The Upwind Finite Element Scheme and Maximum Principle for Nonlinear Convection-Diffusion Problem

Abstract

In this paper, a kind of partial upwind finite element scheme is studied for two-dimensional nonlinear convection-diffusion problem. Nonlinear convection term approximated by partial upwind finite element method considered over a mesh dual to the triangular grid, whereas the nonlinear diffusion term approximated by Galerkin method. A linearized partial upwind finite element scheme and a higher order accuracy scheme are constructed respectively. It is shown that the numerical solutions of these schemes preserve discrete maximum principle. The convergence and error estimate are also given for both schemes under some assumptions. The numerical results show that these partial upwind finite element schemes are feasible and accurate.