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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 language | English |
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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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Dive into the research topics of 'High-Frequency Estimates on Boundary Integral Operators for the Helmholtz Exterior Neumann Problem'. Together they form a unique fingerprint.Projects
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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