{"title":"Higher-Order Convergence of Perfectly Matched Layers in Three-Dimensional Biperiodic Surface Scattering Problems","authors":"Ruming Zhang","doi":"10.1137/22m1532615","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2917-2939, December 2023. <br/> 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.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.8000,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Journal on Numerical Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/22m1532615","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
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.
SIAM数值分析学报，61卷，第6期，2917-2939页，2023年12月。摘要。完美匹配层(PML)是一种非常流行的截断无界域波散射的工具。在[S。N.钱德勒-王尔德和P.蒙克，苹果。号码。数学。， 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在任意紧子集中相对于PML参数呈指数收敛。在作者之前的论文中[R]。张siam J.数字。数学。[j]， 60 (2022)， pp. 804-823]，当波数不是半整数时，该结果已在二维空间中的周期曲面上得到证明。在本文中，我们证明了该方法在三维双周期表面散射问题中具有高阶收敛速度。推广了二维结果，证明了当波数小于0.5时，指数收敛性仍然成立。对于较大的波数，虽然不再证明指数收敛性，但我们能够证明PML方法的高阶收敛性。
期刊介绍：
SIAM Journal on Numerical Analysis (SINUM) contains research articles on the development and analysis of numerical methods. Topics include the rigorous study of convergence of algorithms, their accuracy, their stability, and their computational complexity. Also included are results in mathematical analysis that contribute to algorithm analysis, and computational results that demonstrate algorithm behavior and applicability.
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