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

Abstract

We present a regularization strategy that leads to well-conditioned boundary integral equation formulations of Helm\-holtz equations with impedance boundary conditions in two-dimensional Lipschitz domains. We consider both the case of classical impedance boundary conditions, as well as that of transmission impedance conditions wherein the impedances are certain coercive operators. The latter type of problem 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.