On stability of discretizations of the Helmholtz equation (extended version)

TL;DR

This paper analyzes the stability of various discretization methods for the Helmholtz equation at high frequencies, providing explicit stability and convergence results for finite element, least squares, and DG schemes.

Contribution

It develops a comprehensive $k$-explicit stability and convergence theory for high order finite element methods, including mesh refinement strategies and stability analysis of DG and least squares methods.

Findings

Finite element methods achieve quasi-optimality under specific mesh and polynomial degree conditions.

Stability of least squares and DG methods is reviewed and characterized.

Mesh refinement near vertices improves stability and accuracy.

Abstract

We review the stability properties of several discretizations of the Helmholtz equation at large wavenumbers. For a model problem in a polygon, a complete $k$-explicit stability (including $k$-explicit stability of the continuous problem) and convergence theory for high order finite element methods is developed. In particular, quasi-optimality is shown for a fixed number of degrees of freedom per wavelength if the mesh size $h$ and the approximation order $p$ are selected such that $kh/p$ is sufficiently small and $p = O(\log k)$, and, additionally, appropriate mesh refinement is used near the vertices. We also review the stability properties of two classes of numerical schemes that use piecewise solutions of the homogeneous Helmholtz equation, namely, Least Squares methods and Discontinuous Galerkin (DG) methods. The latter includes the Ultra Weak Variational Formulation.