Higher-Order Convergence of Perfectly Matched Layers in Three-Dimensional Biperiodic Surface Scattering Problems

Abstract

SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2917-2939, December 2023.
Abstract. The perfectly matched layer (PML) is a very popular tool in the truncation of wave scattering in unbounded domains. In [S. N. Chandler-Wilde and P. Monk, Appl. Numer. Math., 59 (2009), pp. 2131–2154], the author proposed a conjecture that for scattering problems with rough surfaces, the PML converges exponentially with respect to the PML parameter in any compact subset. In the author’s previous paper [R. Zhang, SIAM J. Numer. Math., 60 (2022), pp. 804–823], this result has been proved for periodic surfaces in two-dimensional spaces, when the wave number is not a half-integer. In this paper, we prove that the method has a high-order convergence rate in the three-dimensional biperiodic surface scattering problems. We extend the two-dimensional results and prove that the exponential convergence still holds when the wave number is smaller than 0.5. For larger wave numbers, although exponential convergence is no longer proved, we are able to prove a higher-order convergence for the PML method.