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volume 3, issue 2, article 17, 2002.

Received 18 May, 2001;

accepted 8 November, 2001.

Communicated by:R.N. Mohapatra

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Journal of Inequalities in Pure and Applied Mathematics

AN ERROR ESTIMATE FOR FINITE VOLUME METHODS FOR THE STOKES EQUATIONS

A. ALAMI-IDRISSI AND M. ATOUNTI

Université Mohammed V-Agdal Faculté des Sciences

Dépt de Mathématiques & Informatique Avenue Ibn Batouta

BP 1014 Rabat 10000, Morocco EMail:alidal@fsr.ac.ma EMail:atounti@hotmail.com

c

2000Victoria University ISSN (electronic): 1443-5756 045-01

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Norms of certain operators on weighted`pspaces and Lorentz

sequence spaces A. Alami-Idrissi and M. Atounti

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J. Ineq. Pure and Appl. Math. 3(2) Art. 20, 2002

Abstract

In the present paper, we study an error estimate for finite volume methods for the stokes equations. The error is proven to be of orderh, inH₀¹-norm discrete and inL²-norm, wherehrepresents the size of the mesh. The result is new even for the finite volume method.

2000 Mathematics Subject Classification:35Q30, 76D07.

Key words: Finite volume method, Stokes equations, Discrete Poincaré inequality, Error estimate.

1. Introduction

The numerical solution of the Navier-Stokes equations for incompressible viscous fluids has motivated many authors, so much so that giving a complete bibliography has become an impossible task. Therefore, we restrict our atten- tion only to crucial contributions making use of finite element approximations and mixed finite element methods, among them we mention [2, 3,6,8, 12,13, 14,15,16,17,18] (see also the references therein).

The finite volume element method is used in [9], the basic idea is based on the Box method. From the Crouzeix-Raviart element, the authors constructed the mesh of this method since every triangulation is associated to the spaces of finite elements. Later on, they applied the Babuska theorem to the Stokes problem, thus they obtained an analysis of error.

The finite volume projection method for the numerical approximation of two-dimensional incompressible flows on triangular unstructured grids is presented in [4]. The authors considered the unsteady Navier-Stokes equations, the velocity field is approximated by either piecewise constant or piecewise linear functions on the triangles, and the pressure field is approximated by piecewise linear functions. For the discretization of the diffusive flows, a dual grid con- necting the centers of the triangles of the primary grid is introduced there. Using this grid, a stable and accurate discrete Laplacian is obtained.

The finite volume scheme for the Stokes problem is obtained from a mixed finite element method with a well chosen numerical integration diagonalizing the mass matrix which is used in [1]. The analysis of the corresponding finite volume scheme is directly deduced from general results of mixed finite element theory and the authors gave an optimal a priori error estimate.

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The finite volume method on unstructured staggered grids for the Stokes problem is presented in [10]. The authors used an admissible mesh of triangles satisfying the properties required for the finite element method. In the case of acute angles, they proved the existence and the uniqueness of the solution, therefore, if the mesh consist of equilateral triangles, the authors obtained the convergence result.

In this paper, we are interested in the study of an error estimate for finite volume method for the Stokes equations in dimensiond = 2or3, on unstructured staggered grids. The main difficulty of this problem is due to the coupling of the velocity with the pressure. For this reason, we use the Galerkin expansion for the approximation of the pressure such that the pressure unknowns are located at the vertices. The existence and the uniqueness of the solution results are proved by Eymard, Gallouet and Herbin in [10]. We prove here that the error estimate is of order one.

This paper is organized as follows: In Section2, we introduce the continuous Stokes equations under some assumptions. In Section3, we get the numerical scheme and the main results of the existence and the uniqueness of the numerical solution. Finally, in Section4, we present the error estimate for the velocity.

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2. The Continuous Equations

We consider here the Stokes problem:

−ν∆uⁱ(x) + ∂p

∂x_i(x) = fⁱ(x) ∀x∈Ω,∀i= 1, . . . , d, (2.1)

d

X

i=1

∂uⁱ

∂x_i = 0 ∀x∈Ω,

(2.2)

with Dirichlet boundary condition:

(2.3) uⁱ(x) = 0 ∀x∈∂Ω ,∀i= 1, . . . , d, under the following assumption.

Assumption 1. (i) Ωis an open bounded connected polygonal subset ofR^d,d= 2,3.

(ii) ν >0.

(iii) fⁱ ∈L²(Ω);∀i= 1, ..., d.

In the above equation, uⁱ represents the i^th component of the velocity of a fluid, ν the kinematic viscosity andp the pressure. There exist several convenient mathematical formulations of (2.1) – (2.3).

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3. A Finite Volume Scheme on Unstructured Staggered Grids

The finite volume scheme is found by integrating equation (2.1) on a control volume of a discretization mesh and finding an approximation of the fluxes on the control volume boundary in terms of the discrete unknowns. Let us first give the assumptions which are needed on the mesh.

Definition 3.1. Admissible mesh.

LetΩ, be an open bounded polygonal subset ofR^d, (d= 2 or3). An admissible finite volume mesh ofΩ, denoted byT, is given by a family of control volumes, which are open polygonal convex subsets ofΩ¯ contained in hyperplanes ofR^d, denoted by E, (they are the edges (2D), or sides (3D) of the control volumes), with strictly positive(d−1)-dimensional measure and a family of points ofΩ denoted byP satisfying the following properties:

(i) The closure of the union of all the control volumes isΩ.¯

(ii) For any K ∈ T, there exists a subsetE_K of E such that ∂K = ¯K\K =

∪σ∈E_Kσ, let¯ E =∪K∈TE_K.

(iii) For any (K, L) ∈ T², withK 6= L, either the d-dimensional Lebesgue measure ofK¯ ∩L¯is 0 orK¯ ∩L¯ = ¯σfor someσ ∈ E.

(iv) The family P =(x_K)K∈T is such that x_K ∈ K¯ and if σ = K|L it is assumed thatx_k 6= x_L and that the straight lineD_K,L going through x_K andx_Lis orthogonal toK|L.

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(v) For any σ ∈ E such that σ ∈ ∂Ω, let K be the control volume such thatσ ∈ EK, if xK ∈/ σ, let DK,σ be the straight line going throughxK

and orthogonal to σ. Then the condition D_K,σ ∩σ 6= ∅ is assumed, let y_σ =D_K,σ∩σ.

In the sequel, the following notations are used:

• size(T) =sup {diam(K), K ∈ T }.

• m(K)thed−dimensional Lebesgue ofK, for anyK ∈ T.

• m(σ)the (d−1)-dimensional Lebesgue ofσ, for anyσ∈ E.

• E_int ={σ ∈ E, σ6⊂∂Ω}andE_ext ={σ ∈ E, σ⊂∂Ω}.

• Ifσ ∈ E_int, σ = K|Lthend_σ =d_K|L =d(x_K, x_L)and ifσ ∈ E_K∩ E_ext thend_σ =d_K,σ =d(x_K, y_σ).

• For anyσ ∈ E the transmissibility throughσ is defined byτ_σ = ^m(σ)_d

σ if d_σ 6= 0andτ_σ = 0ifd_σ = 0.

In some results and proofs given below, there are summations overσ ∈ E₀ withE₀ ={σ ∈ E;d_σ 6= 0}. For simplicityE₀ =E is assumed.

Let us now introduce the space of piecewise constant functions associated with an admissible mesh and discrete H₀¹-norm for this space. This discrete norm will be used to obtain an estimate of the approximate solution given by a finite volume scheme.

Definition 3.2. Let Ωbe an open bounded polygonal subset ofR^d, (d = 2,3) andT be an admissible mesh. DefineX(T)to be the set of functions fromΩto Rwhich are constant over each control volume of the mesh.

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Definition 3.3. Let Ωbe an open bounded polygonal subset ofR^d, (d = 2,3) andT be an admissible mesh. Foru∈X(T), define the discreteH₀¹-norm by:

kuk1,T = X

σ∈E

τ_σ(D_σu)²

!¹₂ ,

where:

Dσu = |uK −uL|if σ∈ Eint, σ=K|L.

D_σu = |u_K|if σ∈ E_ext∩ E_K

andu_K denotes the value taken byuon the control volumeK.

Lemma 3.1 (Discrete Poincaré inequality). LetΩbe an open bounded polyg- onal subsetR^d,(d= 2,3),T be an admissible mesh andu∈X(T), then:

kuk_L² ≤diam(Ω)kuk1,T,

wherek·k_1,T is the discreteH₀¹-norm.

Proof. See [10, p. 38, 11].

Assume K and L to be two neighboring control volumes of the mesh. A consistent discretization of the normal flux − 5u.n over the interface of two control volumesKandLmay be performed with differential quotient involving values of the unknown located on the orthogonal line to the interface between K andL, on either side of this interface.

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In [10], the authors consider the mesh of Ω, denoted by T, consisting of triangles, satisfying the properties required for the finite element method, see [7], with acute angles only, and defining, for all K ∈ T, the point x_K as the intersection of the orthogonal bisectors of the sides of the trianglesKyields that T is an admissible mesh. Fors ∈ ST, let φs be the shape function associated tosinP₁. A possible finite volume scheme using a Galerkin expansion for the pressure is defined by the following equations:

(3.1) ν X

σ∈E_K

F_K,σⁱ + X

s∈S_K

p_s Z

K

∂φ_s

∂xi

(x)dx

=m(K)f_Kⁱ ∀K ∈ T ,∀i= 1, ..., d,

F_K,σⁱ = τ_σ(uⁱ_K−uⁱ_L), ifσ∈ E_int, σ=K|L , i= 1, ..., d, (3.2)

F_K,σⁱ = τ_σuⁱ_K, ifσ∈ E_ext∩ E_K , i= 1, ..., d, (3.3)

X

K∈T d

X

i

uⁱ_K Z

K

∂φ_s

∂x_i(x)dx = 0 ∀s∈ST, (3.4)

Z

Ω

X

s∈S_T

psφs(x)dx = 0, and (3.5)

f_Kⁱ = 1 m(K)

Z

K

fⁱ(x)dx ,∀K ∈ T. (3.6)

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The discrete unknowns of (3.1) – (3.6) are uⁱ_K, K ∈ T ,∀i = 1, . . . , d, and ps, s∈ST.

The approximate solutions are defined by:

(3.7) uⁱ_K(x) =uⁱ_K a.e x∈K ,∀K ∈ T ,∀i= 1, ..., d and

(3.8) pT = X

s∈ST

p_sφ_s.

The existence and the uniqueness of the solution of the discrete problem (3.1) – (3.6) are proved by Eymard, Gallouet and Herbin in [10]. Moreover, if the element ofT are equilateral triangles then they obtained the following convergence result.

Proposition 3.2. Under Assumption1, there exists an unique solution to (3.1) – (3.6), denoted by{uⁱ_K, K ∈ T, i= 1, ..., d}and{p_s, s ∈ST}. Furthermore, if the elements ofT are equilateral triangle, thenu_T −→ u, assize(T) → 0, whereuis the unique solution to (2.1) – (2.3) anduT = (u¹_T, ..., u^d_T)is defined by (3.7).

Proof. See [10, p. 205].

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4. Error Estimate

In this section, we present the error estimate theorem that is of order one.

Theorem 4.1. Under Assumption 1, let T be an admissible mesh as given by Definition3.1anduⁱ_T ∈X(T), ∀i= 1, ..., d, such thatuⁱ_T =uⁱ_K, ∀i= 1, ..., d for a.e. x ∈ K, for allK ∈ T where(uⁱ_K)K∈T is the solution to (3.1) – (3.6).

Letu= (uⁱ)be the unique variational solution of problem (2.1) – (2.3) and for eachK ∈ T, eⁱ_K =uⁱ(x_K)−uⁱ_K, andeⁱ_T ∈ T defined byeⁱ_T(x) = eⁱ_K for a.e.

x ∈ K, for allK ∈ T. Then there existsC > 0depending only onu,Ωandd such that:

(4.1) keⁱ_Tk1,T ≤Csize(T)

and

(4.2) keⁱ_Tk_L² ≤diam(Ω)Csize(T), wherek·k_1,T is the discreteH₀¹-norm.

Proof. Integrating overK the equation (2.1), then:

(4.3) −ν Z

∂K

∇uⁱ· −→n∂Kdσ∂K + Z

K

∂p

∂x_i(x)dx= Z

K

fⁱ(x) ∀i= 1, . . . , d.

As Z

∂K

∇uⁱ· −→n_∂Kdσ_∂K = X

σ∈E_K

Z

σ

∇uⁱ· −→n_σdσ ∀i= 1, . . . , d.

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We denote by:

Fⁱ_K,σ =− Z

σ

∇uⁱ· −→n_σdσ ∀i= 1, . . . , d,

then:

(4.4) ν X

σ∈E_K

Fⁱ_K,σ+ Z

K

∂p

∂x_i(x)dx= Z

K

fⁱ(x) ∀i= 1, . . . , d.

LetF_K,σ^∗,i be defined by:

F_K,σ^∗,i = τ_σ(uⁱ(x_K)−uⁱ(x_L)) ifσ ∈ E_int, σ =K|L;i= 1, . . . , d, F_K,σ^∗,i = τσuⁱ(xK) ifσ ∈ Eext∩ EK , i= 1, . . . , d,

then the consistency error may be defined as:

Fⁱ_K,σ−F_K,σ^∗,i =m(σ)Rⁱ_K,σ ∀i= 1, . . . , d.

Thanks to the regularity of u, there existsC₁ ∈ R, only depending on u, such that:

(4.5)

Rⁱ_K,σ

≤C₁size(T) ∀K ∈ T andσ∈ E_K ∀i= 1, . . . , d.

Ifσ∈ E_int∩ E_K , σ=K|L, then, we have:

Fⁱ_K,σ−F_K,σⁱ = Fⁱ_K,σ−F_K,σ^∗,i +F_K,σ^∗,i −F_K,σⁱ

= m(σ)Rⁱ_K,σ+F_K,σ^∗,i −F_K,σⁱ

= m(σ)Rⁱ_K,σ+τσ(eⁱ_K−eⁱ_L), (4.6)

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and ifσ ∈ E_ext∩ E_K, then, we have:

(4.7) Fⁱ_K,σ−F_K,σⁱ =m(σ)Rⁱ_K,σ+τ_σeⁱ_K. Subtracting (3.1) from (4.3) then:

(4.8) ν X

σ∈EK

Fⁱ_K,σ−F_K,σⁱ +

Z

K

∂p

∂xi

(x)dx− X

s∈SK

p_s Z

K

∂φ_s

∂xi

(x)dx

= Z

K

fⁱ(x)−m(K)f_K,σⁱ .

Multiplying (4.8) byeⁱ_K, summing forK ∈ T andi, then we obtain:

d

X

i

X

K∈T

X

σ∈EK

Fⁱ_K,σ−F_K,σⁱ eⁱ_K

=

d

X

i

X

K∈T

X

σ∈E_K

τ_σ(eⁱ_K−eⁱ_L)eⁱ_K +

d

X

i

X

K∈T

X

σ∈E_K

m(σ)Rⁱ_K,σeⁱ_K

=

d

X

i

X

σ∈E

τ_σ|D_σeⁱ_T|²+

d

X

i

X

K∈T

X

σ∈E_K

m(σ)Rⁱ_K,σeⁱ_K. (4.9)

Usingdiv(u) = 0and the relation (3.4), we deduce that:

(4.10)

d

X

i

X

K∈T



 Z

K

∂p

∂x_i(x)dx− X

s∈S_K

ps

Z

K

∂φ_s

∂x_i(x)dx



eⁱ_K = 0.

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From the relation (3.6), then:

(4.11)

d

X

i

X

K∈T



 Z

K

fⁱ(x)−m(K)f_K,σⁱ



eⁱ_K = 0.

Replacing (4.9), (4.10), (4.11) in (4.8), hence:

d

X

i

X

σ∈E

τ_σ|D_σeⁱ_K|² =−

d

X

i

X

K∈T

X

σ∈E_K

m(σ)Rⁱ_K,σeⁱ_K,

then:

(4.12)

d

X

i

keⁱ_Tk²_1,T =−

d

X

i

X

K∈T

X

σ∈E_K

m(σ)Rⁱ_K,σeⁱ_K.

Thanks to the propriety of conservativity, one hasRⁱ_K,σ = −Rⁱ_L,σ forσ ∈ Eint, such thatσ =K|L, letRⁱ_σ =|Rⁱ_K,σ|.

Reordering the summation over the edges and using the Cauchy-Schwarz inequality, one obtains:

X

K∈T

X

σ∈EK

≤ X

σ∈EK

m(σ)|R_σⁱ||D_σeⁱ_T|

≤ X

σ∈EK

m(σ)

d_σ |D_σeⁱ_T|²

!¹₂ X

σ∈E

m(σ)d_σ|Rⁱ_σ|²

!¹₂ .

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From the relation (4.5), we have |Rⁱ_σ| ≤ C₁size(T) and we remark that P

σ∈E

m(σ)dσ =m(Ω), then we deduce the existence ofC2, only depending onu andΩ, such that:

(4.13)

X

K∈T

X

σ∈E_K

≤C₂keⁱ_Tk1,Tsize(T).

Then:

(4.14)

d

X

i=1

keⁱ_Tk²_1,T ≤C₂

d

X

i=1

keⁱ_Tk_1,T

!

size(T).

Using Young’s inequality, there existsC₃ only depending on u, Ωandd, such that:

(4.15)

d

X

i=1

keⁱ_Tk²_1,T

!¹₂

≤C3size(T).

We have:

(4.16) keⁱ_Tk1,T ≤

d

X

i=1

keⁱ_Tk²_1,T

!¹₂

≤C₃size(T) ∀i= 1, . . . , d.

Applying the discrete Poincaré inequality, we obtain the relation (4.2).

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