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

Ralf Hiptmair; Andrea Moiola; Euan Spence

doi:10.1137/21M1447052

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

Ralf Hiptmair, Andrea Moiola, Euan Spence

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2 Citations (SciVal)

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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 language	English
Pages (from-to)	1446-1469
Number of pages	24
Journal	SIAM Journal on Applied Mathematics
Volume	82
Issue number	4
Early online date	11 Aug 2022
DOIs	https://doi.org/10.1137/21M1447052
Publication status	Published - 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 ([email protected]. ch). \ddagger Department of Mathematics, University of Pavia, Pavia, Italy ([email protected]). \S Department of Mathematical Sciences, University of Bath, Bath, UK ([email protected]).

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

Access to Document

10.1137/21M1447052

2109.08530.PDFAccepted author manuscript, 1.5 MBLicence: CC BY

https://arxiv.org/abs/2109.08530

Cite this

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title = "Spurious Quasi-Resonances in Boundary Integral Equations for the Helmholtz Transmission Problem",

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.",

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AU - Moiola, Andrea

AU - Spence, Euan

N1 - 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 ([email protected]. ch). \ddagger Department of Mathematics, University of Pavia, Pavia, Italy ([email protected]). \S Department of Mathematical Sciences, University of Bath, Bath, UK ([email protected]). Publisher Copyright: © 2022 Society for Industrial and Applied Mathematics.

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N2 - 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.

AB - 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.

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ER -

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

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Bibliographical note

Keywords

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At the interface between semiclassical analysis and numerical analysis of Wave propogation problems

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

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

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At the interface between semiclassical analysis and numerical analysis of Wave propogation problems

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