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IMA Journal of Numerical Analysis 2003 23(3):507-538; doi:10.1093/imanum/23.3.507
© 2003 by Institute of Mathematics and its Applications
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Convergence of a finite-volume mixed finite-element method for an elliptic–hyperbolic system

J. Droniou1, R. Eymard2, D. Hilhorst3 and X. D. Zhou3

1 Département de Mathématiques, CC 051, Université de Montpellier II, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France 2 Université de Marne-la-Vallée, Marne-la-vallée Cedex 2, 77454, France 3 laboratoire de Mathematiques, CNRS et Université Paris-Sud, 91405, Orsay Cedex, France

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{infty} 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.

Key Words: finite-volume method; mixed finite-element method; system of a hyperbolic and an elliptic equation

Received 6 February 2002. Revised 30 September 2002.


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