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Abstract
We study a commonlyused secondkind boundaryintegral 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 frequencyexplicit bounds on the norms of both the integral operator and its inverse. The bounds on the norm are valid for piecewisesmooth 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 strictlypositive 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) conditionnumber bounds have been proved for this operator for obstacles other than balls.
Original language  English 

Article number  36 
Journal  Integral Equations and Operator Theory 
Volume  94 
Issue number  4 
Early online date  17 Oct 2022 
DOIs  
Publication status  Published  31 Dec 2022 
Bibliographical note
EPSRC EP/R005591/1 AND EP/V001760/1Funding 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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 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