On the convergence of a combined finite volume-finite element method for nonlinear convection-diffusion problems. Explicit schemes

Abstract

This article is a continuation of the work [M. Feistauer et al., Num Methods PDEs 13 (1997), 163–190] devoted to the convergence analysis of an efficient numerical method for the solution of an initial-boundary value problem for a scalar nonlinear conservation law equation with a diffusion term. Nonlinear convective terms are approximated with the aid of a monotone finite volume scheme considered over the finite volume mesh dual to a triangular grid, whereas the diffusion term is discretized by piecewise linear conforming triangular elements. In the previous article [1] the convergence of a semi-implicit scheme was established. Here we are concerned with the analysis of fully explicit schemes. Under the assumption that the triangulations are of weakly acute type, with the aid of the discrete maximum principle, a priori estimates and some compactness arguments based on the use of the Fourier transform with respect to time, the convergence of the approximate solutions to the exact solution is proved, provided that the mesh size tends to zero. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 215–235, 1999