On the Acoustic Single Layer Potential: Stabilization and Fourier Analysis

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 R3. 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. DOI: https://doi.org/10.1137/040615110 Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-21601 Originally published at: Buffa, A; Sauter, S (2006). On the acoustic single layer potential: stabilization and Fourier analysis. SIAM Journal on Scientific Computing (SISC), 28(5):1974-1999. DOI: https://doi.org/10.1137/040615110 SIAM J. SCI. COMPUT. c © 2006 Society for Industrial and Applied Mathematics Vol. 28, No. 5, pp. 1974–1999 ON THE ACOUSTIC SINGLE LAYER POTENTIAL: STABILIZATION AND FOURIER ANALYSIS∗ A. BUFFA† AND S. SAUTER‡ 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 R3. 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. 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 R3. 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.