The discontinuous Galerkin finite element method (DGFEM) using $h$ and $p$ refinement is applied to symmetric elliptic boundary value problems. As example the Poisson equation in a polygonal domain is considered either in two or three space dimensions. The authors focus on the $p$-refinement and investigate the question whether the class of non-overlapping Schwarz preconditioners introduced for the case of $h$-refinement can be extended to $p$- and $hp$-refinement versions of the DGFEM, while preserving their efficiency. Spectral bounds for the preconditioned stiffness matrix are shown to be of order $p^2$ for $p$-DGFEM. Combining these results with those known for $h$-DGFEM, spectral bounds of order $p^2H/h$ are proven for $hp$-DGFEM. Finally, numerical results for the Poisson equation on the unit square illustrate the theoretical estimates.
Reviewer: Kai Schneider (Marseille)