Fast convergent PML method for scattering with periodic surfaces: the exceptional case

TL;DR

This paper investigates the exceptional cases in PML methods for 2D scattering problems with periodic surfaces, proving fast convergence even when wave numbers are half integers, supported by numerical evidence.

Contribution

The paper extends previous convergence results to include the exceptional cases where wave numbers are half integers, providing theoretical proof and numerical validation.

Findings

Fast convergence is observed even in exceptional cases.

Theoretical proof confirms numerical results.

Numerical examples support the extended convergence analysis.

Abstract

In the author's previous paper (Zhang et al. 2022), exponential convergence was proved for the perfectly matched layers (PML) approximation of scattering problems with periodic surfaces in 2D. However, due to the overlapping of singularities, an exceptional case, i.e., when the wave number is a half integer, has to be excluded in the proof. However, numerical results for these cases still have fast convergence rate and this motivates us to go deeper into these cases. In this paper, we focus on these cases and prove that the fast convergence result for the discretized form. Numerical examples are also presented to support our theoretical results.