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Abstract
It is wellknown that when the geometry and/or coefficients allow stable trapped rays, the outgoing solution operator of the Helmholtz equation grows exponentially through a sequence of real frequencies tending to infinity. In this paper we show that, even in the presence of the strongest possible trapping, if a set of frequencies of arbitrarily small measure is excluded, the Helmholtz solution operator grows at most polynomially as the frequency tends to infinity. One significant application of this result is in the convergence analysis of several numerical methods for solving the Helmholtz equation at high frequency that are based on a polynomialgrowth assumption on the solution operator (e.g. hpfinite elements, hpboundary elements, and certain multiscale methods). The result of this paper shows that this assumption holds, even in the presence of the strongest possible trapping, for most frequencies.
Original language  English 

Pages (fromto)  20252063 
Number of pages  39 
Journal  Communications on Pure and Applied Mathematics 
Volume  74 
Issue number  10 
Early online date  31 Jul 2020 
DOIs  
Publication status  Published  1 Oct 2021 
Bibliographical note
Publisher Copyright:© 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.
ASJC Scopus subject areas
 General Mathematics
 Applied Mathematics
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Dive into the research topics of 'For most frequencies, strong trapping has a weak effect in frequencydomain scattering'. Together they form a unique fingerprint.Projects
 1 Finished

At the interface between semiclassical analysis and numerical analysis of Wave propogation problems
Spence, E. (PI)
Engineering and Physical Sciences Research Council
1/10/17 → 30/09/23
Project: Research council
Profiles

Euan Spence
 Department of Mathematical Sciences  Professor
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
Person: Research & Teaching