A new frequency-uniform coercive boundary integral equation for acoustic scattering

Euan A Spence, S N Chandler-Wilde, Ivan G Graham, Valery P Smyshlyaev

Research output: Contribution to journalArticle

28 Citations (Scopus)

Abstract

A new boundary integral operator is introduced for the solution of the soundsoft acoustic scattering problem, i.e., for the exterior problem for the Helmholtz equation with Dirichlet boundary conditions. We prove that this integral operator is coercive in L2(Γ) (where Γ is the surface of the scatterer) for all Lipschitz star-shaped domains. Moreover, the coercivity is uniform in the wavenumber k = ω/c, where ω is the frequency and c is the speed of sound. The new boundary integral operator, which we call the “star-combined” potential operator, is a slight modification of the standard combined potential operator, and is shown to be as easy to implement as the standard one. Additionally, to the authors' knowledge, it is the only second-kind integral operator for which convergence of the Galerkin method in L2(Γ) is proved without smoothness assumptions on Γ except that it is Lipschitz. The coercivity of the star-combined operator implies frequency-explicit error bounds for the Galerkin method for any approximation space. In particular, these error estimates apply to several hybrid asymptoticnumerical methods developed recently that provide robust approximations in the high-frequency case. The proof of coercivity of the star-combined operator critically relies on an identity first introduced by Morawetz and Ludwig in 1968, supplemented further by more recent harmonic analysis techniques for Lipschitz domains. © 2011 Wiley Periodicals, Inc.
Original language English 1384-1415 32 Communications on Pure and Applied Mathematics 64 10 30 May 2011 https://doi.org/10.1002/cpa.20378 Published - Oct 2011

• Projects

Boundary Integral Equation Methods for HF Scattering Problems

Graham, I. & Smyshlyaev, V. P.

Engineering and Physical Sciences Research Council

24/03/0923/09/12

Project: Research council