DG and hp-DG for highly indefinite Helmholtz problems

We develop a stability and convergence theory for a Discontinuous Galerkin formulation (DG) of a highly indefinite Helmholtz problem in ${\rm I\!R}^d, d \in \{2,3\}$. We prove that the DG‐method admits a unique solution under much weaker conditions than for conventional Galerkin methods. It is shown that for the case of hp‐DGFEM the optimal convergence order estimate can be obtained under the conditions that $kh/\sqrt{p}$ is sufficiently small and the polynomial degree p is at least O(log k). (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)