hp-Version discontinuous Galerkin methods with interior penalty for partial differential equations with nonnegative characteristic form.

Abstract

In this paper we consider the a posteriori and a priori analysis of hp-discontinuous Galerkin interior penalty methods for second-order partial differential equations with nonnegative characteristic form. In particular, we discuss the question of error estimation for linear target functionals, such as the outflow flux and the local average of the solution. Based on our a posteriori error bound we design and implement the corresponding adaptive algorithm to ensure reliable and efficient control of the error in the prescribed functional to within a given tolerance. This involves exploiting both local polynomial-degree variation and local mesh subdivision. The theoretical results are illustrated by a series of numerical experiments.