Convergence Analysis of a Discontinuous Galerkin Method with Plane Waves and Lagrange Multipliers for the Solution of Helmholtz Problems

We analyze the convergence of a discontinuous Galerkin method (DGM) with plane waves and Lagrange multipliers that was recently proposed by Farhat et al. [3] for solving twodimensional Helmholtz problems at relatively high wave numbers. We prove that the underlying hybrid variational formulation is well-posed. We also present various a priori error estimates that establish the convergence and order of accuracy of the simplest element associated with this method. We prove that, for k (k h) 2 3 sufficiently small, the relative error in the L-norm (resp. in the H semi-norm) is of order k (k h) 4 3 (resp. of order (k h) 2 3 ) for a solution being in H 5 3 (Ω). In addition, we establish an a posteriori error estimate that can be used as a practical error indicator when refining the partition of the computational domain.