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Abstract
We consider the Helmholtz transmission problem with piecewiseconstant 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 quasiresonances. We give a rigorous explanation of why and when spurious quasiresonances occur, and propose modified boundary integral equations that are not affected by them.
Original language  English 

Pages (fromto)  14461469 
Number of pages  24 
Journal  SIAM Journal on Applied Mathematics 
Volume  82 
Issue number  4 
Early online date  11 Aug 2022 
DOIs  
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 GNCSINDAM, from PRIN project ``NAFROMPDEs"" and from MIUR through the ``Dipartimenti di Eccellenza"" Program (20182022), 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
 quasiresonance
 transmission problem
ASJC Scopus subject areas
 Applied Mathematics
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 1 Finished

At the interface between semiclassical analysis and numerical analysis of Wave propogation problems
Engineering and Physical Sciences Research Council
1/10/17 → 30/09/23
Project: Research council