A general finite volume scheme for an elliptic-hyperbolic system using a variational approach

We study the convergence of general finite volume schemes for the diphasic flow problem in porous media \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $u_t-{\rm div}(u\nabla p)=0$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\Delta p=0$\end{document} in a bounded domain. A general formula for the numerical flux on a triangular mesh is given. The stability and an estimate of the total variation of the approximate solutions are obtained by means of a variational method; the convergence follows easily for general data.