hp–Version Interior Penalty Discontinuous Galerkin Finite

We consider the hp-version interior penalty discontinuous Galerkin finite element method (hp-DGFEM) for linear second-order elliptic reaction-diffusion-advection equations with mixed Dirichlet and Neumann boundary conditions. Our main concern is the extension of the error analysis of the hp-DGFEM to the case when anisotropic (shape-irregular) elements and anisotropic polynomial degrees are used. For this purpose, extensions of well known approximation theory results are derived. In particular, new error bounds for the approximation error of the L2- and H1-projection operators are presented, as well as generalizations of existing inverse inequalities to the anisotropic setting. Equipped with these theoretical developments, we derive general error bounds for the hp-DGFEM on anisotropic meshes, and anisotropic polynomial degrees. Moreover, an improved choice for the (user-defined) discontinuity-penalisation parameter of the method is proposed, which takes into account the anisotropy of the mesh. These results collapse to previously known ones when applied to problems on shape-regular elements. The theoretical findings are justified by numerical experiments, indicating that the use of anisotropic elements, together with our newly suggested choice of the discontinuity-penalisation parameter, improves the stability, the accuracy and the efficiency of the method.