# 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, namely the Regularized Combined Field Integral Equation (RCFIE)1. Writing $\Gamma$ for the boundary of the obstacle, this integral operator map $L^2(\Gamma)$ to itself, contrary to its non-regularized version.

We prove new frequency-explicit bounds on the norms of both the RCFIE and its inverse. The bounds on the norm are valid for piecewise-smooth $\Gamma$ and are sharp, 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 curved.

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 balls2.

1. Bruno and T. Elling and C. Turc, Regularized integral equations and fast high-order solvers for sound-hard acoustic scattering problems. International Journal for Numerical Methods in Engineering, 2012. ↩︎

2. Y. Boubendir and C. Turc, Wave-number estimates for regularized combined field boundary integral operators in acoustic scattering problems with Neumann boundary conditions. IMA Journal of Numerical Analysis, 2013 ↩︎

Date
October 7, 2021
Location
Nice, France