General DG-Methods 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 , . 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 -version of the finite element method explicitly in terms of the mesh width , polynomial degree and wavenumber . It is shown that the optimal convergence order estimate is obtained under the conditions that is sufficiently small and the polynomial degree is at least . On regular meshes, the first condition is improved to the requirement that be sufficiently small.