Discontinuous Galerkin methods on hp-anisotropic meshes I: a priori error analysis

Abstract

In this article we consider the a priori error analysis of hp-version interior penalty discontinuous Galerkin methods for second-order partial differential equations with non-negative characteristic form under weak assumptions on the local mesh design and the local finite element spaces employed. 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. To this end, we prove goal-oriented a priori hp-error bounds for linear target functionals of the solution on (possibly) anisotropic computational meshes with anisotropic tensor-product polynomial basis functions. The theoretical results are illustrated by a series of numerical experiments.