Well-conditioned boundary integral equation formulations and Nystr\"om discretizations for the solution of Helmholtz problems with impedance boundary conditions in two-dimensional Lipschitz domains

TL;DR

This paper introduces a regularization strategy for boundary integral equations that results in well-conditioned formulations for Helmholtz problems with impedance boundary conditions in 2D Lipschitz domains, improving numerical stability and efficiency.

Contribution

The paper develops a new regularization approach for boundary integral equations that enhances conditioning and enables effective Nyström discretizations for Helmholtz problems with impedance conditions.

Findings

Regularized formulations are well-conditioned for Helmholtz impedance problems.

Nyström discretization with graded meshes achieves accurate solutions.

Method accelerates convergence in domain decomposition methods.

Abstract

We present a regularization strategy that leads to well-conditioned boundary integral equation formulations of Helmholtz equations with impedance boundary conditions in two-dimensional Lipschitz domains. We consider both the case of classical impedance boundary conditions, as well as the case of transmission impedance conditions wherein the impedances are certain coercive operators. The latter type of problems is instrumental in the speed up of the convergence of Domain Decomposition Methods for Helmholtz problems. Our regularized formulations use as unknowns the Dirichlet traces of the solution on the boundary of the domain. Taking advantage of the increased regularity of the unknowns in our formulations, we show through a variety of numerical results that a graded-mesh based Nystr\"om discretization of these regularized formulations leads to efficient and accurate solutions of interior and exterior Helmholtz problems with impedance boundary conditions.