Exponential Convergence of Perfectly Matched Layers for Scattering Problems with Periodic Surfaces

Abstract

The main task of this paper is to prove that the perfectly matched layers (PML) method converges exponentially with respect to the PML parameter for scattering problems with periodic surfaces. In [S. N. Chandler-Wilde and P. Monk, Appl. Numer. Math., 59 (2009), pp. 2131--2154], a linear convergence is proved for the PML method for scattering problems with rough surfaces. At the end of that paper, three important questions are asked, and the third question is whether exponential convergence holds locally. In our paper, we answer this open question for a special case, when the rough surface is actually periodic. Due to technical reasons, we have to exclude all the wavenumbers which are half integers. The main idea of the proof is to apply the Floquet--Bloch transform to rewrite the problem as an equivalent family of quasi-periodic problems, and then study the analytic extension of the quasi-periodic problems with respect to the Floquet--Bloch parameters. Then the Cauchy integral formula is applied to avoid linear convergent points. Finally, the exponential convergence is proved from the inverse Floquet--Bloch transform. Numerical results are also presented at the end of this paper.