Windowed Green function method for wave scattering by periodic arrays of 2D obstacles

Abstract: This paper introduces a novel boundary integral equation (BIE) method for the numerical solution of problems of planewave scattering by periodic line arrays of two-dimensional penetrable obstacles. Our approach is built upon a direct BIE formulation that leverages the simplicity of the free-space Green function but in turn entails evaluation of integrals over the unit-cell boundaries. Such integrals are here treated via the window Green function method. The windowing approximation together with a finite-rank operator correction -- used to properly impose the Rayleigh radiation condition -- yield a robust second-kind BIE that produces super-algebraically convergent solutions throughout the spectrum, including at the challenging Rayleigh-Wood anomalies. The corrected windowed BIE can be discretized by means of off-the-shelf Nystr\"om and boundary element methods, and it leads to linear systems suitable for iterative linear-algebra solvers as well as standard fast matrix-vector product algorithms. A variety of numerical examples demonstrate the accuracy and robustness of the proposed methodology