General DG-methods for highly indefinite Helmholtz problems

Abstract

We develop a stability and convergence theory for a Discontinuous Galerkin formulation (DG) of a highly indefinite Helmholtz problem in Rd, d ∈ {1,2,3}. The theory covers conforming as well as non-conforming generalized finite element methods. In contrast to conventional Galerkin methods where a minimal resolution condition is necessary to guarantee the unique solvability, it is proved that the DG-method admits a unique solution under much weaker conditions. As an application we present the error analysis for the hp -version of the finite element method explicitly in terms of the mesh width h, polynomial degree p and wavenumber k. It is shown that the optimal convergence order estimate is obtained under the conditions that kh √p is sufficiently small and the polynomial degree p is at least O(log k). On regular meshes, the first condition is improved to the requirement that kh/p be sufficiently small.