Fast Solver for Quasi-Periodic 2D-Helmholtz Scattering in Layered Media

TL;DR

This paper introduces a fast spectral Galerkin method for solving 2D Helmholtz scattering problems in layered media, achieving high accuracy and efficiency in modeling periodic structures.

Contribution

The paper develops a novel spectral Galerkin scheme with proven super-algebraic convergence for layered Helmholtz problems, excluding Rayleigh-Wood anomalies.

Findings

Super-algebraic error convergence demonstrated

Method performs competitively with Nyström methods

Validated through multiple numerical examples

Abstract

We present a fast spectral Galerkin scheme for the discretization of boundary integral equations arising from two-dimensional Helmholtz transmission problems in multi-layered periodic structures or gratings. Employing suitably parametrized Fourier basis and excluding Rayleigh-Wood anomalies, we rigorously establish the well-posedness of both continuous and discrete problems, and prove super-algebraic error convergence rates for the proposed scheme. Through several numerical examples, we confirm our findings and show performances competitive to those attained via Nystr\"om methods.