We present a study of two residual a posteriori error indicators for the Plane Wave Discontinuous Galerkin (PWDG) method for the Helmholtz equation. In particular we study the h-version of PWDG in which the number of plane wave directions per element is kept fixed. First we use a slight modification of the appropriate a priori analysis to determine a residual indicator. Numerical tests show that this is reliable but pessimistic in that the ratio between the true error and the indicator increases as the mesh is refined. We therefore introduce a new analysis based on the observation that sufficiently many plane waves can approximate piecewise linear functions as the mesh is refined. Numerical results demonstrate an improvement in the efficiency of the indicators.

We consider the numerical solution of an acoustic scattering problem by the Plane Wave Discontinuous Galerkin Method (PWDG) in the exterior of a bounded domain in R 2 . In order to apply the PWDG method, we introduce an artificial boundary to truncate the domain, and we impose a non-local Dirichlet-to-Neumann (DtN) boundary conditions on the artificial curve. To define the method, we introduce new consistent numerical fluxes that incorporate the truncated series of the DtN map. Error estimates with respect to the truncation order of the DtN map, and with respect to mesh width are derived. Numerical results suggest that the accuracy of the PWDG method for the acoustic scattering problem can be significantly improved by using DtN boundary conditions.

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We explain how to use smooth bivariate splines of arbitrary degree to solve the exterior Helmholtz equation based on a Perfectly Matched Layer (PML) technique. In a previous study (cf. [26]), it was shown that bivariate spline functions of high degree can approximate the solution of the bounded domain Helmholtz equation with an impedance boundary condition for large wave numbers k ∼ 1000. In this paper, we extend this study to the case of the Helmholtz equation in an unbounded domain. The PML is constructed using a complex stretching of the coordinates in a rectangular domain, resulting in a weighted Helmholtz equation with Dirichlet boundary conditions. The PML weights are also approximated by using spline functions. The computational algorithm developed in [26] is used to approximate the solution of the resulting weighted Helmholtz equation. Numerical results show that the PML formulation for the unbounded domain Helmholtz equation using bivariate splines is very effective over a range of examples.