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

We study here the convergence of a finite volume scheme for a coupled system of an hyperbolic and an elliptic equations defined on an open bounded set of R(2). On the elliptic equation, a four points finite volume scheme is used then an error estimate on a discrete H-1 norm of order h is proved, where h defines the size of the triangulation. On the hyperbolic equation, 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.