Eigenvalues of the truncated Helmholtz solution operator under strong trapping

TL;DR

This paper demonstrates that strong trapping in the exterior Helmholtz problem leads to near-zero eigenvalues in the truncated variational formulation, impacting the convergence of iterative solvers like GMRES.

Contribution

It establishes a rigorous link between trapped rays and near-zero eigenvalues in the truncated Helmholtz operator, advancing understanding of solver behavior in high-frequency trapping scenarios.

Findings

Existence of near-zero eigenvalues under trapping conditions

Connection between quasimodes and eigenvalues in the truncated problem

Implications for iterative solver performance in Helmholtz computations

Abstract

For the Helmholtz equation posed in the exterior of a Dirichlet obstacle, we prove that if there exists a family of quasimodes (as is the case when the exterior of the obstacle has stable trapped rays), then there exist near-zero eigenvalues of the standard variational formulation of the exterior Dirichlet problem (recall that this formulation involves truncating the exterior domain and applying the exterior Dirichlet-to-Neumann map on the truncation boundary). Our motivation for proving this result is that a) the finite-element method for computing approximations to solutions of the Helmholtz equation is based on the standard variational formulation, and b) the location of eigenvalues, and especially near-zero ones, plays a key role in understanding how iterative solvers such as the generalised minimum residual method (GMRES) behave when used to solve linear systems, in particular those arising from the finite-element method. The result proved in this paper is thus the first step towards rigorously understanding how GMRES behaves when applied to discretisations of high-frequency Helmholtz problems under strong trapping (the subject of the companion paper [Marchand, Galkowski, Spence, Spence, 2021]).