The authors are concerned with the convergence analysis of a discontinuous Galerkin (DG) method with plane wave basis functions for the solution of Helmholtz problems. The continuity of the solution at the interelement boundaries is enforced by Lagrange multipliers. The method was proposed by C. Farhat, I. Harari and U. Hetmaniuk [Comput. Methods Appl. Mech. Eng. 192, No.11-12, 1389–1419 (2003; Zbl 1027.76028)]. The analysis is done for a two-dimensional low-order element with four plane waves in a computational domain that can be decomposed into rectangular elements.
The authors prove that the hybrid problem obtained by applying the DG method in the continuous case is well-posed. They also prove existence and uniqueness of the solution for the discrete problem and the following a priori error estimate: for a solution in \(H^{5/3}(\Omega)\) the relative error in the \(L^2\)-norm is of order \(k(kh)^{4/3}\) and the \(H^1\)-seminorm of order \((kh)^{2/3}\) being \(k\) the wavenumber. An a posteriori error estimate is also derived.