An error estimate for finite volume methods for the Stokes equations on equilateral triangular meshes

An Error Estimate for Finite Volume Methods forthe Stokes Equations on Equilateral TriangularMeshesPhilippe Blanc, Robert Eymard, Raphaele HerbinUniversite Aix-Marseille, Marseille 1, France

Received 20 February 2004; accepted 4 December 2004Published online 8 April 2004 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/num.20020

We give here an error estimate for a finite volume discretization of the Stokes equations in two spacedimensions on equilateral triangular meshes. This work was initiated by an analogous result presented byAlami-Idrissi and Atounti for general triangular meshes. However, in this latter article, the result is notactually proven. We state here the restricting assumptions (namely equilateral triangles) under which theerror estimate holds, using the tools which were introduced by Eymard, Gallouet and Herbin and used byAlami-Idrissi and Atounti. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 907–918,2004

Keywords: finite volume scheme, Stokes equation

1. INTRODUCTION

Consider the flow of an incompressible fluid in a two-dimensional domain �, with velocity u �(u(1), u(2))t and pressure p. Recall that the Stokes equations may be written

��u�i ��x� ��p

�xi�x� � f�i ��x�, x � �, @i � 1, 2, (1.1)

div u�x� � �i�1

2�u�i �

�xi�x� � 0, x � �, (1.2)

with Dirichlet boundary condition

Correspondence to: Raphaele Herbin, Universite Aix-Marseille, Marseille 1, France (e-mail: [email protected])1991 Mathematics Subject Classification: 35K65, 35K55

© 2004 Wiley Periodicals, Inc.

u�i ��x� � 0, x � ��, @i � 1, 2, (1.3)

under the following assumption:

Assumption 1

(i ) � is an open bounded connected polygonal subset of �2,(ii ) � � 0,

(iii ) f (i ) � L2(�), @i � 1, 2.

In order to ensure uniqueness of the pressure, we also impose the following condition:

��

p�x�dx � 0. (1.4)

For the sake of completeness, let us recall the definition of an admissible finite elementtriangular mesh denoted by �, see e.g., Ciarlet [1], with the following notations, see also Fig. 1.

The set of edges of the mesh is denoted by �; for any triangle K of the mesh, one denotesby xK its circumcenter and by �K the set of the three edges of the triangle. For two neighboringtriangles K and L, one denotes by � � K�L their common edge. For any � � �K, one denotesby z� the orthogonal projection of xK on �.

The mesh size is defined as follows: size(� ) � sup{diam(K ), K � �}.For any K � � and � � �, m(K ) is the area of K and m(� ) the length of �.The set of interior (resp. boundary) edges is denoted by �int (resp. �ext), that is, �int � {� �

�; � � ��} (resp. �ext � {� � �; � � ��}).The set of neighbors of K is denoted by �K, that is, �K � {L � �; ?� � �K, �� K� � L�}.

FIG. 1. Notations for a control volume K.

908 BLANC, EYMARD, AND HERBIN

If � � K�L, we denote by d� or dK�L the Euclidean distance between xK and xL (which ispositive) and by dK,� the distance from xK to �.

If � � �K � �ext, let d� denote the Euclidean distance between xK and x� (then, d� � dK,�).For any � � �; the “transmissibility” through � is defined by �� m(� )/d� if d� � 0.The set of vertices of � is denoted by ��. For S � ��, let �S be the shape function

associated to S in the piecewise linear finite element method for the mesh �. For all K � �, let�K � �� be the set of the vertices of K.

The finite volume scheme given in Eymard, Gallouet and Herbin [2], which is cell-centeredfor the velocities and uses a Galerkin expansion for the pressure, reads

� ��K

FK,��i � � �

S��K

pS �K

��S

�xi�x�dx � m�K �fK

�i �, @K � �, @i � 1, 2, (1.5)

FK,��i � � ��uK

�i � � uL�i ��, if � � �int, � � K�L, i � 1, 2,

FK,��i � � ��uK

�i �, if � � �ext � �K, i � 1, 2, (1.6)

�K��

�i�1

2

uK�i � �

K

��S

�xi�x�dx � 0, @S � ��, (1.7)

��

�S��

pS�S�x�dx � 0, (1.8)

fK�i � �

1

m�K � �K

f �x�dx, @K � �. (1.9)

The discrete unknowns of (1.5)–(1.9) are uK(i ), K � �, i � 1, 2, and pS, S � ��.

In the next section, we give some results that were proven in [2] for this scheme, and explainwhy the results of Alami-Idrissi and Atounti [3] are not satisfying. We then give in Section 3an error estimate under restrictive assumptions.

2. PREVIOUS RESULTS

Existence and uniqueness of uK(i ), K � �, i � 1, 2 and pS, S � �� were proven in [2] and

therefore the approximate solution may be defined by

p� � �S��

pS�S, (2.10)

u��i ��x� � uK

�i �, a.e. x � K, @K � �, @i � 1, 2. (2.11)

FINITE VOLUMES FOR THE STOKES EQUATIONS 909

The proof of the convergence of the scheme is not straightforward in the general case. In [2],the convergence of the discrete velocities given by the finite volume scheme (1.5)–(1.9) isproven in the case of a mesh consisting of equilateral triangles.

In [3], the authors state a theorem (Theorem 4.1, p. 5) which gives an error estimate with norestriction on the admissible mesh nor on the regularity of the exact solution. The statement oftheir theorem seems to be at least incomplete (the property of consistency of the fluxes [3] (4.5)cannot be obtained without some regularity hypotheses on u), and the conclusion of theirtheorem remains, in our opinion, unproved. Let us focus on Relation [3] (4.10), which writes

�i�1

2 �K��

��K

�p

�xi�x�dx � �

S�SK

pS �K

�S

�xi�x�dx�eK

�i � � 0.

We do not believe that the brief argumentation of this equality, given in [3], which states

“Using div �u� � 0 and the relation (3.4), we deduce that:”

(relation [3] (3.4) is Relation (1.7) of this article) holds. First, the term

�i�1

2 �K��

��K

�p

�xi�x�dx�eK

�i �

is not in general equal to 0 and should be adequately controled by the interpolation error on thepressure.

Second, the use of Relation [3] (3.4) does not seem to be sufficient in order to yield

�i�1

2 �K��

� �S�SK

pS �K

�S

�xi�x�dx�eK

�i � � 0, (2.12)

with the definition eK(i ) � uK

(i ) � u(i )(xK) given in the statement of Theorem [3] 4.1 (which is alsothat of Ref. 2, 4, or 5, recall that the family of points (xK)K�� is such that if the control volumesK and L have a common edge �, then the line segment xKxL is orthogonal to �). Indeed, Relation[3] (3.4) is only sufficient to prove that

�i�1

2 �K��

� �S�SK

pS �K

�S

�xi�x�dx�uK

�i � � 0,

but not that

�i�1

2 �K��

� �S�SK

pS �K

�S

�xi�x�dx�u�i ��xK� � 0.


Note that in fact, (2.12) can be obtained under the following sufficient conditions:

(1) eK(i ) � uK

(i ) � 1/[m(K )] K u(i )(x)dx instead of eK(i ) � uK

(i ) � u(i )(xK),(2) the discretization mesh is made of triangles, so that the following equality is satisfied:

�K

��S

� xi� x�dx

1

m�K� �K

u�i�� x�dx � �K

��S

� xi� x�u�i�� x�dx. (2.13)

But, if eK(i ) is chosen to be as uK

(i ) � 1/[m(K )] K u(i )(x)dx rather than uK(i ) � u(i )(xK), then the

consistency of the fluxes condition [3] (4.5) (which remains a necessary step of the proof) is nolonger true in the general case: this is why the meshes were restricted to the equilateral case inthe convergence theorem of [2].

In the next section, we state the error estimate and give its proof according to the previousremarks.

3. ERROR ESTIMATE

Theorem 3.1. Under assumptions 1, we assume that � is such that there exists an admissibletriangular mesh on � such that all triangles are equilateral. Let � be such a mesh. Furthermorewe assume that there exists (u, p) � (H2(�))2 � H1(�) solution to (1.1)–(1.3), (1.4). Let (u, p)be such a solution. Define

uK�i � �

1

m�K � �K

u�i ��x�dx, for i � 1, 2, (3.14)

and

eK�i � � uK

�i � � uK�i �, for i � 1, 2, (3.15)

and let e be the piecewise constant function defined a.e. from � to �2 by e�K � eK. Then thereexists a constant C � � (independent of all the data and of the diameter h of the control volumesof the mesh) such that

�e�1,� Ch��u��H2��2 �1

�� p�H1�� ,

where

�e�1,� � � �i�1,2

�e�i ��1,�2 � 1/2

and �e�i ��1,� � � ��

��D�e�i ��2�1/2

, for i � 1, 2,

(3.16)

with �� m(�)/d�, D�e(i ) � �eK(i ) � eL

(i )� if � � �int, � � K/L, and D�e(i ) � �eK(i )� if � � �ext

� �K; we recall that �int (resp. �ext, �K) denotes the set of interior edges (resp. edges of theboundary ��, edges of the boundary �K ).


Proof. Let K � � be a given control volume. Let

FK,��i � � ��

m��

dK,L�uL

�i � � uK�i �� if � � K�L,

m��

dK,�uK

�i � if � � �ext., for i � 1, 2.

Integrating (1.1) over K for i � 1, 2, substracting off (1.5) and introducing FK,�(i ) yields

� ��K

EK,��i � � �

K

�p�

�xi�x�dx � � �

��K

m�� K,��i � �u� � �

K

�p

�xi�x�dx, (3.17)

with

EK,��i � � FK,�

�i � � FK,��i � ,

K,��i � �u� � �

1

dK,L�uK

�i � � uL�i ��

1

m��

�u�i ��x� � nK,�d��x� if � � K�L,

1

dK,�uK

�i � �1

m��

�u�i ��x� � nK,�d��x� if � � �ext.

Multiplying (3.17) by eK(i ) and summing over i and K leads to

��e�1,�2 � X � �A � B, (3.18)

with

X � � �i�1,2

�K��

eK�i � �

K

�p�

�xi�x�dx,

A � �i�1

2 �K��

��K

m�� K,��i � �u�eK

�i �,

and

B � ��i�1

2 �K��

�K

�p

�xi�x�dxeK

�i �.

Let us first show that


X � 0 (3.19)

holds (note that (3.19) is exactly Relation (2.12) presented in the previous section). Indeed, bydefinition of p�, one has

�K

�p�

�xi�x�dx � �

S��

�K,S�i � pS, with �K,S

�i � � �K

�S

�xi�x�dx.

Hence,

X � �S��

� �K��

�i�1,2

�K,S�i � uK

�i � � �K��

�i�1,2

�K,S�i � uK

�i ��pS.

Now by (1.7), ¥i�1,2 �K,S(i ) uK

(i ) � 0 for all S � ��. Furthermore, since �S /�xi is constant on eachtriangle K, (2.13) holds and therefore, one has

�K��

�i�1,2

�K,S�i � uK

�i � � �K��

�K

�S�x� � u�x�dx.

Thanks to this last equality and to (1.2)–(1.3), one then gets

�K��

�i�1,2

�K,S�i � uK

�i � � ��

div u�x�S�x�dx � ��

S�x�u�x� � n�x�d��x� � 0.

Hence (3.19) is proven. yLet us now study the term A. Reordering the summation over the edges of the mesh, one

obtains

A �i�1

2 ��

m�� i ��u�D�e�i �,

where �(i )(u) � �K,�

(i ) (u)� for � � K�L � �int or � � �K � �ext. Therefore, by the Cauchy-Schwarz inequality,

A � �i�1

2 ��

m�� d��i ��u��2� 1/2

�e�1,�.

Now by Lemma 3.2 given below, there exists a constant C0 � � such that


�i�1

2 ��

m�� d��i ��u��2 C0h

2 �i�1

2 � ��int,��K�L

�u�i ��H2�K�L�2 � �

��ext

�u�i ��H2�K �2 �,

that is,

�i�1

2 ��

m�� d��i ��u��2 3C0h

2� �i�1

2 �K��

�u�i ��H2�K �2 � 3C0h

2�u��H2��22 .

Hence there exists a constant C1 � � such that

A C1�u��H2��2h�e�1,�. (3.20)

Let us now turn to the term B. Let

p� � �S��

pSS (3.21)

be an interpolation of the exact pressure p defined by

pS �1

� S�x�dx ��

p�x�S�x�dx. (3.22)

Again, thanks to (1.7) [the computations are similar to those in the proof of (3.19)] one has

�i�1

2 �K��

�K

�p�

�xi�x�dxeK

�i � � 0,

and therefore

B � �i�1

2 �K��

�K

��p� � p�

�xi�x�dxeK

�i � � �i�1

2 �K��

eK�i � �

��K

��

�p��x� � p�x��cK,��i � d��x�,

where cK,�(i ) is the dot product between the normal unit vector to � outward to K and the ith vector

of the orthonormal basis of �2. Reordering the summation over the edges of the mesh, oneobtains

B �i�1

2 � ��

� ��p��D�e�i �� ,

where


��p� � ��

�p��x� � p�x��d��x�.

By the Cauchy-Schwarz inequality, one obtains that

B2 2�e�1,�2 �

��

d�

m�� p��2.

Thanks to Lemma 3.3 given below, there exists a constant C2 � � such that

��

d�

m�� p��2 C2h

2 ��

� p�H1�V��2 , (3.23)

where V� denotes the set of triangles K � � to which at least one of the vertices of � is a vertex.Noting that a given control volume K � � is a subset of at most 15 different V� sets, there

exists a constant C3 � � such that

B C3h�e�1,�� p�H1��. (3.24)

From (3.18), (3.19), (3.20), and (3.24), one concludes the proof of the theorem. yWe now give the proof of the consistency results which are used in the above proof. In

Lemma 3.2, we first control the error on the discretization of the flux when considering the meanvalue of the exact solution, which was used in order to obtain (3.19). Note that in [2], theconsistency was proven with respect to the value of the solution at the points xK and not withrespect to the mean value. It is because of this mean value that we have to restrict to equilateraltriangles here.

Lemma 3.2 (Consistency of the flux). Under the hypotheses of Theorem 3.1, let K � �, � ��K, there exists a constant C such that the following inequality holds:

� 1

dK,� � 1

m�K � �K

w �x�dx �1

m��

w �x�d��x�� 1

m��

�w �x� � nK,�d��x�� C�w�H2�K �,

(3.25)

where

�w�H2�K � � � �i,j�1

2 �K

� �2w

�xi�xj�x�� 2

dx� 1/2

.

Proof. Let � (H2(K )) [dual space of H2(K )] be defined for any w � H2(K ) by


�w� �h

dK,� � 1

m�K � �K

w �x�dx �1

m��

w �x�d��x�� h

m��

�w �x� � nK,�d��x�.

Let K be a given equilateral triangle with diameter 1, let � be the one to one mapping fromK to K, and let � (H2(K)) be defined for any w � H2(K ) by

�w� �h

dK,� � 1

m�K � �K

w �� x��dx �1

m��

w �� x��d��x��

h

m��

��w � ��x� � nK,�d��x�,

Performing the change of variable x � ��1( x), and using the fact that �(w � �)(��1( x)) �(1/h)�w( x), we obtain

�w� �h

dK,� � h2

m�K � �K

w �x�dx �h

m��

w �x�d��x�� h

m��

�w �x� � nK,�d��x�,

and therefore,

�w� �1

dK,� � 1

mK�

K

w �x�dx �1

m��

�

w �x�d��x�� 1

m��

�

�w �x� � nK,�d��x�.

Since � (H2(K)) and vanishes on the set of polynomials of degree 1 on K, by theBramble-Hilbert lemma, there exists C � � independent of h such that

��w�� C�w�H2�K�.

Hence we get, with w � w � ��1,

��w�� Ch�w�H2�K�, @w � H2�K �.

This yields (3.25) and concludes the proof. yNext, we give in Lemma 3.3 the consistency result needed to establish (3.23).

Lemma 3.3. Under the assumptions of Theorem 3.1, let p� be the interpolation of p definedby (3.21), (3.22), then there exists a constant C � � such that, for any � � �,

��

�p�x� � p��x��d��x� Ch�p�H1�V��,


where V� denotes the set of triangles K � � to which at least one of the vertices of � is a vertex,and �p�H 1(V�) � ��p�(L2(V�))2.

Proof. Let � � �. We first remark V� is chosen such that the restriction of p� to � onlydepends on the values of p on V�. Let us then remark that there exists a finite number of possiblegeometries for V�, which we number from 1 to P.

For each k � 1, . . . , P, let V�(k) denote a given such set corresponding to h � 1.

Let � be the one-to-one mapping from V� to V�(k), and let � (H1(V�

(k))) be defined by

�p� �1

h ��

�p�� x�� p�� x��d��x�, Vp � H1�V��k��.

where p denotes the piecewise linear interpolation, defined as in (3.21), (3.22), of p on V�(k).

Performing the change of variable x � ��1( x), we obtain

�p� � ��

�p�x� � p�x��d��x�.

The linear form vanishes for constant functions and therefore by the Bramble-Hilbert lemma,there exists c(k) depending only on the geometry of V�

(k) such that � p� c�k��p�H1�V��k��. Then,

noting that if p � p � ��1 one has � ( p( x) � p( x))d�( x) � (1/h) � (p(x) � p�(x))d�(x), oneconcludes the proof of the lemma with C � maxk�1,. . .,Pc(k). y

4. CONCLUSION

We underlined here some restrictive assumptions under which an error estimate for the discretevelocities may be obtained by the finite volume scheme that was introduced in [2]. Indeed, if themesh is made of equilateral triangles, then an error estimate holds. It is easily seen that the errorestimate still holds for meshes which are “close to” equilateral, that is, if the distance betweenthe circumcenter and the center of gravity is of order two with respect to the size of the mesh:in this case, the consistency error will remain of order one if one takes xK to be the center ofgravity rather than the circumcenter, and the rest of the above proof remains unchanged. Wemay note, however, that both equilateral or “close to” equilateral meshes are rather hard to usein practice. To our knowledge, it is still an open question to know if the result stated in [3] (i.e.,without the assumption of equilateral triangles) holds. In fact, we recently obtained somenumerical results on general meshes which seem to indicate poor convergence of the presentscheme. One might conclude that the finite volume based on piecewise constant velocities andpiecewise linear pressures is not a good idea; we wish to emphasize, however, that on the onehand, the piecewise constant velocities are equipped with a discrete H1 norm. On the other hand,the piecewise linear approximation of the pressure must be viewed as a tool to discretize thedivergence of the velocity at the vertices in an adequate way. In fact, in [6], we introduce amodified scheme, for which we may obtain a property of consistency of the discrete divergence,along with a “weak” estimate on the discrete pressure, and for which we are thus able to provean error estimate on general triangular meshes (see also [7]). We further note that this modifiedscheme is equivalent to the scheme studied here in the case of equilateral triangles.


References

1. P. G. Ciarlet, The finite element method for elliptic problems North-Holland, Amsterdam, 1978.

2. R. Eymard, T. Gallouet, and R. Herbin, Finite volume methods. Handbook for numerical analysis, Ph.Ciarlet and J. L. Lions, editors, North Holland, Amsterdam, 2000, Vol. VII, pp 715–1022.

3. A. Alami-Idrissi and M. Atounti, An error estimate for finite volume methods for the Stokes equations,J Inequalities Pure Appl Math 3(1) (2002), Article 17.

4. R. Herbin, An error estimate for a finite volume scheme for a diffusion-convection problem on atriangular mesh, Num Meth PDE 11 (1995), 165–173.

5. T. Gallouet, R. Herbin, and M. H. Vignal, Error estimate for the approximate finite volume solutionsof convection diffusion equations with general boundary conditions, SIAM J Numer Anal 37(6) (2000),1935–1972.

6. R. Eymard and R. Herbin, A cell-centered finite volume scheme on general meshes for the Stokesequations in two dimensions, CRAS, Mathematiques 337(2) (2003), 125–128.

7. R. Eymard and R. Herbin, A finite volume scheme on general meshes for the steady Navier-Stokesproblem in two space dimensions, LATP preprint, submitted (2003). http://www.cmi.univ-mrs.fr/�herbin/


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An error estimate for finite volume methods for the Stokes equations on equilateral triangular meshes