A Boundary Integral Method for Generalized Impedance Boundary Conditions in 2D

Generalized impedance boundary conditions for the Helmholtz equation arise in a number of acoustic and electromagnetic models that approximate boundary effects, including thin coatings and the viscous and thermal losses that can occur in boundary layers. Here we present a well-conditioned numerical method for this class of boundary conditions based on a boundary integral re-formulation of the equations. The formulation applies a combined layer representation together with a surface preconditioner to obtain an integral equation of the second kind. Additionally, by using appropriate image sources, we construct well-conditioned representations of impedance-to-impedance maps for subdomains carrying these boundary conditions on part of their boundary, so that the method can be used within a domain-decomposition framework. Several numerical examples in geometries inspired by acoustic applications demonstrate the efficacy of the method.