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

TL;DR

This paper presents a new boundary integral equation method using window Green functions for accurately and efficiently solving wave scattering problems involving periodic arrays of 2D obstacles, including at challenging spectral points.

Contribution

It introduces a robust second-kind BIE formulation with a window Green function approach that achieves super-algebraic convergence for periodic wave scattering problems.

Findings

Achieves super-algebraic convergence including at Rayleigh-Wood anomalies

Compatible with Nyström and boundary element discretizations

Demonstrates high accuracy and robustness through numerical examples

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