High-Frequency Estimates on Boundary Integral Operators for the Helmholtz Exterior Neumann Problem

Euan Spence, J. Galkowski, Pierre Marchand

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Abstract

We study a commonly-used second-kind boundary-integral equation for solving the Helmholtz exterior Neumann problem at high frequency, where, writing Γ for the boundary of the obstacle, the relevant integral operators map L 2(Γ) to itself. We prove new frequency-explicit bounds on the norms of both the integral operator and its inverse. The bounds on the norm are valid for piecewise-smooth Γ and are sharp up to factors of log k (where k is the wavenumber), and the bounds on the norm of the inverse are valid for smooth Γ and are observed to be sharp at least when Γ is smooth with strictly-positive curvature. Together, these results give bounds on the condition number of the operator on L 2(Γ) ; this is the first time L 2(Γ) condition-number bounds have been proved for this operator for obstacles other than balls.

Original languageEnglish
Article number36
JournalIntegral Equations and Operator Theory
Volume94
Issue number4
Early online date17 Oct 2022
DOIs
Publication statusPublished - 31 Dec 2022

Bibliographical note

EPSRC EP/R005591/1 AND EP/V001760/1

Funding Information:
EAS thanks Zydrunas Gimbutas (NIST) and Leslie Greengard (New York University and Flatiron Institute) for useful discussions about the operators B k , η , R and B k , η , R ′during a visit to New York University in November 2012. The authors thank the anonymous referee for their careful reading of the paper and constructive comments. PM and EAS were supported by EPSRC Grant EP/R005591/1 and JG by EPSRC Grant EP/V001760/1. The authors have no competing interests to declare that are relevant to the content of this article.

Funding Information:
EAS thanks Zydrunas Gimbutas (NIST) and Leslie Greengard (New York University and Flatiron Institute) for useful discussions about the operators and during a visit to New York University in November 2012. The authors thank the anonymous referee for their careful reading of the paper and constructive comments. PM and EAS were supported by EPSRC Grant EP/R005591/1 and JG by EPSRC Grant EP/V001760/1. The authors have no competing interests to declare that are relevant to the content of this article.

Keywords

  • Boundary integral equation
  • Helmholtz
  • High frequency
  • Neumann problem
  • Pseudodifferential operator
  • Semiclassical analysis

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory

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