Eigenvalues of the truncated Helmholtz solution operator under strong trapping
| Autor: | Euan A. Spence, Pierre Marchand, Jeffrey Galkowski |
|---|---|
| Přispěvatelé: | University College of London [London] (UCL), University of Bath [Bath] |
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Helmholtz equation
Existential quantification 010103 numerical & computational mathematics Trapping 01 natural sciences Dirichlet distribution Computer Science::Robotics symbols.namesake Mathematics - Analysis of PDEs FOS: Mathematics Mathematics - Numerical Analysis 0101 mathematics 35J05 35P15 35B34 35P25 Eigenvalues and eigenvectors Mathematics Applied Mathematics Operator (physics) 010102 general mathematics Mathematical analysis Numerical Analysis (math.NA) Mathematics::Spectral Theory Computational Mathematics Helmholtz free energy Obstacle symbols Analysis [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] Analysis of PDEs (math.AP) |
| Popis: | 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]). |
| Databáze: | OpenAIRE |
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