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Journal Articles SIAM Journal on Mathematical Analysis Year : 2021

Eigenvalues of the truncated Helmholtz solution operator under strong trapping

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). The significance of this result is 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]).

Dates and versions

hal-03114612 , version 1 (19-01-2021)

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Cite

Jeffrey Galkowski, Pierre Marchand, Euan A. Spence. Eigenvalues of the truncated Helmholtz solution operator under strong trapping. SIAM Journal on Mathematical Analysis, 2021, 53 (6), ⟨10.1137/21M1399658⟩. ⟨hal-03114612⟩

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