Resumen
In this paper, we prove the convergence of a numerical method for solving two-phase immiscible, incompressible flow in porous media. The method combines an upwind time implicit finite volume scheme for the saturation equation (hyperbolic-parabolic type) and a centered finite volume scheme for the Chavent global pressure equation (elliptic type). The capillary pressure is not neglected, and we study the case when the diffusion term in the saturation equation is weakly degenerated. Estimates on the approximate solution are proven; then by using compactness theorems we obtain a limit when the size of the discretization goes to zero, and we prove that this limit is the unique weak solution of the problem that we study.

Resumen
We study the convergence of two coupled numerical schemes, which are a discretization of a so-called elliptic-hyperbolic system. Only weak convergence properties are proved on the discrete diffusion of the elliptic problem, and an adaptation of the H-convergence method gives a convergence property of the elliptic part of the scheme. The limit of the approximate solution is then the solution of an elliptic problem, the diffusion of which is not in the general case the H-limit of the discrete diffusion. In a particular case, a kind of weak limit is then obtained for the hyperbolic equation.