Spurious Quasi-Resonances in Boundary Integral Equations for the Helmholtz Transmission Problem

Ralf Hiptmair, Andrea Moiola, Euan Spence

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Abstract

We consider the Helmholtz transmission problem with piecewise-constant material coefficients, and the standard associated direct boundary integral equations. For certain coefficients and geometries, the norms of the inverses of the boundary integral operators grow rapidly through an increasing sequence of frequencies, even though this is not the case for the solution operator of the transmission problem; we call this phenomenon that of spurious quasi-resonances. We give a rigorous explanation of why and when spurious quasi-resonances occur, and propose modified boundary integral equations that are not affected by them.
Original languageEnglish
Pages (from-to)1446-1469
Number of pages24
JournalSIAM Journal on Applied Mathematics
Volume82
Issue number4
Early online date11 Aug 2022
DOIs
Publication statusPublished - 31 Dec 2022

Bibliographical note

Funding Information:
\ast Received by the editors September 17, 2021; accepted for publication (in revised form) April 26, 2022; published electronically August 11, 2022. https://doi.org/10.1137/21M1447052 Funding: The work of the second author was supported by GNCS-INDAM, from PRIN project ``NAFROM-PDEs"" and from MIUR through the ``Dipartimenti di Eccellenza"" Program (2018-2022), Department of Mathematics, University of Pavia. The work of the third author was supported by EPSRC grant EP/R005591/1. \dagger Seminar for Applied Mathematics, ETH Zu\"rich, Zu\"rich, Switzerland (hiptmair@sam.math.ethz. ch). \ddagger Department of Mathematics, University of Pavia, Pavia, Italy (andrea.moiola@unipv.it). \S Department of Mathematical Sciences, University of Bath, Bath, UK (E.A.Spence@bath.ac.uk).

Publisher Copyright:
© 2022 Society for Industrial and Applied Mathematics.

Keywords

  • Helmholtz equation
  • boundary integral equations
  • quasi-resonance
  • transmission problem

ASJC Scopus subject areas

  • Applied Mathematics

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