Convergence of a finite‐volume mixed finite‐element method for an elliptic–hyperbolic system

This paper gives a proof of convergence for the approximate solution of an elliptic–hyperbolic system, describing the conservation of two immiscible incompressible phases flowing in a porous medium. The approximate solution is obtained by a mixed finite‐element method on a large class of meshes for the elliptic equation and a finite‐volume method for the hyperbolic equation. Since the considered meshes are not necessarily structured, the proof uses a weak total variation inequality, which cannot yield a BV‐estimate. We thus prove, under an L^∞ estimate, the weak convergence of the finite‐volume approximation. The strong convergence proof is then sketched under regularity assumptions which ensure that the flux is Lipschitz continuous.