Hp-finite Element Methods for Hyperbolic Problems A

This paper is devoted to the a priori and a posteriori error analysis of the hp-version of the discontinuous Galerkin nite element method for partial differential equations of hyperbolic and nearly-hyperbolic character. We consider second-order partial diierential equations with nonnegative characteristic form, a large class of equations which includes convection-dominated diiusion 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 diiusion-dominated, respectively. In the case of a rst-order hyperbolic equation the error bound is hp-optimal in the DG-norm. For rst-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 ndings are illustrated by numerical experiments.