High-frequency estimates on boundary integral operators for the Helmholtz exterior Neumann problem - Archive ouverte HAL Access content directly
Journal Articles Integral Equations and Operator Theory Year : 2022

High-frequency estimates on boundary integral operators for the Helmholtz exterior Neumann problem

Abstract

We study a commonly-used second-kind boundary-integral equation for solving the Helmholtz exterior Neumann problem at high frequency, where, writing $\Gamma$ for the boundary of the obstacle, the relevant integral operators map $L^2(\Gamma)$ 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 $\Gamma$ 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 $\Gamma$ and are observed to be sharp at least when $\Gamma$ is smooth with strictly-positive curvature. Together, these results give bounds on the condition number of the operator on $L^2(\Gamma)$; this is the first time $L^2(\Gamma)$ condition-number bounds have been proved for this operator for obstacles other than balls.

Dates and versions

hal-03810414 , version 1 (11-10-2022)

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Jeffrey Galkowski, Pierre Marchand, Euan A. Spence. High-frequency estimates on boundary integral operators for the Helmholtz exterior Neumann problem. Integral Equations and Operator Theory, 2022, 94 (4), ⟨10.1007/s00020-022-02715-2⟩. ⟨hal-03810414⟩

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