Mixed finite volume methods

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING

Int. J. Numer. Meth. Engng. 46, 1351–1366 (1999)

MIXED FINITE VOLUME METHODS

J.-M. THOMAS ∗ AND D. TRUJILLO

Laboratoire de Math�ematiques Appliqu�ees; CNRS UPRES A 5033; Universit�e de Pau et des Pays de l’Adour; France

SUMMARY

We present in this paper a new Finite Volume Methods for elliptic equation, based on a mixed primal-dualformulation. In this approach the uxes are introduced as unknowns of the problem and we use two dualmeshes. This method is called ‘mixed �nite volume method (MFV)’. We recall �rst the abstract theory ofgeneralized mixed formulation and then we develop error estimates in the case where two dual rectangularmeshes or two dual triangular meshes are used. Finally, we present some numerical results and we calculatefor each example the L2-error relative to the primal and dual unknowns. Copyright ? 1999 John Wiley &Sons, Ltd.

KEY WORDS: second-order elliptic problem; �nite volume; vertex centered scheme; mixed �nite element; generalizedinf–sup conditions; a priori error estimates

1. INTRODUCTION

Finite Volume Element Methods were developed as an attempt to use �nite element ideas to createa more systematic Finite Volume methodology. First convergence results have been obtained fordi�usion equations on triangular meshes with linear �nite element spaces (see [1; 2]). However, itseems di�cult to generalize these results to general self-adjoint elliptic boundary value problemsand general triangulation. In order to avoid these di�culties, we propose in this paper a newmethod based on a mixed primal–dual formulation of the problem. Here we make the choice todescribe it in the context of Neumann boundary condition

−div(A gradp) =f in

(A gradp) ·n =0 on @(1)

The basic idea is to introduce simultaneously p and A gradp as unknowns of the problem and touse dual meshes for the unknown p and for the unknown uxes. This method is called a MixedFinite Volume Method (MFV). In [3; 4], we established the convergence of decomposition methodscoupled with the MFV method where the mesh associated with the unknown p was triangularand the mesh associated with the uxes rectangular. But this approach is not generalizable toquadrangular meshes. We develop hereafter a method presented in [5] on triangular unstructuredmeshes. Here, we study error estimates whenever two dual rectangular–triangular meshes are used.This method is a new approach to �nite volumes methods which gives, in simple cases and via a

∗ Correspondence to: J.-M. Thomas, Laboratoire de Math�ematiques Appliqu�ees, CNRS UPRES A 5033, Universit�e de Pauet des Pays de l’Adour, Av. de l’Universit�e, 64000 Pau, France. E-mail: [email protected]

CCC 0029-5981/99/331351–16$17.50 Received 1 September 1998Copyright ? 1999 John Wiley & Sons, Ltd. Revised 1 February 1999

1352 J.-M. THOMAS AND D. TRUJILLO

numerical quadrature formula, an already known scheme in the triangular case and a new nine–point scheme in the rectangular case. Theoretical generalization of this method to 3-D situationis straightforward. A distinct approach of Finite Volume Methods on unstructured meshes can befound in [6]. For an analysis of some Finite Volume Method in the spirit of Finite Di�erenceMethods, we refer to [7].

2. CONTINUOUS PROBLEM

Let us consider the following boundary value problem:

div u=f in

u= −A gradp in

(A gradp) ·n=0 on @

(2)

where is a connected open bounded domain of R2; n the external unit normal vector along @; fa function of L2() and A a 2×2 matrix. We assume that f satis�es the compatibility condition∫ fdx=0. We suppose that A is symmetrical and veri�es the uniform ellipticity condition

∃�¿0; ^TA(x)^¿�^T^ for all x∈ �; ^∈R2

We introduce the classical spaces

H 1() ={p∈L2(); gradp∈ (L2())2}

H 2() ={p∈H 1(); gradp∈ (H 1())2}

and

H(div; )={u∈ (L2())2; div u∈L2()}

equipped with the Hilbertian graph norm

‖p‖H 1() =(∫

{|gradp|2 + p2} dx)1=2and

‖u‖H(div;) =(∫

{|u|2 + (div u)2

}dx)1=2

2.1. A new mixed formulation

Let us consider the spaces

U= {u∈H(div;); u · n=0 on @}and

P=H 1()=R

A function p of the space P is determined nearby a constant; we can �x the constant by thechoice of the condition of zero mean value of p:

∫ p dx=0:

Copyright ? 1999 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 46, 1351–1366 (1999)

MIXED FINITE VOLUME METHODS 1353

We introduce here a new mixed formulation ‘�a la Petrov–Galerkin’ in order to obtain afterdiscretization an approximation uh of u in U and an approximation ph of p in P.The primal–dual formulation of Problem (1) is given by

(u; p)∈U×P∀v∈ (L2())2;

∫u · v dx +

∫A gradp ·v dx=0

∀q∈L2();∫q div u dx=

∫fq dx

(3)

Remark. The existence and the uniqueness of the solution of (3) are trivial, since (u; p) issolution of (3) if and only if p is the standard variational solution of (2) and then u= −A gradp.Here, we are interested in the discretization of (3). To this end, let us introduce a more generalframework.

3. ABSTRACT THEORY OF GENERALIZED MIXED FORMULATION

Let us consider the abstract problem

(u; p)∈U×P∀v∈V; m(u; v) + a(p; v)= 0

∀q∈Q; b(u; q)= 〈f; q〉where (U; ‖ · ‖U); (P; ‖ · ‖P); (V; ‖ · ‖V) and (Q; ‖ · ‖Q) are four Hilbert spaces, m; a and b are threebilinear forms de�ned, respectively, on U×V; P×V and U×Q and f is an element of Q′, the dualspace of Q.Let us de�ne Uh;Vh; Ph and Qh four �nite-dimensional subspaces of U; V; P and Q, respectively.

Let mh and ah be two bilinear forms de�ned on Uh×Vh and Ph×Vh. In the applications, mh andah will be obtained from m and a by numerical quadratures. Let us now consider the discreteproblem

(uh; ph)∈Uh×Ph∀vh ∈Vh; mh(uh; vh) + ah(ph; vh)= 0

∀qh ∈Qh; b(uh; qh)= 〈f; qh〉Q′ ; Q

(4)

Let U0h and V1h be the spaces de�ned by

U0h= {uh ∈Uh;∀qh ∈Qh; b(uh; qh)= 0}V1h= {vh ∈Vh;∀uh ∈U0h; mh(uh; vh)= 0}

We assume that there exists three constants M;A and B independent of h such that

∀u∈U; ∀v∈V; m(u; v)6M‖u‖U‖v‖V∀uh ∈Uh; ∀vh ∈Vh; mh(uh; vh)6M‖uh‖U‖vh‖V∀p∈P; ∀v∈V; a(p; v)6A‖p‖P‖v‖V



∀ph ∈Ph; ∀vh ∈Vh; ah(ph; vh)6A‖ph‖P‖vh‖V∀u∈U; ∀q∈Q; b(u; q)6B‖u‖U‖q‖Q

For the sake of brevity, we should not give in our examples the veri�cation of these naturalconditions of continuity.Our mixed �nite volume methods is built on the following abstract result:

Theorem 1. Let us assume that the next three Babu�ska–Brezzi conditions are satis�ed:

(i) infuh ∈U0h

supvh ∈Vh

mh(uh; vh)‖uh‖U‖vh‖V¿�¿0

(ii) infph ∈ Ph

supvh ∈V1h

ah(ph; vh)‖ph‖P‖vh‖V¿�¿0

(iii) infqh∈Qh

supuh ∈Uh

b(uh; qh)‖uh‖U‖qh‖Q¿�¿0

and that

dim(Uh) + dim(Ph)= dim(Vh) + dim(Qh)

Then Problem (4) has a unique solution (uh; ph):Moreover; if �; � and � are independent of h; then there exists a constant C independent of

h such that

‖p− ph‖P + ‖u − uh‖U6C{infrh ∈ Ph

(‖p− rh‖P + sup

vh ∈Vh

|a(rh; vh)− ah(rh; vh)|‖vh‖V

)+ inf

wh ∈Uh

(‖u − wh‖U + sup

vh ∈Vh

|m(wh; vh)− mh(wh; vh)|‖vh‖V

)}The detailed proof was given in [8]. With a similar assumption an analogous result was obtained

by Nicola��des [9], see also [10].In the next section, we shall use this result to analyse discretizations of Problem (3). For that

we have to build �nite-dimensional spaces which verify all the conditions of Theorem 1.

4. H(DIV)-INTERPOLATION RESULTS

In the following, we will consider two triangulations on : a triangulation Th for the approximatedscalar unknown ph; a triangulation T #h for the approximated vectorial unknown uh. The triangulationT #h will be in the next examples a subtriangulation of Th: We will note Kl; l=1; : : : ; L; the cells ofTh and K#m; m=1; : : : ; M; the cells of T

#h . In this presentation, we assume that the two triangulations

are formed only by quadrangles or only by triangles.In the case of triangulations with quadrangles, for any Kl ∈Th and any K#m ∈T #h ; there exist

invertible bilinear (meaning a�ne with respect to each variable) mappings Fl and F#m such that

Fl(K)=Kl; F#m(K)=K#m

where K is the reference unit square [0; 1]×[0; 1]. In the particular case where the quadrangleis a rectangle or more generally a parallelogram, the bilinear mapping is in fact a�ne. In the



case of triangulations with triangles, for any Kl ∈Th and any K#m ∈T #h ; there exist invertible a�nemappings Fl and F#m such that

Fl(K)=Kl; F#m(K)=K#m

where K is now the reference unit triangle with points (0; 0); (0; 1) and (1; 0) as vertices.We note DFl and DF#m the Jacobian matrices of Fl and F

#m, respectively, Jl and J

#m the deter-

minant of DFl and DF#m, respectively.We suppose that the families of triangulations (Th)h and (T #h )h are regular as usual in the analysis

of conforming �nite element methods, see for example [11]. So we have

‖DFl(x)‖6ChKl ; ‖DF#m(x)‖6ChKm‖DF−1

l (x)‖6 ChKl; ‖(DF#m)−1(x)‖6

ChKm

and

C1h2Kl6|JKl |6C2h2Kl ; C1h2K#m6|JK#m |6C2h2K#mwhere C; C1 and C2 are three constants independent of h.Hereafter we recall some properties of the H(div)-interpolation operator which will be useful

in the sequel. For more details, see [12; 13]. Let w be an element of H(div;): For any K#m, weconsider the function wm of H(div; K) such that

w|K#m ◦F#m=1

|J #m|DF#m wm

Let E(wm)∈H(div; K) be the interpolate function de�ned byE(wm)∈P(K)∫�iE(wm) · ni d�=

∫�iwm · ni d�; 16i6IK

with P(K)=P1; 0(K)×P0;1(K) and IK =4 in the case where K is the reference square whileP(K)=RT0(K) and IK =3 in the case where K is the reference triangle. Here P1;0 is de�ned asthe space of polynomial functions linear in the �rst variable, constant in the second variable, P0;1 isde�ned in a similar way; P0(K) is the space of functions constant on K and RT0(K)= {(P0(K))2⊕xP0(K)} is the Raviart–Thomas space of lowest order [12]. We note �i 16i6IK ; the edges of Kand ni the external unit normal vector along �i : Since E(wm) is constant along �i ; we obtain

E(wm) · ni= 1

meas(�i)

∫�iwm · ni d�; 16i6IK

Now, we de�ne the function Eh(w)∈H(div; ) such that

Eh(w)|K#m ◦F#m=1

|J #m|DF#m E(wm)

The function Eh(w) veri�es ∫�#m; i

Eh(w) ·nm; i d�=∫�#m; i

w · nm; i d�



where �#m; i; 16i6IK , are the edges of K#m and nm; i the external unit normal vector along

�#m; i: With these de�nitions of wm;E(wm) and Eh(w), we have

|wm|1; K6Ch|w|1; Km ; ‖w− E(w)‖0; K6C‖w− Eh(w)‖0; K#mFurthermore, we remark that Eh(w) is constant along �#m; i; 16i6IK ; thus we obtain

Eh(w) · nm; i= 1meas(�m; i)

∫�m; iw · nm; i d�; 16i6IK

Remark. In the case of a rectangle K#m with edges parallel to the co-ordinates axis one hasEh(w)∈P1;0(K#m)×P0;1(K#m) and the four uxes through the edges of K#m: Eh(w)·nm; i for 16i64;are the degrees of freedom of Eh(w): In the case of a triangle K#m, one has Eh(w) ∈RT0(K#m) andthe three uxes through the edges of the triangle K#m are the degrees of freedom of Eh(w).

We choose the numeration of the edges of K#m in order to have �#m; i=F

#m(�i): So we get

meas(�#m; i)Eh(w) · nm; i=meas(�i) E( wm) · niMoreover, it is easy to see that

div(E( wm))∈P0(K); div(Eh( w)|K#m)∈P0(K#m)and

div(E(wm)) = div(Eh( w)|K#m)

=1

meas(K#m)

∫K#m

divw dx

=1

meas(K)

∫Kdiv wmdx

Finally, when the triangulation T #h is built with parallelograms or triangles, we have the follow-ing interpolation error bound: If w∈ (H 1())2 with div w∈H 1() then there exists a constantindependent of h such that

‖w− Eh(w)‖H(div;)6Ch(|w|1; + |divw|1; )

5. DEVELOPMENT OF THE METHOD FOR RECTANGULAROR TRIANGULAR MESHES

Let us begin this section by de�ning the di�erent meshes. Let (Th)h be a regular family oftriangulations formed by triangles or rectangles with edges which are parallel to the axis. Sucha triangulation will be associated to the approximation of the space P: We note Kl (l=1; : : : ; L)the elements of Th: For any rectangle Kl ∈Th; let Gl be the centre of gravity (i.e. the intersectionof medians) of Kl: By connecting for l=1; : : : ; L; the centre Gl by straight-line segments to theedge midpoints of Kl; we build a subtriangulation T #h of �. The element of T #h will be denotedK#m; 16m6M . The family (T

#h )h is also a regular family of triangulations of �.



Let I be the number of vertices of the primal triangulation Th: To any vertex Si; 16i6I; of thetriangulation Th; we associate a control volume, the box Ki; de�ned as the union of the elementsK#m which admit Si as vertex. We obtain in this manner a new triangulation T ∗

h which is calledthe dual one of Th; of course, the cells Ki of the triangulation T ∗

h are polygonal, not necessarilyconvex. This dual triangulation has I cells.Let J # be the number of edges of the subtriangulation T #h which are not situated on the boundary

@: Let �#j ; 16 j6 J #; be such an edge: then �#j is the common edge of two elements K#m and

K#n of T#h ; we will denote K

#j the set K

#j =K

#m ∪K#n . The unit vector n#m;n normal to K#m ∩K#n =�#j

is directed from K#m towards K#n ; so one has n

#m;n= − n#n;m; let n#j be an arbitrary choice between

n#m;n and n#n;m: For any j; 16 j6 J #; we will introduce the piecewise constant vectorial function

w#j de�ned by

w#j =n#j on K#j0 on �− K#j

Finally, for the triangulation Th (resp. T #h ), we note Th; t (resp. T#h; t ) the set of elements of the

triangulation which are triangular and Th; r (resp.T #h; r) the set of those which are rectangular.

5.1. Discretization spaces

We can now introduce the discretization spaces. First, we choose the space associated with thescalar unknown ph:

Ph= {ph ∈P;∀Kl ∈Th; l; ph|Kl ∈P1(Kl);∀Kl ∈Th; rph|Kl ∈Q1(Kl)}

where P1(Kl) is the space of linear functions on Kl and Q1(Kl)is the space of functions whichare linear in each variable.Then we take as the space of the vectorial unknown uh

Uh= {uh ∈U;∀K#m ∈T #h; tuh|K#m ∈RT0(K#m);∀K#m ∈T #h; ruh|K#m ∈P1;0(K#m)×P0;1(K#m)}

We now introduce the spaces associated with the test functions. First we choose for the vectorialtest functions the following space:

Vh=Span{w#j ; 16 j6 J #

}and �nally we take for the scalar test functions the space

Qh={qh ∈L2() with

∫qh dx=0;∀Ki ∈K∗

h ; qh|Ki ∈P0(Ki)}

where P0(Ki) is the space of constant functions on Ki.It is clear that with these choices of �nite-dimensional spaces we have

dim(Ph)= dim(Qh)

dim(Uh)= dim(Vh)= J #



5.2. Approximated bilinear forms

Let us begin by de�ning the bilinear form m(: ; :) on H (div; )×(L2())2 by

m(u; v)=∫u · v dx

In order to obtain a Finite Volume Method, we de�ne the bilinear form mh(: ; :) on Uh×Vh by∀uh ∈Uh; ∀16 j6 J #; mh(uh;w#j )=meas(K

#j )(uh ·n#j )|�#j

One recalls that for uh in Uh; uh · n#j is constant on �#j . Since the quantity m(uh;w#j ) is an integralover the domain K#j ; mh(uh;w

#j ) can be considered as an approximation of m(uh;w

#j ) by a one

point numerical quadrature rule over K#j .The bilinear form a(: ; :) on H 1()×(L2())2 is next de�ned by

a(p; v)=∫A gradp ·v dx

and this last expression is replaced in the discrete problem by

∀ph ∈Ph; ∀vh ∈Vh; ah(ph; vh)=∫Ah gradph · vhdx;

where Ah is the matrix constant on each triangle Tl ∈Th such that

Ah|Tl =Ah(Gl)

Finally, we introduce the bilinear form b(: ; :) de�ned on H (div; )×L2() by

b(u; q)=∫q div u dx

which translates the conservation law. In order to obtain a conservative scheme, it is essential touse no numerical integration on these quantities; therefore we have

∀uh ∈Uh; ∀qh ∈Qh; b(uh; qh)=∫qh div uh dx

5.3. Discrete problem

With all these notations; the discrete problem can now be written as follows:

(uh; ph)∈Uh×Ph∀vh ∈Vh; mh(uh; vh) + ah(ph; vh)= 0∀qh ∈Qh; b(uh; qh)= 〈f; qh〉Q′ ;Q

(5)

Before any other consideration, let us notice that the last row of (5) translates into

∀Ki ∈T ∗h ;∫Kidiv uh dx=

∫Kifdx



or equivalently,

∀Ki ∈T ∗h ;∫@Kiuh ·n@Ki d�=

∫Kifdx

where n@Ki is the unit outward normal along @Ki.Next, during the choice of the test functions vh and of the approximated form mh(: ; :); one can

remark that the second row of (5) allows an explicit elimination of the degrees of freedom ofuh. This leads to a scheme using only the degrees of freedom of ph; i.e. the values of ph at thevertices of the triangulations.

6. A PRIORI ERROR ESTIMATES

We will use in this section the abstract result given by Theorem 2. For that, we introduce theHilbert space norms

‖u‖U= ‖u‖H (div;) ={‖u‖20; + ‖ div u‖20;

}1=2and

‖p‖P = ‖p‖H 1() ={‖p‖20; + ‖ gradp‖20;

}1=2The spaces Q and V are equipped with the norms ‖ · ‖0; of L2() and (L2())2, respectively.

6.1. Inf–sup properties

Lemma 2. There exists a constant �¿0 independent of h such that

infqh∈Qh

supuh ∈Uh

b(qh; uh)‖qh‖0;‖uh‖H(div;)¿�¿0

Proof. Let us consider the �nite-dimensional space

Qh={qh ∈L2();∀K#m ∈T #h qh|K#m ∈P0(K#m)

}We know (cf. [13]) that there exists a positive constant � such that

infqh∈Qh

supuh∈Uh

b(qh; uh)‖qh‖0; ‖uh‖H(div;)¿�¿0

Then since Qh is a subspace of Qh; Lemma 2 is evident.

Lemma 3. There exists � independent of h such that

infph ∈ Ph

supvh∈V1h

ah(ph; vh)‖vh‖0;|ph|1;¿�¿0



Proof. Let us consider a function ph ∈Ph: To this function ph we associate the vectorial function�vh ∈Vh de�ned by �vh=

∑16j6J # �j w

#j with

�j =meas(�#j )

meas(K#j ){ph(Si1 )− ph(Si2 )} if �#j = @Ki1 ∩ @Ki2 and w#j = n@Ki1

�j =meas(�#j )

meas(K#j ){ph(Si2 )− ph(Si1 )} if �#j = @Ki1 ∩ @Ki2 and w#j = n@Ki2

�j =0 if �#j is not included in⋃

16i6I@Ki

Let us suppose �rst that Kl is a triangle. For this choice of �vh, we have

ah(ph; �vh) = − ph(Si)∑

16i6I

{∫@Ki

(Ah gradph) n@Ki d�}

Let Xl= −∑16i6I {ph(Si)∫@Ki ∩ Tl (Ah gradph) · n@Ki d�} be the contribution due to the element

Kl ∈Th:Let us assume for a moment that Kl is equilateral of diameter h. Then an elementary computation

gives us

− ∑16i6I

{∫@Ki∩Kl

ph(Si) n@Ki d�}=32meas(Kl) (gradph)|Kl

and so it comes in this particular case

Xl = 32 meas(Kl) (Ah gradph)|Kl · (gradph)|Kl

¿ a ‖ gradph‖20; KlWe treat now the general case. Firstly we transform by an a�ne mapping the triangle Kl into anequilateral triangle K ; if DFl is the Jacobian matrix of the transformation Fl from K onto Kl and(DFl)−T the transposed inverse matrix, then we have

Xl=−(Ah grad ph)|K ·∑

16i6I

{∫[@Ki∩Kl

ph(S i) n [@Ki∩Kl d�}

with

Ah= |det(DFl)| (DFl)−1Ah(DFl)−T

Then we deduce

Xl¿al ‖ grad ph‖20; Kand coming back to the triangle Tl, it results that

Xl¿a ‖ gradph‖20; Klfor some constant a¿0.If we consider now the rectangular case, we can show in this case that∫

Kl

Ah gradph · �vh dx¿�‖ �vh‖20; Kl¿�‖ �vh‖0; Kl‖ gradph‖0; Kl

where � has been de�ned in Section 2.



However, to prove the lemma we still have to verify that �vh ∈V1h. We remark that for anyuh ∈U0h we have

mh(uh; �vh)=−I∑i=1

∫Kiph(Si) div(uh) dx

and using the fact that uh belongs to U0h, we obtain the result.

Lemma 4. There exists � independent of h such that

infuh∈U0h

supvh∈Vh

mh(uh; vh)‖uh‖H(div;)‖vh‖0;¿�¿0

Proof. Let uh be an element of Uh. We can take vh ∈Vh such that �vh=∑

16j6J # �j w#j with

vj =(uh ·n#j )|�#jWe notice that

mh(uh; vh)=J #∑j=1

measKj�2j = ‖vh‖20;

Moreover, proceeding element by element we obtain by a direct calculation

‖uh‖20;6CJ #∑j=1

measKj�2j6C‖vh‖20;

where C is a constant independent of h.

6.2. Approximation properties

The proof of the following lemma is without any particular di�culty:

Lemma 5. Assume that p∈H 2() and that the coe�cients of A belong in C1( �): Thenthere exists a constant C independent of h such that

infrh∈Ph

(‖p− rh‖1; + sup

vh∈Vh

|a(rh;; vh)− ah(rh; vh)|‖vh‖0;

)6Ch

We equally have:

Lemma 6. Assume that u∈ (H 1())2 and that div u∈H 1(). There exists a constant Cindependent of h such that

infwh∈Uh

(‖u − wh‖H(div;) + sup

vh∈Vh

|m(wh; vh)− mh(wh; vh)|‖vh‖0;

)6Ch

Proof. Let K#m and K#n be the two neighbouring elements of T

#h . Let � be the edge of K such

that

F#m(�)=�#m;n



where K is the basis element introduced in Section 4. It is easy to verify that

n#m;n=meas(K#m)

meas(K)

meas( �)meas(�#m;n)

(DF#−1m )t n

where n is the external unit normal along �.Moreover, we recall that

Eh(u) ·n#m;n|�#m; n =meas(�)meas(�#m;n)

(E( u) · n)| �

So, we can deduce that∣∣∣∣∣∫K#m

Eh(u) · n#m;n dx − meas(K#m)Eh(u) ·n#m;n|�#m; n

∣∣∣∣∣=

∣∣∣∣∣meas(K#m)meas(K)

meas(�)meas(�#m;n)

∫K#m

1|J #m|

DF#m E( u) · (DF#−1m )t n∣∣J #m∣∣ d x

−meas(K#m)meas( �)meas(�m)

E( u) · n|�∣∣∣∣∣

=meas(K#m)

meas(K)

meas( �)meas(�#m;n)

∣∣∣∣∫KE( u) · n d x −meas(K) (E( u) · n)| �

∣∣∣∣where u satis�es

u ◦F#m=1

|J #m|DF#m u

Thus, comes the next inequality∣∣∣∣∣∫K#m

Eh(u) ·n#m;n dx −meas(K#m)(Eh(u) ·n#m;n)|�#m; n∣∣∣∣∣

6Cmeas (K#m)meas (�#m;n)

∣∣∣∣∫K(E( u)− u) · n d x

∣∣∣∣+ ∣∣∣∣∫K( u · n − �) d x

∣∣∣∣where � is the constant de�ned by �=

∫� u · n dx=(E( u) · n)| �. We have∣∣∣∣∫

K( u · n − �) dx

∣∣∣∣6C‖ u · n − �‖0; K

6C(| u · n − �|1; K +

∣∣∣∣∫�( u · n − �) d�

∣∣∣∣)Since

∫�( u · n − �) d�=0 and | u · n − �|1; K = | u · n|1; K , we obtain∣∣∣∣∣

∫K#m

Eh(u) · n#m;n dx −meas(K#m)(Eh(u) ·n#m;n)|�#m; n∣∣∣∣∣

6C(meas(K#m))1=2(‖E( u)− u‖0; K + | u|1; K)

6Ch(meas(K#m))1=2|u|1; K#m



Then, for all vh=∑

K#m; n|vm; n|n#m;n ∈Vh, we have

|m(Eh(u); vh)− mh(Eh(u); vh)|

6

(∑K#m; n

|vm; n|∣∣∣∣∣∫K#m; n

Eh(u) ·n#m;n dx −meas(K#m;n)Eh(u) · n#m;n|�m; n∣∣∣∣∣)

6Ch

(∑K#m; n

(meas(K#m;n))1=2|vm; n||u|1; K#m

)6Ch ‖vh‖0;|u|1;

Moreover, the H(div)-interpolation results give us

‖u − Eh(u)‖H (div;)6Ch

6.3. Error bounds

As a conclusion, we now give the result which implies the convergence of the MFV.

Theorem 7. Assume the coe�cients of the matrix A continuously di�erentiable on �. If thesolution (u; p) of Problem (3) is su�ciently smooth and if (uh; ph) is the solution of Problem(5); then we have

‖uh − u‖H(div;) + ‖p− ph‖1;6Ch(|u|1; + |div u|1; + |p|2;)

7. GENERALIZATION TO QUADRANGULAR MESHES

In this section we brie y describe the generalization of the above method to quadrangular trian-gulations. For more details, see [4].Let us suppose that Th is a quadrangular mesh. In a similar way as in the preceding sections

we construct the cells K#m for 16m6M; K#j for 16 j6 J and the �nite volumes Ki 16i6I .

By considering the median of K , we obtain four squares K1; K2; K3 and K4. In order to de�nethe discrete space Uh, let us introduce the space

R(K)={u∈H(div; K); u|K i ∈P1;0(K i)× P0;1(K i); i=1; 2; 3; 4

}We classically de�ne the space for the pressure by

Ph={ph ∈H 10 (); ∀Kl ∈Th ∃p∈Q1(K); ph|Kl ◦Fl= p

}In order to conserve the continuity of the uxes, we want the normal component of the velocity

function to be constant on each edge. Then we can associate degrees of freedom with the uxeson edges, as in the rectangular case. So, for the de�nition of the space Uh; the Piola’s transformis appropriate (see [13]) and we put

Uh={uh ∈H(div;); ∀Kl ∈Th ∃ u∈R(K); uh ◦Fl= 1

|Jl|DFl u}



In a usual way, we associate to pressure the next space

Ph={ph ∈H 10 (); ∀Kl ∈Th ∃p∈Q1(K); ph|Kl ◦Fl= p

}Moreover, we de�ne on the vectorial functions d1 and d2 by

∀Kl ∈Th; d1|Kl =(DF−1l )t

(10

)and

∀Kl ∈Th; d2|Kl =(DF−1l )t

(01

)So, if K#m and K

#n are the two neighbouring cells of T

#h then we can de�ne the function n

#m;n on

K#m;n by n#m;n=d1|K#m; n if F−1

l (�m;n) is vertical and n#m;n=d2|K#m; n otherwise.Let Vh be the space spanned by these functions.

Finally we de�ne

Qh={qh ∈L2(); ∀Ki ∈T ∗

h ; qh|Ki ∈P0(Ki)}

We can do the same remarks as in the preceding section. So we have

dim Uh= dim Vh

and

dim Ph= dim Qh

All the results concerning the convergence of the method, which were obtained in the precedingsections, still hold in this case when using the �nite-dimensional spaces de�ned above.

8. NUMERICAL RESULTS

In this section, we present some numerical examples which illustrate the theoretical results obtainedin the preceding sections. We begin by considering the case of rectangular meshes, next we presentthe quadrangular case and �nally we give an example with triangular mesh. For each example,the errors in the L2-norm for the scalar unknown p and the vectorial unknown u are calculated.In the following, we note these errors ep and eu, respectively.

Remark. For the sake of simplicity, only the problem with homogeneous Dirichlet condition isconsidered.

8.1. Rectangular case

We are interested in the rectangular case. The domain is the square [0; 1] × [0; 1] and theexact solution is given by

p(x; y)= sin (�x) sin (�y)



Figure 1. Mesh 1 Figure 2. Mesh 2

Table I. L2-error for p and u in the quadrangularcase

Mesh ep eu

Mesh 1 1·12× 10−2 2·8× 10−2Mesh 2 2·74× 10−3 1·4× 10−2

We present below the errors in the L2-norm relatively to p and u. These results are obtained using10×10 and 20×20 regular meshes.

Mesh ep euMesh 10×10 1·03×10−2 9·3×10−3Mesh 20×20 2·57×10−3 3·1×10−3

One can remark that the order of convergence seems equal to 2 for the scalar unknown p andthat the order of convergence for the vectorial unknown u seems greater than 3

2 .

8.2. Quadrangular case

Let us now consider the same domain and the same function p as in the preceding example,but this time the quadrangular meshes are as shown in Figures 1 and 2.The errors in the L2-norm relatively to p and u obtained using meshes 1 and 2 are shown in

Table I.It is easily seen that the order of convergence for the scalar function p seems to be conserved

(equal to 2) while the one corresponding to the vectorial function u seems equal to 1 now.

8.3. Triangular case

Finally, we consider the two triangular meshes shown in Figures 3 and 4. We present the errorsin the L2-norm relatively to p and u obtained using the above meshes in Table II. The order ofconvergence seems preserved, in agreement with the theoretical results.



Figure 3. Mesh 1 Figure 4. Mesh 2

Table II. L2-error for p and u in the triangularcase

Mesh ep eu

Mesh 1 2·99× 10−2 2·3× 10−2Mesh 2 5·85× 10−3 1·06× 10−2

REFERENCES

1. Cai Z. On the �nite volume element method. Numerical Mathematics 1991; 58:713–735.2. Cai Z, Mandel J, McCormick S. The �nite volume element method for di�usion equations on general triangulations.SIAM Journal of Numerical Analysis 1991; 28:392–402.

3. Thomas J-M, Trujillo D. Finite volume variational formulation. Application to domain decomposition methods.In Domain Decomposition Methods in Sciences and Engineering, Sixth International Conference on DomainDecomposition Methods in Science and Engineering, Quarteroni A et al. (eds). AMS Series, ContemporaryMathematics, 1994; 157:127–132.

4. Thomas J-M, Trujillo D. Analysis of �nite volume methods and application to reservoir simulation. In Proceedingsof the Conference: Mathematical Modelling of Flow through Porous Media, Bourgeat A, Carasso C, Luckhaus S,Mikelic A (eds). World Scienti�c: Singapore, 1995; 318–336.

5. Thomas J-M, Trujillo D. Finite volume methods for elliptic problems; Convergence on unstructured meshes. InNumerical Methods in Mechanics, Conca C, Gatica G (eds). Pitman Research Notes in Mathematics Series,vol. 371. Addison-Wesley Longman: Reading MA, 1997; 163–174.

6. Cai Z, Jones J, McCormick S, Russell T. Control-volume mixed �nite element methods. Computational Geosciences,1997; 1:289–315.

7. Eymard R, Gallou�et T, Herbin R. The �nite volume method. In Handbook of Numerical Analysis, vol. VII.Ciarlet PG, Lions JL (eds). Elsevier Science: Amsterdam, to appear.

8. Trujillo D. Couplage espace-temps de sch�emas num�eriques en simulation de r�eservoir. Th�ese, Universit�e de Pau et desPays de l’Adour, 1994.

9. Nicola��des RA. Existence, uniqueness and approximation for generalized saddle-point problems. SIAM Journal ofNumerical Analysis 1982; 19:349–357.

10. Bernardi C, Canuto C, Maday Y. Generalized inf-sup conditions for Chebychev spectral approximation of Stokesproblem, SIAM Journal of Numerical Analysis 1988; 25:1237–1271.

11. Ciarlet PG. Basic error estimates for elliptic problems. In Handbook of Numerical Analysis, Finite Element Methods,vol. II, Finite Element Methods (Part 1). Ciarlet PG, Lions JL (eds). North-Holland: Amsterdam, 1991; 17–351.

12. Raviart P-A, Thomas J-M. A Mixed Finite Element Method for Second Order Elliptic Problems. In MathematicalAspects of the Finite Element Method, Lecture Notes in Mathematics, vol. 606. Springer: Berlin, 1977; 292–315.

13. Roberts JE, Thomas J-M. Mixed and hybrid methods. In Handbook of Numerical Analysis, Finite Element Methods,vol. II, Finite Element Methods (Part 1). Ciarlet PG, Lions JL (eds). North-Holland: Amsterdam, 1991; 523–639.


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