三维准周期光子晶体的无发散投影法

Zixuan Gao, Zhenli Xu, Zhiguo Yang
{"title":"三维准周期光子晶体的无发散投影法","authors":"Zixuan Gao, Zhenli Xu, Zhiguo Yang","doi":"arxiv-2409.05528","DOIUrl":null,"url":null,"abstract":"This paper presents a point-wise divergence-free projection method for\nnumerical approximations of photonic quasicrystals problems. The original\nthree-dimensional quasiperiodic Maxwell's system is transformed into a periodic\none in higher dimensions through a variable substitution involving the\nprojection matrix, such that periodic boundary condition can be readily\napplied. To deal with the intrinsic divergence-free constraint of the Maxwell's\nequations, we present a quasiperiodic de Rham complex and its associated\ncommuting diagram, based on which a point-wise divergence-free quasiperiodic\nFourier spectral basis is proposed. With the help of this basis, we then\npropose an efficient solution algorithm for the quasiperiodic source problem\nand conduct its rigorous error estimate. Moreover, by analyzing the decay rate\nof the Fourier coefficients of the eigenfunctions, we further propose a\ndivergence-free reduced projection method for the quasiperiodic Maxwell\neigenvalue problem, which significantly alleviates the computational cost.\nSeveral numerical experiments are presented to validate the efficiency and\naccuracy of the proposed method.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"55 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A divergence-free projection method for quasiperiodic photonic crystals in three dimensions\",\"authors\":\"Zixuan Gao, Zhenli Xu, Zhiguo Yang\",\"doi\":\"arxiv-2409.05528\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This paper presents a point-wise divergence-free projection method for\\nnumerical approximations of photonic quasicrystals problems. The original\\nthree-dimensional quasiperiodic Maxwell's system is transformed into a periodic\\none in higher dimensions through a variable substitution involving the\\nprojection matrix, such that periodic boundary condition can be readily\\napplied. To deal with the intrinsic divergence-free constraint of the Maxwell's\\nequations, we present a quasiperiodic de Rham complex and its associated\\ncommuting diagram, based on which a point-wise divergence-free quasiperiodic\\nFourier spectral basis is proposed. With the help of this basis, we then\\npropose an efficient solution algorithm for the quasiperiodic source problem\\nand conduct its rigorous error estimate. Moreover, by analyzing the decay rate\\nof the Fourier coefficients of the eigenfunctions, we further propose a\\ndivergence-free reduced projection method for the quasiperiodic Maxwell\\neigenvalue problem, which significantly alleviates the computational cost.\\nSeveral numerical experiments are presented to validate the efficiency and\\naccuracy of the proposed method.\",\"PeriodicalId\":501162,\"journal\":{\"name\":\"arXiv - MATH - Numerical Analysis\",\"volume\":\"55 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Numerical Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.05528\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Numerical Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.05528","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

摘要

本文提出了一种用于光子准晶体问题数值近似的无点发散投影方法。通过涉及投影矩阵的变量替换,原三维准周期麦克斯韦系统被转化为高维周期系统,从而可以随时应用周期边界条件。为了处理麦克斯韦序列的内在无发散约束,我们提出了一个准周期德拉姆复数及其相关的交换图,并在此基础上提出了一个点向无发散的准周期傅里叶谱基础。在此基础上,我们提出了准周期源问题的高效求解算法,并对其进行了严格的误差估计。此外,通过分析特征函数傅里叶系数的衰减率,我们进一步提出了准周期最大傅里叶特征值问题的无辐合还原投影方法,大大降低了计算成本。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A divergence-free projection method for quasiperiodic photonic crystals in three dimensions
This paper presents a point-wise divergence-free projection method for numerical approximations of photonic quasicrystals problems. The original three-dimensional quasiperiodic Maxwell's system is transformed into a periodic one in higher dimensions through a variable substitution involving the projection matrix, such that periodic boundary condition can be readily applied. To deal with the intrinsic divergence-free constraint of the Maxwell's equations, we present a quasiperiodic de Rham complex and its associated commuting diagram, based on which a point-wise divergence-free quasiperiodic Fourier spectral basis is proposed. With the help of this basis, we then propose an efficient solution algorithm for the quasiperiodic source problem and conduct its rigorous error estimate. Moreover, by analyzing the decay rate of the Fourier coefficients of the eigenfunctions, we further propose a divergence-free reduced projection method for the quasiperiodic Maxwell eigenvalue problem, which significantly alleviates the computational cost. Several numerical experiments are presented to validate the efficiency and accuracy of the proposed method.
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
自引率
0.00%
发文量
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信