{"title":"二维和三维椭圆型偏微分方程的局部傅里叶配置方法:理论和MATLAB代码","authors":"Yan Gu, Zhuojia Fu, Mikhail V. Golub","doi":"10.1002/msd2.12061","DOIUrl":null,"url":null,"abstract":"<p>A localized Fourier collocation method is proposed for solving certain types of elliptic boundary value problems. The method first discretizes the entire domain into a set of overlapping small subdomains, and then in each of the subdomains, the unknown functions and their derivatives are approximated using the pseudo-spectral Fourier collocation method. The key idea of the present method is to combine the merits of the quick convergence of the pseudo-spectral method and the high sparsity of the localized discretization technique to yield a new framework that may be suitable for large-scale simulations. The present method can be viewed as a competitive alternative for solving numerically large-scale boundary value problems with complex-shape geometries. Preliminary numerical experiments involving Poisson, Helmholtz, and modified-Helmholtz equations in both two and three dimensions are presented to demonstrate the accuracy and efficiency of the proposed method.</p>","PeriodicalId":60486,"journal":{"name":"国际机械系统动力学学报(英文)","volume":null,"pages":null},"PeriodicalIF":3.4000,"publicationDate":"2022-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/msd2.12061","citationCount":"3","resultStr":"{\"title\":\"A localized Fourier collocation method for 2D and 3D elliptic partial differential equations: Theory and MATLAB code\",\"authors\":\"Yan Gu, Zhuojia Fu, Mikhail V. Golub\",\"doi\":\"10.1002/msd2.12061\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>A localized Fourier collocation method is proposed for solving certain types of elliptic boundary value problems. The method first discretizes the entire domain into a set of overlapping small subdomains, and then in each of the subdomains, the unknown functions and their derivatives are approximated using the pseudo-spectral Fourier collocation method. The key idea of the present method is to combine the merits of the quick convergence of the pseudo-spectral method and the high sparsity of the localized discretization technique to yield a new framework that may be suitable for large-scale simulations. The present method can be viewed as a competitive alternative for solving numerically large-scale boundary value problems with complex-shape geometries. Preliminary numerical experiments involving Poisson, Helmholtz, and modified-Helmholtz equations in both two and three dimensions are presented to demonstrate the accuracy and efficiency of the proposed method.</p>\",\"PeriodicalId\":60486,\"journal\":{\"name\":\"国际机械系统动力学学报(英文)\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":3.4000,\"publicationDate\":\"2022-12-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1002/msd2.12061\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"国际机械系统动力学学报(英文)\",\"FirstCategoryId\":\"1087\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/msd2.12061\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"ENGINEERING, MECHANICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"国际机械系统动力学学报(英文)","FirstCategoryId":"1087","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/msd2.12061","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"ENGINEERING, MECHANICAL","Score":null,"Total":0}
A localized Fourier collocation method for 2D and 3D elliptic partial differential equations: Theory and MATLAB code
A localized Fourier collocation method is proposed for solving certain types of elliptic boundary value problems. The method first discretizes the entire domain into a set of overlapping small subdomains, and then in each of the subdomains, the unknown functions and their derivatives are approximated using the pseudo-spectral Fourier collocation method. The key idea of the present method is to combine the merits of the quick convergence of the pseudo-spectral method and the high sparsity of the localized discretization technique to yield a new framework that may be suitable for large-scale simulations. The present method can be viewed as a competitive alternative for solving numerically large-scale boundary value problems with complex-shape geometries. Preliminary numerical experiments involving Poisson, Helmholtz, and modified-Helmholtz equations in both two and three dimensions are presented to demonstrate the accuracy and efficiency of the proposed method.