Wavenumber-explicit regularity estimates on the acoustic single- and double-layer operators

TL;DR

This paper establishes sharp, wavenumber-explicit bounds on the Helmholtz boundary-integral operators, which are crucial for analyzing the stability and accuracy of boundary element methods in high-frequency wave scattering problems.

Contribution

The paper provides the first sharp, wavenumber-explicit estimates for the Helmholtz single- and double-layer operators in boundary Sobolev spaces, advancing the theoretical understanding of high-frequency wave problems.

Findings

Sharp bounds on boundary-integral operators derived

Estimates depend explicitly on the wavenumber

Facilitates wavenumber-explicit analysis of numerical methods

Abstract

We prove new, sharp, wavenumber-explicit bounds on the norms of the Helmholtz single- and double-layer boundary-integral operators as mappings from $L^2(\partial \Omega)\rightarrow H^1(\partial \Omega)$ (where $\partial\Omega$ is the boundary of the obstacle). The new bounds are obtained using estimates on the restriction to the boundary of quasimodes of the Laplacian, building on recent work by the first author and collaborators. Our main motivation for considering these operators is that they appear in the standard second-kind boundary-integral formulations, posed in $L^2(\partial \Omega)$, of the exterior Dirichlet problem for the Helmholtz equation. Our new wavenumber-explicit $L^2(\partial \Omega)\rightarrow H^1(\partial \Omega)$ bounds can then be used in a wavenumber-explicit version of the classic compact-perturbation analysis of Galerkin discretisations of these second-kind equations; this is done in the companion paper [Galkowski, M\"uller, Spence, arXiv 1608.01035].