hp-Finite Element Methods for Hyperbolic Problems

Abstract

Presented as Invited Lecture at the 10th Conference on the Mathematics of Finite Elements and Applications, Brunel University, June 1999.

This paper is devoted to the a priori and a posteriori error analysis of the hp-version of the discontinuous Galerkin finite element method for partial differential equations of hyperbolic and nearly-hyperbolic character. We consider second-order partial differential equations with nonnegative characteristic form, a large class of equations which includes convection-dominated diffusion problems, degenerate elliptic equations and second-order problems of mixed elliptic-hyperbolic-parabolic type. An a priori error bound is derived for the method in the so-called DG-norm which is optimal in terms of the mesh size h; the error bound is either 1 degree or 1/2 degree below optimal in terms of the polynomial degree p, depending on whether the problem is convection-dominated, or diffusion-dominated, respectively. In the case of a first-order hyperbolic equation the error bound is hp-optimal in the DG-norm. For first-order hyperbolic problems, we also discuss the a posteriori error analysis of the method and implement the resulting bounds into an hp-adaptive algorithm. The theoretical findings are illustrated by numerical experiments.