Residual based adaptivity and PWDG methods for the Helmholtz equation

TL;DR

This paper investigates residual-based a posteriori error indicators for the PWDG method applied to the Helmholtz equation, proposing improvements in efficiency and reliability through new analysis and numerical validation.

Contribution

It introduces a new residual error indicator for the PWDG method with fixed plane wave directions, enhancing its reliability and efficiency.

Findings

The residual indicator is reliable but initially pessimistic.

A new analysis improves indicator efficiency.

Numerical tests confirm improved performance.

Abstract

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.