The discontinuous Galerkin method for semilinear parabolic problems

Abstract

We prove a priori error estimates for a space-time finite element method for semilinear parabolic problems. The finite element method has basis functions that are continuous in space and discontinuous in time, and variable spatial meshes and time steps are allowed. The effect of numerical quadrature is emphasized. R' esum' e. Nous montrons des estimations d'erreur a priori pour une m'ethode des 'el'ements finis en espace et en temps pour des probl`emes paraboliques semi-lin'eaires. La m'ethode des 'el'ements finis consider'ee a des fonctions de base continues en espace et discontinues en temps, et admet des maillages spatiales et des pas de temps variables. L'effet de quadrature num'erique est accentu'e. 1. Introduction. In this paper we consider the numerical solution of the semilinear parabolic equation (1.1) u t \Gamma \Deltau = f(x; t; u); in\Omega \Theta (0; t ); u = 0; on @\Omega \Theta (0; t ); u(\Delta; 0) = u 0 ; in\Omega ; by using a finite element method w..