@article{cf3f9c38fd08452e8f48d7332c0821ad,

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.",

keywords = "Helmholtz equation, boundary integral equations, quasi-resonance, transmission problem",

author = "Ralf Hiptmair and Andrea Moiola and Euan Spence",

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: {\textcopyright} 2022 Society for Industrial and Applied Mathematics.",

year = "2022",

month = dec,

day = "31",

doi = "10.1137/21M1447052",

language = "English",

volume = "82",

pages = "1446--1469",

journal = "SIAM Journal on Applied Mathematics",

issn = "0036-1399",

publisher = "Society for Industrial and Applied Mathematics Publications",

number = "4",

}