Convergence of a finite volume scheme for an elliptic-hyperbolic system

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M. H. V IGNAL

Convergence of a finite volume scheme for an elliptic-hyperbolic system

M2AN - Modélisation mathématique et analyse numérique, tome 30, n^o7 (1996), p. 841-872

<http://www.numdam.org/item?id=M2AN_1996__30_7_841_0>

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

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MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS MODÉLISATION MATHÉMATIQUE ET ANALYSE NUMÉRIQUE

(Vol. 30, n° 7, 1996, p. 841 à 872)

CONVERGENCE OF A FINITE VOLUME SCHEME FOR AN ELLIPTIC-HYPERBOLIC SYSTEM (*)

by M. H. VlGNAL ( 0

Résumé. — Dans cet article, on étudie la convergence d'un schéma de type volumes finis pour un système couplé formé d'une équation elliptique et d'une équation hyperbolique linéaires, définies sur un ouvert borné de IR2.

Pour l'équation elliptique un schéma volumes finis à quatre points est utilisé, puis une inégalité discrète de Poincaré pour les fonctions à moyenne nulle, est montrée, ce qui permet d'établir une estimation d'erreurs de l'ordre de h, en norme H discrète, où h définit la taille du maillage.

Sur l'équation hyperbolique, on utilise un schéma décentré vers l'amont de l'écoulement, puis en utilisant une estimation faible sur la variation de la solution approchée, on montre, sous une condition de stabilité, la convergence de cette solution vers la solution faible du problème.

Abstract. — We study hère the convergence of a finite volume scheme for a coupled System of an hyperbolic and an elliptic équations defined on an open bounded set of IFK .

On the elliptic équation, a four points finite volume scheme is used then an error estimate on a discrete H1 norm of order h is proved, where h defines the size of the triangulation.

On the hyperbolic équation, one uses an upstream scheme with respect to the flow, then using an estimate on the variation of the approximate solution, the convergence of the approximate solution toward a solution of the coupled system is shown, under a stability condition.

1. INTRODUCTION

One considers a problem coming from the modelization of a diphasic flow in a porous medium. In a simplified case it leads to the détermination of the velocity u of one of the two phases and of the pressure R

Let Q be an open bounded polygonal connected set of IR², one sets Let g e L~(F), u^ö e L°°(Q) and ue L°°(F⁺ x R⁺) , be given, with F+ ~ {y e F\g{y) 5= 0}, one assumes g(y) dy = 0 \ then one considers the problem defined by :

(1) AP(x)=Q, xe Q

(2) ",(*»*) -div («O, t) VP(JC)) =0, xe Q, t e R.

(*) Manuscript received July 31, 1995.

C1) UMPA. CNRS-UMR No 128, E.N.S. Lyon, 46, Allée d'Italie, 69364 Lyon Cedex 07.

M2 AN Modélisation mathématique et Analyse numérique 0764-583X/96/07/S 7.00

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with following boundary conditions and initial condition : (3) V P ( y ) . / i ( y ) = 0(y), ye F

(4) M(y,O = M(y,o. y e r⁺, te u⁺

(5) M(JC, 0) = WO(X), ^ G I 2

where w is the outward unit normal to F.

More precisely, one searches u in L~(Q x IR⁺ ) and P in H^X(Q) solutions of (l)-(5) in the following weak sensé :

f V P ( x ) . V<Z>(x) dx- f g(y)&(y)dy = O for ail <Z> G H\Ü)

JQ J r

and

J ƒ

II(JC,

r) | f (x, t)dxdt~\ J

M(X,

r)

VP(JC)

. Vp(x, 0

f ,

I IJ l

I W( A tl Q

iriu^A

for ail ^ e C ^ ( ^⁺x [ R⁺) with Q⁺ = Q ^ F⁺.

Note that the test functions ^ are equal to zero on 7^" = {y e^JT ; g ( y ) < 0 } but not necessarily on F⁺.

To discretize these équations, a finite volume scheme is used, then the results presented by R. Eymard and T. Gallouët in [5] and R. Herbin in [8] are generalized. Indeed the System considered in [5] is the same as the one presented hère but the authors use a coupled finite element-finite volume scheme, then discrete unknowns are localized at the vertices of the meshes whereas in this note they are localized at the cell centers. For this scheme they prove a convergence property toward a weak solution of System (l)-(5). The scheme used on the pressure is a finite element scheme, hence the convergence of the approximate solution of the elliptic équation follows from the finite element framework. In this note one uses a four points finite volume scheme for the pressure, then a convergence proof is given by generalizing the results of R. Herbin in [8]. One of the two essential différences cornes from boundary condition, since in [8] the boundary condition is a Dirichlet condition whereas hère a Neumann condition is considered, then estimâtes are changed by

M AN Modélisation mathématique et Analyse numérique

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CONVERGENCE OF A FINITE VOLUME SCHEME 843

boundary terms. In particular, for our estimâtes it is necessary to count the number of triangles which are "after" a given triangle in ail directions, while in [8] only one direction is sufficient In a same way, to prove the error estimate in discrete H1 norm, the discrete L2 norm of the error is majorized by its discrete Ho norm, assuming the solution's mean value equals to zero since problem's solutions differ from a constant, whereas in [8] the discrete L2 norm of the error is majorized by the sum of its discrete ƒƒ* norm and of its discrete L norm on the boundary.

The second différence cornes from the assumptions on the meshes. In [8] all the meshes must have a measure of same order, whereas here déformations of the meshes are authorized (see section 2.1).

On the velocity an upstream finite volume scheme with respect to the flow is used, then, under a stability condition, the convergence of the approximate solution toward a solution of the hyperbolic équation is shown. To prove this result, one needs an estimate on the variation of the approximate solution, it uses an estimate on discrete Hl norm of the approximate solution of the elliptic équation, this result in [5] is given by the finite element framework. Other results on the existence and the uniqueness of solutions of hyperbolic équa- tions are given in [7], [3], [1], [10].

Numerical experiments on the comparison between finite element scheme and finite volume scheme, done by J. M. Fiard and R. Herbin in [6] for a conduction problem and by R. Herbin and O. Labergerie in [9] for a diffusion- convection problem, have shown that the approximation of fluxes is better for the finite volume scheme. Furthermore, comparison between the scheme presented here and the weighted finite volume scheme of R. Eymard and T. Gallouët have also been done in [6] on a system more gênerai than (l)-(5), where (1) is changed by div (f(u(x, t) ) VP(x) ) = 0, x e Q, t e U+. This numerical test shows that the scheme presented here gives better results than those given by the weighted finite volume scheme. Other authors have been interested by finite volume scheme on triangular meshes, see for instance [11].

In [11], results are restricted to particular meshes, whereas, here, as it has been already remark, the assumptions are most gênerai.

2. DISCRETIZATION

Before discretizing the elliptic and the hyperbolic équations, one first gives assumptions on the triangulation.

2.1. Assumptions on the triangulation

Let ST = (Kj)i *sj^^L be a triangulation of Q which satisfies : there exists rj such that for any 6 angle of an element of 3", one has :

(6) rj < 9 < | - rj .

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One deflnes h. = \/S{Kj), where S{K.) dénotes the 2D Lebesgue measure of Kj, then the assumption (6) gives the following resuit : there exists a^l > 0 and a² > 0, depending only on rj, such that for ail side a of the triangulation, the length l(a) of a vérifies :

(7) a^x.h.*k l(a) ^ c^.hj

if a is an edge of the cell Kj.

L

Then one dermes h e M*+ by h = max h..

7 = 1

Some notations will be useful to describe the numerical scheme :

NOTATIONS

9"ext = {J 6 {1, .... L} ; Kj e<T,Kjnr* 0}

stf^&xt the set of the edges of the triangulation which are on the boundary r of Q

£&^int the set of the edges of the triangulation which are in Q

= m a x ( 0 ( y ) , O ) and g~ ( y ) = ( - g )+ . For ail Kj e 3", 1 ^ j ^ L one sets :

c-(j) the edges of Kp i = 1,2 or 3

x7 the intersection of the orthogonal bisectors of the edges of Kj gtj = 9⁺ (y)dy and ^ . = g" (y)dy,i =

Je &) Jc;U)

= I , 2 o r 3 , i f c(. 0 ' ) e ^e x t

flf^iy = flfⁱ;-^,i=l,2or3,if^C(.(7)^e ^^{e x t}

Text(j) the set of the suffix i = 1, 2 or 3 such that c^O') E j ^e x t

T. the set of the suffix of the neighbours of K- cjk = dKj n a ^ for all k e r;

d.fc = d(xj, cJk) + Ö?(X^,, c^), for all k e z\, where d is the euclidian distance of U2

x-^k the center of the side c^jk.

2.2. Discretization of the elliptic équation

To discretize (1), a four points finite volume scheme is used ; the principle of the finite volume schemes, (see [4]), is to integrate équations on each control volume (here (Kj G 3~), so one has :

f

J ö

VP(y).n^K(y)dy = 0

j J

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CONVERGENCE OF A FINITE VOLUME SCHEME 845 where nK dénotes the outward unit normal to dK..

One approximates P by P^ with P^(x) — Pj if x e K'., so the discretized équation can be given by approximating the flux of P through one edge ci(j)ofKJby:

l(cJk)—^—— if 3* e {l,...,L}suchthatc/(7*) = c/fc

g (y) dy if c^f-(y ) e *fl^^ext

Then one has :

_ (P^k-Pj) ^

(8) 2 /(c7*> 5^ + / 62(.)^ = ° for ail 7 e { 1 , - , L }

with the convention 2 = 0-

One can remark that on the domain's boundary the approximation of the0

flux is exact.

2.3. Discretization of the hyperbolic équation

Before discretizing (2), one defines the time step Ô, so let J be a triangu- lation of Q which satisfies the assumption (6) and a e ] 0 , 1[, then one chooses S e M*+ which satisfies the following conditions :

(Pj-pk) (9) t - v " * / djk

where £f = {(7, /:) G { l , ..., L}² ; ( ^ . , AT^fc) e 2T x ST, A; e ^ and One sets ?" = m5 for all n e M.

To discretize (2), first an Euler scheme explicit in time is used, and as for the elliptic équation, one intégrâtes (2) on each control volume :

f u(x,r-)-M(x,O

One approximates u by u^^è with u^^â(x, t) — uj if x G KJ and r e [t*\ tn+l[. Then an upstream discrete value with respect to the flow is chosen for u at the interfaces of meshes, and at the boundary.

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One defines M? for ail j e {1, ..., L) by uj = { , uo(x) dx and unyi for

1 J e ^ e x t a n d f o r l e Te x t ( i ) b y « ;

So, the discretized équation is given by :

i

where

for ail 7 G {1, ..., L} and all

ui else .

3. CONVERGENCE OF THE FOUR POINTS FINITE VOLUME SCHEME FOR THE ELLIPTIC EQUATION

Let 2T be a triangulation of Q which satisfies the property (6).

One proves in this section the existence of solutions (P)x <= ^ L of (8) and that these solutions differ only from a constant, proving the following resuit :

PROPOSITION 1 : Let g G LT{r), one defines for ail j e {l,..., L } : g = \ g(y)dy if c.(j) e <x t.

Then :

1) if S 9U = 0 V/ e {1, ..., L} and ( ƒ> ), ,.

P^y = P^t Y /^f* E { l , . . . , L }

3) (ƒ

satisfy (8> then

ü)

solutions of (8) and solutions differ only from a constant.

Furthermore one proves the numerical scheme's convergence proving an error estimate on discrete H1 norm of order h :

(8)

THEOREM 1 : Let g e L~( F), one dénotes by P the weak solution of(l), (3)

L

such that 2 S(Kj) P(Xj) — 0, where x. is the intersection of the orthogonal bisectors of the edges of Kn one supposes g such that P is in C2(Q).

3 L

Let (Pj)] çj^L satisfy (8) and 2 S(Kj) Pj ~ 0, one defines the error by e. = Pj - P(Xj) for ail je {1, /.~ï}.

Then there exists C{ and C2 positive, independent of 2T such that :

' - - S E

<

* - '

^d) >

**« c > r < C * and f- -**

V

-

C2.h.

3.1. Proof of Proposition 1

Proof of the first part of the Proposition 1 : One supposes that V / e {1...L} E 0. = O and (PJ)l^J^L satisfy (8).

Let j0 e {1, ...,L} such that Py0 = min {^ ; k e {l, ..., L}}.

Thanks to (8), one has :

But according to assumptions 2) fif(/0 = 0 i a nd Kcok) > 0»

i e tc x l(yo )

<i.o ^ > 0 and P^ - PJo ^ 0 VA: e T -o, so one has :

P = p \fk e T. .

Using the fact that Q is connected, one obtains by induction : fV = Pk

V7; / : G {1 L}.

Proof of the second part of the Proposition 1 : Supposing that (Pj)x ^j^L satisfy (8) and summing these équations one gets :

I,

Proof of the third part of the Proposition 1 : One supposes that g(y) dy = Q.

r

Then thanks to the first part of this Proposition, (8) is a linear System which has a kernel of dimension 1.

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So, thanks to the second part of this Proposition, the image space of this linear System is the set of the B e IRL, B = \bv b2, ..., bL) such that

£ bj = 0.

Then, as ^ b.= \ g(y) dy = 0, there exists (P;)i <ZJ<Z^L solutions of (8) and these solutions differ only from a constant.

3.2. Proof of Theorem 1

Définition of the consistency error on the fluxes : As it has been remark in Section 2.2, fluxes are exact on the domain's boundary, then one defines the consistency error only at the interfaces of meshes.

The exact flux on the side c.k in the direction of K. to Kk is :

and the approximate flux on the side c}k in the same direction is :

One defines the consistency error, denoted by Rc^(K), by : P(xk)-P(x)

One can remark that the conservativity of exact and approximate fluxes implies :

Now the following result is proved :

LEMMA 1 : Under the assumptions of Theorem 1, there exists a constant CR 5= 0 independent of 2T such that :

IV*y>l ^CR'h Y / , * E { 1 , . . . , L } .

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CONVERGENCE OF A FINITE VOLUME SCHEME 849 Proof of Lemma 1 : Using a first order Taylor expansion one proves that there exists C, ^ 0 and C2 3= 0 depending only on av a2 and on the second order derivative of P such that :

~ V P ( y ) . nK dy - VP(x.,) . nK

Cjk)Jck j J J

VP(xik).nK- P(xk)~

xjk K. C2.h So the proof of Lemma 1 is completed.

Let e^k - P^k - P(x^k) be the error on the cell K^k for ail k G {1, ..., L}, then one proves the first resuit of Theorem 1, i.e. the following property :

There exists C ^ 0 independent of ET such that :

1/2

( H )

Proof of the inequality (11) : Thanks to (1) one has :

(12)

One subtracts (8) from (12), one multiplies by e. and one sums over j , then using the conservativity of the exact and approximate fluxes, the properties (6) and (7), the Lemma 1 and the Young inequality (for more details see [8]), one gets :

where S(Q) is the 2D Lebesgue measure of the domain Q.

Then the proof of the inequality (11) is completed. Let's complete the proof of Theorem 1.

Proof of Theorem 1 : The L2 discrete error estimate is shown by using a discrete Poincaré-Wirtinger inequality, i.e. the following result :

There exists C > 0, independent of 2T, such that : (13)

L

Ka)

(11)

with m =

= îS(K.) eti where and ev- are the errors on the both

X ) k î *

éléments of 2T for which a is an edge and d^a is defined as follows : for ail a G se^XTA there exists y and k in {l, ..., L} such that a = c^Jk then d^ü = d^.

This resuit will be proved in four steps : Step 1 :

Let SP be a square included in Q and the both directions 2X and @>2 defined by SP, see figure 1.

Figure 1.

One dénote by d = | b — c | and one chooses for coordinate System the coordinate System defined by any point of [R² and the both directions 2^x and

Let se m

Let K

be the set of the sides a of j /i n t such that

and in and Kkc\ 0 ,

by 2T such that K}r\0» *

x = ( xp x2) G K.C\ SP and y = (yv y2) G Kkc\&y one dénotes [x, y] the line segment delimited by x and y, and one defines :

^xx xn y, (respectively ( se'Xi ) the set of the sides of se'int such that the intersection with the line segment [ ( xp x2), ( xp v2)] (respectively ( [ ( xp y2), (yv y2)]) is a point.

j3/^2 ) (respectively (stf^) the set of the sides of srfint ^ such that the intersection with the line defined by ( xp0 ) (respectively ( ( 0 , y2) ) and parallel to Sè2 (respectively {2X) is a point.

Then one has :

e. -

a.e y e ^ n Vx e _n

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CONVERGENCE OF A FTNITE VOLUME SCHEME 851

Integrating over KkC\ SP and summing over k, one obtains for ail x e @> r\Q:

l'ï

_{v b v b}

with m#=jç^ ^

For ail a e s/inV let da be the angle between a and £%v then swapping the summation and the intégral in the second terms, one has :

d\e. - eK: - eK- \ + ^ \ex; ~ €K~ I l(a) c o s

Using the Cauchy Schwarz inequality, one gets :

and then :

/ a2dM

\

4a2h

a

Integrating over [b, c] with respect to xx and swapping the summation and the intégral, one has for ail x2 e [b, c ] :

(14) E

(13)

where 2è^x^ is the line defined by the point (0, x²) and the direction

<%VM{^) is the set of the triangles which are eut by £^1 } and 6(a2)22

C\ =d+2a~h + - ( ^ + 4aJ).

1 2 ai 2

With the same arguments, one can prove the following result for all xx e [b, c] :

(15) 2 KKji^^l2)) | ^ - m ^ |2 ^ C , ^ 2 ^ K a ) -

One concludes by integrating (14) with respect to x2 so :

L . . . . 2

(16) ^S(Kjn0>)\ej-m3i\2^Cld2 ^ ^ /(a)- Step 2 :

In this step the inequalities (15), (14) and (16) are proved over half of the square 0> denoted T, i.e. the triangle (A,B, D) (see fig. 1),

Using the orthogonal symmetry in relation to [BD], the problem is the same as the one of the step 1, then with the same notations, one has the following results :

I — I²

and :

Step 3 :

One proves the result over an ordinary triangle of Q.

So let an ordinary triangle Tof Q and a right-angle triangle T, see figure 2.

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CONVERGENCE OF A FINITE VOLUME SCHEME 853 Then there exists a linear application F which transforms T in T defined as follow :

F: ^{x \ \l}^x^{/, tan 6}

l2 sin 0 As T=\J(K.nT) then T = \J (F(K.

j = i J'= i

/2 si

O I lx Vi O

Figure 2.

Let the function ë defined from T to IR by if

j

One has the following resuit : there exists // such that for ail triangulation

~ {KJ)^X^J^^L of Q one has, for ail angle of an element of 2T :

Then there exists rj(rj,F) such that for ail angle of an element of

but one can have : 0 ^ ~.

So one defines ( 7 ^ ) 1 ^ ^ ^ ^ , ^ triangulation of F(Q) such that 1) \/k G { l , ..., L'}, there exists 7 e { l , ..., L} such that :

2) for ail 9 angle of an element of ( T^k)^x ^^k ^^L,⁹ one has :

(15)

According to the step 2, one has :

nT)) \ej-m^?\²=^S(T^kr^T) \e^k-m^?\²

j=\ k=\

where m^ is the mean value of e over T.

Remarking that :

~~ eKT

and that

one gets :

Thus

So, one obtains :

= cv

/2 sm

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CONVERGENCE OF A FINITE VOLUME SCHEME 855 With the same arguments, one proves the following inequalities :

(17) 2 (18) 2

where ^7 (respectively â?y) is the set of the triangles K of 2T such that K r^ I (respectively K r\ J) is not emptyset.

Step 4 :

Let a subset T oï Q which is the union of two triangles TA and T2, one dénotes dTx n dT2 by /.

S( Tx ) mj ( 2 ) 2

As m = c / ^ x , one can write :

2, ^ , „ , „ . _ , . . . „,„,S(T2)

( y\s(K.

According to the step 3 one has :

Then it just remains to estimate the différence between m2 and mv

Let x = (JCP JC2) e ƒ, so :

I ^ **~ *~\ f 1 / \ I 2

lm2 ~ mi I ^ 2( | ^ ( J CP x2) -

Integrating over ƒ and thanks to the inequalities (17) and (18), one has

1(1) \m2-m \2

(17)

Then :

C

( 2

With the same arguments, one can prove the following result :

and then :

L

7 = 1 a e. stfiTil

As we have supposed that Q is a finite union of triangles, one has :

L

where C does not depend of the triangulation ST.

One concludes remarking that ~ =S —1— for all a E ^/int, so :

Remark 1 : If the boundary condition is a Dirichlet condition, then this proof can be generalized, it is closed to those given by R. Herbin in [8].

In this case one considers a direction £è which is parallel to none edges of the mesh.

Let j G {1,..., L] and x e Kj9 then one dénotes by s/jxi the set of the edges such that the intersection between these sides and the line which contains x and parallel to Se is not empty set.

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CONVERGENCE OF A FINITE VOLUME SCHEME 857 One can write :

|2

where 0a is the angle between a and 2.

Integrating over K., summing over j and swapping summations and inté- grais, one gets :

7=1 \ a e

and then one concludes with the error estimate in discrete HlQ norm (see [8]).

4. CONVERGENCE OF THE NUMERICAL SCHEME FOR THE HYPERBOLIC EQUATION In this section one will prove the convergence of the solution of (10) toward a weak solution of problem (2), (4), (5), proving the following Theorem :

THEOREM 2 : Let ( 3~ , 3q)q&H be a séquence of triangulations of Q and time steps which satisfy the properties (6) and the stability condition (9).

One dénotes by Vq = (Kj)\q> .* L«> and tn{q) = nSq.

Let the séquence ( % ^ ) ^G^ with Uay ô (JC, t) = u"q)nj if x e K^q) and ( e [tn{qytn{q)\, where { M(^ " , J e {l L ^ } , n 6 M}, is solution of the discretized équation (10) associated to the triangulation 9"' and the time step

Then :

1) there exists a subsequence still denoted (u^ s ) e N, which converges toward u when q —> °°, Le. when h goes to 0, in L°°(f3 x R+) for the weak * topology, i.e. one has :

lim Uay ô (x, t) (p(x, t) dxdt - \\ u(x, t) (p(x, t) dx dt

for ail (pe Ll(Qx U+ ).

2) u is the weak solution of problem (2), (4), (5), i.e. one has :

u(x, t) ^ (x, t)dxdt- u(x, t) VP(JC) . VÇ?(JC, t) dx dt JQJU+ ö t in Ju +

+ uo(x)(p(x,O)dx+\ u(y,t

JQ JrJu+

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for ail <p<= C~(Q+ xU+) with Q+ = Q u F+.

In order to prove the first part of this Theorem, one will prove in the subsection 4.1 an LT(Q X R⁺ ) estimate on the approximate solution.

Then to prove the second part, one will first show, in subsection 4.2, a weak estimate on the variation of the approximate solution, and then, using this estimate, one will prove that u is a weak solution of problem (2), (4), (5).

4.1. L°°(Q x IR⁺ ) estimate on the approximate solution

Hère one proves that the family (wg- ô ) g e N is bounded in L^{Q x R⁺) , then, thanks to the sequential weak ic relative compactness of the bounded sets of LT(Q x R+) , it shows the existence of a subsequence, still denoted (u^^ô )^N, which converges in LT(Q x R⁺) for the weak * topology when h goes to 0.

Let 2T be a triangulation of Q and ô G R*⁺ which satisfy the property (6) and the stability condition (9). Another expression of the équation (10) is :

for ail j e { l , . . . , L } .

So ufj + x is a linear combination of the unv 1 ^ k ^ L and w£.

i = 1,2 or 3.

Then like in [5], thanks to (8) and to the stability condition (9), one has the following properties :

1) the sum of the coefficients of the combination is equal to 1

2) the coefficients of the combination are ail positive and one can write :

|M?|, sup |û?.|\

m a x / sup |w^;°|, | | M | |^L-( r + x R + )

(20)

CONVERGENCE OF A FINITE VOLUME SCHEME 859 Then (u$ ô )q G M is bounded in LT(Q x R J .

4.2. Convergence of the numerical scheme

The LT(Q x K?⁺ ) stability gives the existence of a subsequence which converges to u in L°°(Q xR⁺) for the weak * topology.

One will show that u is a solution of (2), (4), (5) in a weak sensé.

Let 2T be a triangulation of Q and ö e U*+ which satisfy the property (6) and the stability condition (9).

One considers q> e C™(Q+ xR+) with Q+ = Q u F+, and one defines T such that, Vjce £2+, supp(<p(x,. ) ) c [ 0 , 7 - 1 ] and such that there exists i V e N such that A/g ^ T < ( A^ + 1 ) S.

Multiplying (10) by "ô?1FT^(*>f") a n d integrating over Kp then summing over j and n, one gets :

with :

In a classical way one could show that :

JQJU^{+ ÖT} JQ

The proof of the following resuit will be given :

lim £m £^{2 / ï}2 / ï = + u(x, t) VP(x) . V<p(x, t) dx dt = + u(x,

*O JQJM⁺

JrJu +

t)<p(y,t)dydt.

JrJu⁺

(21)

One defines E^3h and E^4h by :

2 E («;-«;) <p(y.ta)g(.y)dy

7=1 / e T^cxt(y) t'c/O')^;

« = 0 l Ji3

Jr '

The resuit will be proved in three steps.

The first step is the proof of the existence of a constant C 5= 0, independent of 2T and <5, such that :

\F —F I < Ch^U2 According to (8) one has :

„ = 0 y

(22)

CONVERGENCE OF A FINITE VOLUME SCHEME

thus :

861

T I 3 / i l

[ i

7 = 1 I E Texl(y)

One deflnes :

<p(x9tn)dx

-1

<p(y,tn)dy- ^ T ^ ç>Utn

icit ^Kk) JK,

-L

)dx

Remarking that if ^ = C is constant then Z^ = 0, one gets

1.

^{;. ) f}

Similarly one could show that :

and then :

( P —

d

^ 0

(23)

To conclude, the following Lemma is proved :

LEMMA 2 : Let 2T be a triangulation of Q and ö e R*+ which satisfy the property (6) and the stability condition (9). Let T > 0, one defines N by Nô ^ T< (N + 1)3 and one supposes N 5= 1, then one sets

je SText ie rext(y)

Then there exists a constant C ^ 0 independent of £T and S such that EFh(T) ^ Chm ,

Proof of Lemma 2 : Let we M, K. e 2T, one multiplies (10) by w" and one sums over n and j , then using the following property :

M. ^{+ l \ 2}

one gets :

n = Q j=

Let BA and ^2 be defined by :

= 0 j = 1

(24)

CONVERGENCE OF A FINÏTE VOLUME SCHEME 8 6 3

Then one has :

L \

J = ' ' 6 rMü ) /

Remarking that :

( n \ 2 n n 1 / n n \ 2 . 1 / n \ 2 1 / n \ 2 Uj) -Uj uk = ^(uj -uk) + 2 ( " / ) ~ 2 ^ w ^

/ p

k

0 , J k ) 6 i ^ ajk j=\ leer; ajk

and thanks to (8) :

one can write :

(P,-P)

Furthermore according to (10) one has :

(25)

The number of the neighbours of a triangle K. is lower than 3, and the number of its sides on the boundary is lower than 2, then thanks to the Cauchy- Schwarz inequality :

5 / u x x2 / n « \2 ( P — P

Thus using this inequality and the last expression of B2 in (19) one gets

+ > > ( u ) ü.. I ^ o > > > ( u ) ü + >

^ J ^J v 7 ' y y ƒ ^ - ^ - ^ ^ - / v y' ' y y - "

7 = 1 i e rc x,(y) ƒ H = O ; = 1 i G TextO) 7 = 1

with :

So one can write the following inequality :

2 ^ ^ K O £ i

0 0', k) € ^ W7^ « - 0 7 = 1 l e

Using the Cauchy Schwarz inequality and the previous property one gets : EFh(T) ^ h»\£)m x (ô ^

/V L \ 1/2

„ = 07=1 / G Tcxt(7)

(26)

CONVERGENCE OF A FINITE VOLUME SCHEME 865 But

n = Oj=l ,'

with T and G + independent of ST and ô.

Still using the Cauchy Schwarz inequality, one has :

y

1/2

a, Kc]k)

But j^T ^ ~J~ ** C V(7'T J

a,, a9 and r/7 so :

\ where C only dépends on

V

\I2

ihk)*sr

Thus to complete the proof of Lemma 2, it just remains to show that there exists a constant C 5= 0, independent of £T and Ô, such that :

(20)

2 ^

This resuit will be proved in four steps (another proof is possible, using the error estimate (11), see Remark 2) :

Step 1 :

One shows that there exists a constant C\ ^ 0 such that : (21)

"y*

1/2

Multiplying (8) by P and summing over ƒ one gets :

L

7 = 1

>>—^r

7=1 iET

^{2 S}

e x l(j)

(27)

Then thanks to Cauchy-Schwarz :

7 = 1

~iü ) ) ^1/2

1 /2. # x. ^ 9 \

a2 meas(r) \\g\\L-(n[ 2J hAPj] )

1/2

where meas(F) is the 1D Lebesgue measure of F.

Furthermore one has :

and then :

2 2-

7=1 kezj

\ g \ \^L~

Step 2 :

One shows that there exists C² and C³ positive such that :

(22)

2

7 = 1

2

7 = 1

Q is a bounded polygonal open set, first £2 will be supposed convex. If Q is convex, then one defines two directions @x and £iï2 and one breaks up the boundary of Q in four parts, not necessarily disjoint as on the figure 3.

Figure 3.

(28)

CONVERGENCE OF A FINITE VOLUME SCHEME 867 One dénotes by £Tr (respectively 3"^, 2Tr3, 2Tr ) the set of the suffix of the meshes which have sides on Fx (respectively on F2, F3, F4).

Let j G STri and x e dKj, then one defines : {(xvx2)e Q;

(x,, x2 ) is in the line which contains x and parallel to 3>,}

Let ç? G C~(£2) such that \/x e Qy 0 ^ <p(x) ^ 1, <p(x) = 1 Vx e i ^ , and çp(jc) = 0 Vx e F³.

Then for all 7 G 3~^r one has :

where j^o~ is the suffix of the element of 9"^r such that there exists i e {1, 2, 3} such that c.(jox) n B^ * 0 , so :

:

f\ + K ^ ^ f

But (p is in C°°(i2) and belongs to [0,1], then one has :

where

^^C2 = II ^11 L-(ö) Integrating over Fv one obtains :

- P\-

+ C2 a2

(29)

According to the Young inequality :

P\-

then as for ail a e se'^int one obtains :

, + C2)

2

Similarly one could prove the same resuit for 9"^A, 2T^r and 9"^r. Then summing these inequalities, one complètes the second step with Q convex.

If Q is not convex then one breaks up its boundary F in p parts (-T-), <;,S / ) disjoint and for ail i= 1, ...,p, one defines F] as on the figure4.

r,

Figure 4.

Then one defines for all i = 1, ...,/>, ^^c'^; e C ~ ( ö ) such that V* e r3, 0 ^ ^( 0( J C ) ^ 1, p( 0( y ) = l if y e Tf. and ^c o( y ) = 0 if y e r ; . Then similarly to the convex case one could prove (22).

(30)

CONVERGENCE OF A HNITE VOLUME SCHEME 869

Step 3 :

One supposes that :

2 ^ ^

7 = 1

it is always possible since solutions of (8) differ from a constant.

Then as one has proved (11), one could show that there exists a constant C4 3= 0 such that :

j=\ j = 1 k e Tj "jk

Step 4 :

One concludes, remarking that :

2 ^ v f ï t z

U,k)e& Jk j=l kezj Ujk

then thanks to (21), (22) and (23) one has :

L l(c..) / L l(c,

v y (p ~p f < c \/c +c c ( y y

7=1 kG zj "jk \ j = i ke Tj "jk

So the proof of the inequality (20) and of the Lemma 2 are completed, and then one has :

Remark 2 : As P is supposed to be in C2{Q), the estimate (20) can also be given by the error estimate in discrete HXQ norm (11), but our proof of (20) does not use the property P e C ( Q ) and then it could be extended to more complex cases.

The second step is the proof of the following resuit :

(31)

E3 h can also be written :

j=\

( p p f f

Tj Ujk JcJk

c,U)

and :

« = 0 j=\ \ kt / e rcx[(y )

L

then one gets :

n = 0 j = 1 k e Zj

B u t a c c o r d i n g t o ( 1 ) a n d ( 8 ) if <p = (p{xjk, tn) i s c o n s t a n t ,

\E3h~ E4h\ = 0> t n U S :

J4 / J I

- * )

ujk

where C only dépends on a2 and the first order derivative of (p.

(32)

Then one has :

3 ( c , fty2

One concludes remarking that :

limiE^4h=\ f u(x,t)VP(x).V<p(x,t)dxdt-

- f f «(y,*

which complètes the proof of the numerical scheme's convergence.

Remark 3 : One can prove the same results for a System a little bit different than (l)-(5), where (3) is changed by a Fourier condition :

where X > 0.

Then one must introducé some unknowns at the boundary. As fluxes are not exact on the boundary, the error estimate, in discrete H] norm, includes boundary terms.

The technic used to prove the error estimate in discrete Hl0 norm is closed to this used in this note.

Acknowledgements : I wish to thank Thierry Gallouët and Raphaële Herbin for their attention to this work and their precious remarks.

REFERENCES

[1] S. BENHARBIT, A. CHALABI, L P. VILA, Numerical Viscosity and Convergence of Finite Volume Methods for Conservation Laws with Boundary Conditions, to appear in SIAM Journal of Num. Anal

[2] S. CHAMPIER, T. GALLOUËT, R. HERBIN, 1993, Convergence of an Upstream finite volume scheme on a Triangular Mesh for a Nonlinear Hyperbolic Equation, Numer. Math., 66, pp. 139-157.

[3] R. DlPERNA, 1985, Measure-Valued Solutions to Conservation Laws, Arch. Rat.

Mech. Anal., 88, pp. 223-270.

[4] R. EYMARD, T. GALLOUËT, R. HERBIN, The finite volume method, in préparation for the Handbook of Numerical Analysis, Ph. Ciarlet and J. L. Lions eds.

vol. 30, n° 7, 1996

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[5] R. EYMARD, T. GALLOUËT, 1993, Convergence d'un schéma de type Eléments Finis-Volumes Finis pour un système formé d'une équation elliptique et d'une équation hyperbolique, M2AN, 27, 7, pp. 843-86 L

[6] J. M. FIARD and R. HERBIN, 1994, Comparison between finite volume and finite element methods for the numerical computation of an elliptic problem arising in electrochemical engineering, CompuL Math. Appi Mech. Engin., 115, pp. 315- 338.

[7] T. GALLOUËT, R. HERBIN, 1993, A uniqueness Resuit for Measure Valued Solutions of Non Linear Hyperbolic Equations, Differential Intégral Equations, 6, 6, pp. 1383-1394.

[8] R. HERBIN, 1994, An error estimate for a finite volume scheme for a diffusion convection problem on a triangular mesh, Num. Me th. in P.D.E.

[9] R. HERBIN, O. LABERGERIE, Finite volume and finite element schemes for elliptic-hyperbolic problems, submitted.

[10] A. SZEPESSY, 1989, Measure valued solution of scalar conservation laws with boundary conditions, Arch. Rat Mech. Anal, 107, 2, pp. 182-193.

[11] P. S. VASSILEVSKI, S. I. PETROVA, R. D. LAZAROV, 1992, Finite Différence Schemes on Triangular Cell-Centered Grids With Local Refinement, SIAM J. Sci.

Stat. Comput., 13, 6, pp. 1287-1313.