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For Most Frequencies, Strong Trapping Has a Weak Effect in Frequency‐Domain Scattering
Author(s) -
Lafontaine David,
Spence Euan A.,
Wunsch Jared
Publication year - 2021
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21932
Subject(s) - mathematics , infinity , helmholtz equation , operator (biology) , mathematical analysis , boundary (topology) , trapping , convergence (economics) , boundary value problem , domain (mathematical analysis) , helmholtz free energy , polynomial , scattering , sequence (biology) , physics , quantum mechanics , ecology , biochemistry , chemistry , repressor , biology , transcription factor , economics , gene , economic growth , genetics
It is well‐known 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 polynomial‐growth assumption on the solution operator (e.g. hp ‐finite elements, hp ‐boundary 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. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.