H-Convergence and Numerical Schemes for Elliptic Problems

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. AMS Subject Classifications: 35K65, 35K55.