For most frequencies, strong trapping has a weak effect in frequency-domain scattering

David Lafontaine, Euan A. Spence, Jared Wunsch

Research output: Contribution to journalArticle

7 Citations (SciVal)

Abstract

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.

Original languageEnglish
Pages (from-to)2025-2063
Number of pages39
JournalCommunications on Pure and Applied Mathematics
Volume74
Issue number10
Early online date31 Jul 2020
DOIs
Publication statusPublished - 1 Oct 2021

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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