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

Research output: Contribution to journalArticlepeer-review

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 languageEnglish
Article number37
Number of pages63
JournalAdvances in Computational Mathematics
Volume48
Issue number4
Early online date4 Jun 2022
DOIs
Publication statusPublished - 31 Aug 2022

Keywords

  • GMRES
  • Helmholtz equation
  • High frequency
  • Trapping

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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