Domain decomposition for quasi-periodic scattering by layered media via robust boundary-integral equations at all frequencies

TL;DR

This paper introduces a robust domain decomposition method for scalar wave scattering in layered media, effectively handling all frequencies including Wood frequencies through advanced boundary-integral formulations and shifted Green functions.

Contribution

The paper presents a novel DDM approach using stable boundary-integral equations and shifted Green functions to ensure convergence at all frequencies, including problematic Wood frequencies.

Findings

Stable computations of RtR operators across all frequencies.

Successful application to multi-layered media in 2D and 3D.

Efficient solution of large-layer systems with recursive Schur complements.

Abstract

We develop a non-overlapping domain decomposition method (DDM) for scalar wave scattering by periodic layered media. Our approach relies on robust boundary-integral equation formulations of Robin-to-Robin (RtR) maps throughout the frequency spectrum, including cutoff (or Wood) frequencies. We overcome the obstacle of non-convergent quasi-periodic Green functions at these frequencies by incorporating newly introduced shifted Green functions. Using the latter in the definition of quasi-periodic boundary-integral operators leads to rigorously stable computations of RtR operators. We develop Nystr\"om discretizations of the RtR maps that rely on trigonometric interpolation, singularity resolution, and fast convergent windowed quasi-periodic Green functions. We solve the tridiagonal DDM system via recursive Schur complements and establish rigorously that this procedure is always completed successfully. We present a variety of numerical results concerning Wood frequencies in two and three dimensions as well as large numbers of layers.