Well-posed boundary integral equation formulations and Nystr\"om discretizations for the solution of Helmholtz transmission problems in two-dimensional Lipschitz domains

TL;DR

This paper compares Nyström discretization solvers for various boundary integral formulations of Helmholtz transmission problems in Lipschitz domains, highlighting the advantages of newer formulations especially at high frequencies and contrasts.

Contribution

It introduces and analyzes the well-posedness of new boundary integral formulations and demonstrates their computational benefits over classical methods for Helmholtz transmission problems.

Findings

Formulations (3) and (4) are more computationally efficient.

New formulations are well-posed in Lipschitz domains.

Advantages are pronounced at high frequencies and contrasts.

Abstract

We present a comparison between the performance of solvers based on Nystr\"om discretizations of several well-posed boundary integral equation formulations of Helmholtz transmission problems in two-dimensional Lipschitz domains. Specifically, we focus on the following four classes of boundary integral formulations of Helmholtz transmission problems (1) the classical first kind integral equations for transmission problems, (2) the classical second kind integral equations for transmission problems, (3) the {\em single} integral equation formulations, and (4) certain direct counterparts of recently introduced Generalized Combined Source Integral Equations. The former two formulations were the only formulations whose well-posedness in Lipschitz domains was rigorously established. We establish the well-posedness of the latter two formulations in appropriate functional spaces of boundary traces of solutions of transmission Helmholtz problems in Lipschitz domains. We give ample numerical evidence that Nystr\"om solvers based on formulations (3) and (4) are computationally more advantageous than solvers based on the classical formulations (1) and (2), especially in the case of high-contrast transmission problems at high frequencies.