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

Abstract

We analyze the convergence of a discontinuous Galerkin method (DGM) with plane waves and Lagrange multipliers that was recently proposed by Farhat, Harari, and Hetmaniuk [Comput. Methods Appl. Mech. Engrg., 192 (2003), pp. 1389–1419] for solving two-dimensional Helmholtz problems at relatively high wavenumbers. 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-1.5pt element associated with this method. We prove that, for $k\,(k\,h)^{\frac{2}{3}}$ sufficiently-1.5pt small, the relative error in the $L^{2}$-norm (resp. in the $H^1$ seminorm) is of order $k\,(k\,h)^{\frac{4}{3}}$ (resp. of order $(k\,h)^{\frac{2}{3}}$) for a solution being in $H^{\frac{5}{3}}(\Omega)$. 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.