Numerical investigation of the conditioning for plane wave discontinuous Galerkin methods

TL;DR

This paper investigates the conditioning of plane wave discontinuous Galerkin methods for the Helmholtz problem, showing how basis properties affect numerical stability and proposing orthogonalization to improve solver efficiency.

Contribution

It provides empirical analysis of the spectral condition number dependence on mesh size, wave number, and basis directions, and demonstrates a basis orthogonalization technique to enhance solver performance.

Findings

Condition number depends algebraically on mesh size and wave number.

Condition number depends exponentially on the number of plane wave directions.

Orthogonalization reduces GMRES iterations significantly.

Abstract

We present a numerical study to investigate the conditioning of the plane wave discontinuous Galerkin discretization of the Helmholtz problem. We provide empirical evidence that the spectral condition number of the plane wave basis on a single element depends algebraically on the mesh size and the wave number, and exponentially on the number of plane wave directions; we also test its dependence on the element shape. We show that the conditioning of the global system can be improved by orthogonalization of the local basis functions with the modified Gram-Schmidt algorithm, which results in significantly fewer GMRES iterations for solving the discrete problem iteratively.