On the acoustic single layer potential: stabilization and Fourier analysis

Abstract

In this paper, we propose a general approach for stabilizing the single layer potential for the Helmholtz boundary integral equation and prove its stability. We consider Galerkin boundary element discretizations and analyze their convergence. Furthermore, we derive quantitative error bounds for the Galerkin discretization which are explicit with respect to the mesh width and the wave number for the special case that the surface is the unit sphere in $\mathbb{R}^3$. We perform then a qualitative analysis which allows us to choose the stabilization such that the (negative) influence of the wave number in the stability and convergence estimates attains its minumum.