Applying GMRES to the Helmholtz equation with strong trapping: how does the number of iterations depend on the frequency?

Pierre Marchand; Jeffrey Galkowski; Alastair Spence; Euan A. Spence

doi:10.1007/s10444-022-09931-9

Applying GMRES to the Helmholtz equation with strong trapping: how does the number of iterations depend on the frequency?

Pierre Marchand, Jeffrey Galkowski, Alastair Spence, Euan A. Spence

University College London

Research output: Contribution to journal › Article › peer-review

4 Citations (SciVal)

Abstract

We consider GMRES applied to discretisations of the high-frequency Helmholtz equation with strong trapping; recall that in this situation the problem is exponentially ill-conditioned through an increasing sequence of frequencies. Our main focus is on boundary-integral-equation formulations of the exterior Dirichlet and Neumann obstacle problems in 2- and 3-d. Under certain assumptions about the distribution of the eigenvalues of the integral operators, we prove upper bounds on how the number of GMRES iterations grows with the frequency; we then investigate numerically the sharpness (in terms of dependence on frequency) of both our bounds and various quantities entering our bounds. This paper is therefore the first comprehensive study of the frequency-dependence of the number of GMRES iterations for Helmholtz boundary-integral equations under trapping.

Original language	English
Article number	37
Number of pages	63
Journal	Advances in Computational Mathematics
Volume	48
Issue number	4
Early online date	4 Jun 2022
DOIs	https://doi.org/10.1007/s10444-022-09931-9
Publication status	Published - 31 Aug 2022

Bibliographical note

Funding Information:
EAS gratefully acknowledges discussions with Alex Barnett (Flatiron Institute) that started his interest in eigenvalues of discretisations of the Helmholtz equation under strong trapping. In addition, all the authors thank Barnett for giving them insightful comments on an earlier version of this paper. PM thanks Pierre Jolivet (Institut de Recherche en Informatique de Toulouse, CNRS) and Pierre-Henri Tournier (Sorbonne Université, CNRS) for their help with the software FreeFEM. The authors thank the referees for their careful reading of the paper and numerous suggestions for improvement. This research made use of the Balena High Performance Computing (HPC) Service at the University of Bath. PM and EAS were supported by EPSRC grant EP/R005591/1.

Funding Information:
EAS gratefully acknowledges discussions with Alex Barnett (Flatiron Institute) that started his interest in eigenvalues of discretisations of the Helmholtz equation under strong trapping. In addition, all the authors thank Barnett for giving them insightful comments on an earlier version of this paper. PM thanks Pierre Jolivet (Institut de Recherche en Informatique de Toulouse, CNRS) and Pierre-Henri Tournier (Sorbonne Université, CNRS) for their help with the software FreeFEM. The authors thank the referees for their careful reading of the paper and numerous suggestions for improvement. This research made use of the Balena High Performance Computing (HPC) Service at the University of Bath. PM and EAS were supported by EPSRC grant EP/R005591/1.

Publisher Copyright:
© 2022, The Author(s).

Keywords

GMRES
Helmholtz equation
High frequency
Trapping

ASJC Scopus subject areas

Computational Mathematics
Applied Mathematics

Access to Document

10.1007/s10444-022-09931-9Licence: CC BY

Cite this

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title = "Applying GMRES to the Helmholtz equation with strong trapping: how does the number of iterations depend on the frequency?",

abstract = "We consider GMRES applied to discretisations of the high-frequency Helmholtz equation with strong trapping; recall that in this situation the problem is exponentially ill-conditioned through an increasing sequence of frequencies. Our main focus is on boundary-integral-equation formulations of the exterior Dirichlet and Neumann obstacle problems in 2- and 3-d. Under certain assumptions about the distribution of the eigenvalues of the integral operators, we prove upper bounds on how the number of GMRES iterations grows with the frequency; we then investigate numerically the sharpness (in terms of dependence on frequency) of both our bounds and various quantities entering our bounds. This paper is therefore the first comprehensive study of the frequency-dependence of the number of GMRES iterations for Helmholtz boundary-integral equations under trapping.",

keywords = "GMRES, Helmholtz equation, High frequency, Trapping",

author = "Pierre Marchand and Jeffrey Galkowski and Alastair Spence and Spence, {Euan A.}",

note = "Funding Information: EAS gratefully acknowledges discussions with Alex Barnett (Flatiron Institute) that started his interest in eigenvalues of discretisations of the Helmholtz equation under strong trapping. In addition, all the authors thank Barnett for giving them insightful comments on an earlier version of this paper. PM thanks Pierre Jolivet (Institut de Recherche en Informatique de Toulouse, CNRS) and Pierre-Henri Tournier (Sorbonne Universit{\'e}, CNRS) for their help with the software FreeFEM. The authors thank the referees for their careful reading of the paper and numerous suggestions for improvement. This research made use of the Balena High Performance Computing (HPC) Service at the University of Bath. PM and EAS were supported by EPSRC grant EP/R005591/1. Funding Information: EAS gratefully acknowledges discussions with Alex Barnett (Flatiron Institute) that started his interest in eigenvalues of discretisations of the Helmholtz equation under strong trapping. In addition, all the authors thank Barnett for giving them insightful comments on an earlier version of this paper. PM thanks Pierre Jolivet (Institut de Recherche en Informatique de Toulouse, CNRS) and Pierre-Henri Tournier (Sorbonne Universit{\'e}, CNRS) for their help with the software FreeFEM. The authors thank the referees for their careful reading of the paper and numerous suggestions for improvement. This research made use of the Balena High Performance Computing (HPC) Service at the University of Bath. PM and EAS were supported by EPSRC grant EP/R005591/1. Publisher Copyright: {\textcopyright} 2022, The Author(s).",

year = "2022",

month = aug,

day = "31",

doi = "10.1007/s10444-022-09931-9",

language = "English",

volume = "48",

journal = "Advances in Computational Mathematics",

issn = "1019-7168",