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

摘要

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