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.

中文翻译：

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三维双周期表面散射问题中完美匹配层的高阶收敛

《SIAM 数值分析杂志》，第 61 卷，第 6 期，第 2917-2939 页，2023 年 12 月。
摘要。完美匹配层（PML）是无界域中波散射截断中非常流行的工具。在 [SN Chandler-Wilde 和 P. Monk，Appl。数字。Math., 59 (2009), pp. 2131–2154]，作者提出了一个猜想，对于粗糙表面的散射问题，PML 相对于任何紧凑子集中的 PML 参数呈指数收敛。在作者之前的论文中[R. 张，SIAM J.数字。Math., 60 (2022), pp. 804–823]，当波数不是半整数时，这个结果已被证明适用于二维空间中的周期表面。在本文中，我们证明了该方法在三维双周期表面散射问题上具有高阶收敛速度。我们扩展了二维结果并证明当波数小于0.5时指数收敛仍然成立。对于更大的波数，虽然不再证明指数收敛，但我们能够证明 PML 方法的高阶收敛。