (x) · v(x)dx ¯ ∀ ¯ v ∈ H ₀ ¹ (Ω) ^d ,

(4.2) nstocontfss

where the trilinear form b(., ., .) is defined by ( 1.10). ^deftricont

We now give the finite volume scheme for this problem. Under hypotheses ( ^hypomega 3.2)- ( ^hypfg 3.4), let D be an admissible discretization of Ω in the sense of Definition ^adisc 2.1. We introduce Bernoulli’s pressure p + ¹ ₂ u ² instead of p, again denoted by p, and for any real value λ > 0 and α ∈ (0, 2), we look for (u, p) such that



 

 

 

 

 

 

 

 

 

 

 

 

 

 

(u, p) ∈ H _D (Ω) ^d × H _D (Ω) with Z

Ω

p(x)dx = 0,

η Z

Ω

u(x) · v(x)dx + ν[u, v] _D + 1 2

Z

Ω

u(x) ² div _D (v)(x)dx

− Z

Ω

p(x)div _D (v)(x)dx + b _D (u, u, v) = Z

Ω

f (x) · v(x)dx ∀ v ∈ H _D (Ω) ^d Z

Ω

div _D (u)(x)q(x)dx = − λ size( D ) ^α h p, q i D ∀ q ∈ H _D (Ω)

(4.3) schvfnlss

17

where, for u, v, w ∈ H _D (Ω), we define the following approximation for b(u, v, w) b _D (u, v, w) = 1

2 X

K ∈M

X

L ∈N

K

(A KL · (u K + u L )) ((v L − v K ) · w K )

(4.4) deftridc

System ( 4.3) is equivalent to finding the family of vectors (u ^schvfnlss K ) K ∈M ⊂ R ^d , and scalars (p K ) _K∈M ⊂ R solution of the system of equations obtained by writing for each control volume K of M :



 

 

 

 

 

 

 

 

 

 

 

 

 

 

η m K u K − ν X

L∈N

K

m K | L

d _K|L (u L − u K ) − ν X

σ∈E

K

∩E

ext

m σ

d K,σ (0 − u K )

+ X

L ∈N

K

(A KL · ( 1

2 (u K + u L ))) (u L − u K )

+ X

L ∈N

K

A KL (p L − p K ) − 1 2

X

L ∈N

K

A KL (u ² _L − u ² _K ) = Z

K

f (x)dx X

L ∈N

K

A KL · (u K + u L ) − λ size( D ) ^α X

L ∈N

K

m K | L

d K | L

(p L − p K ) = 0

(4.5) schvfNSS

supplemented by the relation:

X

K∈M

m K p K = 0

Defining ˜ p K = p K − u ² _K /2 and ˜ p σ = (˜ p K + ˜ p L )/2 if σ = K | L, ˜ p σ = ˜ p K if σ ∈ E ^ext ∩ E ^K , and using the fact that P

σ∈E

K

m σ n K,σ = 0, one again notices that:

P

L∈N

K

A KL (˜ p L − p ˜ K ) is in fact equal to P

σ∈E

K

m σ p ˜ σ n K,σ , thus yielding a conserva- tive form for the fifth and sixth terms of the left handside of the discrete momentum equation in ( 4.5). Defining ^schvfNSS u σ = (u K + u L )/2 if σ = K | L, u σ = 0 if σ ∈ E ^ext ∩E ^K , one obtains that the nonlinear convective term P

L ∈N

K

(A KL · ( ¹ ₂ (u K + u L ))) (u L − u K ) is equal to P

σ ∈E

K

m σ (n K,σ · u σ )u σ − m K u K (div _D u) K ; one may note that (div _D u) K = P

σ ∈E

K

m σ n K,σ · u σ . Hence the nonlinear convective term is the sum of a conserva- tive form and a source term due to the stabilization (this source term vanishes for a discrete divergence free function u).

Let us then study some properties of the trilinear form b _D . First note that the quantity b _D (u, v, w) also writes

b _D (u, v, w) = 1 2

X

K | L ∈E

int

(A KL · (u K + u L )) ((v L − v K ) · (w L + w K ))

(4.6) deftridcbis

We thus get that, for all u, v ∈ H _D (Ω) ^d , b _D (u, v, v) = 1

2 X

K | L ∈E

int

(A KL · (u K + u L ))((v L ) ² − (v K ) ² )

= − 1 2

Z

Ω

v(x) ² div _D (u)(x) dx

(4.7) ^superb

We get in particular, that, for all u ∈ E _D (Ω), b _D (u, u, u) = 0, which is the discrete equivalent of the continuous property.

18

Remark 4.1. [Upstream weighting versions of the scheme] All the results of this paper are available, setting F KL (u) = A KL · (u K + u L ) and considering, for u, v, w ∈ H _D (Ω),

b ^ups _D (u, v, w) = b _D (u, v, w) + 1 2

X

K|L∈E

int

Θ KL | F KL (u) | (v L − v K ) · (w L − w K ),

with, for example, Θ KL = max(1 − 2ν ^m _d

^K|L

K|L

/ | F KL (u) | , 0). We then get, for all u, v ∈ H _D (Ω), the inequality

b ^ups _D (u, v, v) ≥ − 1 2

Z

Ω

v(x) ² div _D (u)(x)dx,

which is sufficient to get all the estimates of this paper, together with the convergence properties of the scheme. The use of such a local upwinding technique may be useful to avoid the development of nonphysical oscillations only where meshes are too coarse.

The following technical estimates are crucial to prove the convergence properties of the scheme.

Lemma 4.2 (Estimates on b _D (., ., .) by discrete Sobolev norms). Under hypotheses l4l2h1

( _adisct 1.3)-( ^hypomegat 1.7), let ^hypfgt D be an admissible discretization of Ω × (0, T ) in the sense of definition 4.8, and θ > 0 such that regul( D ) ≥ θ. Then there exists C 18 bd4 > 0 and C 19 bd > 0, only depending on d, θ and Ω, such that

b _D (u, v, w) ≤ C 18 ^bd4 k u k L

⁴

(Ω)

^d

k v k D k w k L

⁴

(Ω)

^d

≤ C 19 ^bd k u k D k v k D k w k D .

(4.8) ^estib

Proof. The quantity b _D (u, v, w) reads b _D (u, v, w) = 1

4 X

K∈M

X

L∈N

K

(w K · (v L − v K )) m K | L

d _K|L ((x L − x K ) · (u K + u L )) Applying the Cauchy-Schwarz inequality twice and using the fact that (x L − x K ) ² = d ² _KL and that, for any admissible discretization P

L ∈N

K

m

K|L

d

K|L

d ² _KL ≤ d ^m _θ

^K

yield:

b _D (u, v, w) ² ≤ C 20 dse X

K∈M

X

L∈N

K

m K | L

d _K|L (w K ) ² (x L − x K ) ² (2(u K ) ² + 2(u L ) ² )

!

X

K∈M

X

L∈N

K

m K | L

d _K|L (v L − v K ) ²

!

≤ C 21 dse3 X

K∈M

m K | w K | ⁴

! ^1/2 X

K∈M

m K | u K | ⁴

! ^1/2 k v k ² D .

The inequality ( 4.8) is now a straightforward consequence of the following discrete ^estib Sobolev inequality, which holds under the same regularity assumptions on the mesh (see proof in [10] or ^coudiere [15, pp. 790-791]): ^book

k u k ^L

⁴

^(Ω) ≤ C 22 k u k D .

(4.9) sob1

19