On singular Galerkin discretizations for three models in high-frequency scattering

TL;DR

This paper demonstrates that for high-frequency scattering models, certain mesh constructions cause the Galerkin discretization matrices to become singular, showing that mesh resolution conditions are essential for stability.

Contribution

It constructs specific meshes leading to singular matrices in Galerkin discretizations, proving the necessity of resolution conditions for stability in high-frequency scattering models.

Findings

Discrete system matrices can be singular with certain meshes.

Resolution conditions are necessary for discrete stability.

Mesh design impacts well-posedness of discretized models.

Abstract

We consider three common mathematical models for time-harmonic high frequency scattering: the Helmholtz equation in two and three spatial dimensions, a transverse magnetic problem in two dimensions, and Maxwell's equation in three dimensions with dissipative boundary conditions such that the continuous problem is well posed. In this paper, we construct meshes for popular (low order) Galerkin finite element discretizations such that the discrete system matrix becomes singular and the discrete problem is not well posed. This implies that a condition "the finite element space has to be sufficiently rich" in the form of a resolution condition - typically imposed for discrete well-posedness - is not an artifact from the proof by a compact perturbation argument but necessary for discrete stability of the Galerkin discretization.