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 $$\Gamma $$ Γ for the boundary of the obstacle, the relevant integral operators map $$L^2(\Gamma )$$ 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 $$\Gamma $$ Γ and are sharp up to factors of $$\log k$$ log k (where k is the wavenumber), and the bounds on the norm of the inverse are valid for smooth $$\Gamma $$ Γ and are observed to be sharp at least when $$\Gamma $$ Γ is smooth with strictly-positive curvature. Together, these results give bounds on the condition number of the operator on $$L^2(\Gamma )$$ L 2 ( Γ ) ; this is the first time $$L^2(\Gamma )$$ L 2 ( Γ ) condition-number bounds have been proved for this operator for obstacles other than balls.

Does the Helmholtz Boundary Element Method Suffer from the Pollution Effect?

“…Instead, each is the sum of a semiclassical pseudodifferential operator and an operator acting only on frequencies \leq k that transports mass between points on the boundary connected by rays; this decomposition was recently established in[38, Chapter 4], with[38, Lemma 4.27] explicitly writing out the decomposition when \Gamma is curved. The estimates on boundary layer operators at high frequency in Theorem 5.1 were then proved using the ideas from[38, Chapter 4] in[42, Theorem 4.3].…”