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

This article is a continuation of the work [M. Feistauer et al., Num Method s PDEs 13 (1997), 163-190] devoted to the convergence analysis of an effici ent numerical method for the solution of an initial-boundary value problem for a scalar nonlinear conservation law equation with a diffusion term. Non linear convective terms are approximated with the aid of a monotone finite volume scheme considered over the finite volume mesh dual to a triangular g rid, whereas the diffusion term is discretized by piecewise linear conformi ng triangular elements. In the previous article [1] the convergence of a se mi-implicit scheme was established. Here we are concerned with the analysis of fully explicit schemes. Under the assumption that the triangulations ar e of weakly acute type, with the aid of the discrete maximum principle, a p riori estimates and some compactness arguments based on the use of the Four ier transform with respect to time, the convergence of the approximate solu tions to the exact solution is proved, provided that the mesh size tends to zero. (C) 1999 John Wiley & Sons, Inc.