Discontinuous Galerkin method for the Helmholtz transmission problem in two-level homogeneous media

Abstract

In this paper, the discontinuous Galerkin (DG) method is developed and analyzed for solving the Helmholtz transmission problem (HTP) with the first order absorbing boundary condition in two-level homogeneous media. This whole domain is separated into two disjoint subdomains by an interface, where two types of transmission conditions are provided. The application of the DG method to the HTP gives the discrete formulation. A rigorous theoretical analysis demonstrates that the discrete formulation can retain absolute stability without any mesh constraint. We prove that the errors in $ H^{1} $ and $ L^{2} $ norms are bounded by $ C_{1}kh + C_{2}k^{4}h^{2} $ and $ C_{1}kh^{2} + C_{2}k^{3}h^{2} $, respectively, where $ C_1 $ and $ C_2 $ are positive constants independent of the wave number $ k $ and the mesh size $ h $. Numerical experiments are conducted to verify the accuracy of the theoretical results and the efficiency of the numerical method.