Fredrik Fryklund, Leslie Greengard, Shidong Jiang, Samuel Potter
{"title":"用于评估层势和体积势的轻量级、几何灵活的快速算法","authors":"Fredrik Fryklund, Leslie Greengard, Shidong Jiang, Samuel Potter","doi":"arxiv-2409.11998","DOIUrl":null,"url":null,"abstract":"Over the last two decades, several fast, robust, and high-order accurate\nmethods have been developed for solving the Poisson equation in complicated\ngeometry using potential theory. In this approach, rather than discretizing the\npartial differential equation itself, one first evaluates a volume integral to\naccount for the source distribution within the domain, followed by solving a\nboundary integral equation to impose the specified boundary conditions. Here,\nwe present a new fast algorithm which is easy to implement and compatible with\nvirtually any discretization technique, including unstructured domain\ntriangulations, such as those used in standard finite element or finite volume\nmethods. Our approach combines earlier work on potential theory for the heat\nequation, asymptotic analysis, the nonuniform fast Fourier transform (NUFFT),\nand the dual-space multilevel kernel-splitting (DMK) framework. It is\ninsensitive to flaws in the triangulation, permitting not just nonconforming\nelements, but arbitrary aspect ratio triangles, gaps and various other\ndegeneracies. On a single CPU core, the scheme computes the solution at a rate\ncomparable to that of the fast Fourier transform (FFT) in work per gridpoint.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"32 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A Lightweight, Geometrically Flexible Fast Algorithm for the Evaluation of Layer and Volume Potentials\",\"authors\":\"Fredrik Fryklund, Leslie Greengard, Shidong Jiang, Samuel Potter\",\"doi\":\"arxiv-2409.11998\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Over the last two decades, several fast, robust, and high-order accurate\\nmethods have been developed for solving the Poisson equation in complicated\\ngeometry using potential theory. In this approach, rather than discretizing the\\npartial differential equation itself, one first evaluates a volume integral to\\naccount for the source distribution within the domain, followed by solving a\\nboundary integral equation to impose the specified boundary conditions. Here,\\nwe present a new fast algorithm which is easy to implement and compatible with\\nvirtually any discretization technique, including unstructured domain\\ntriangulations, such as those used in standard finite element or finite volume\\nmethods. Our approach combines earlier work on potential theory for the heat\\nequation, asymptotic analysis, the nonuniform fast Fourier transform (NUFFT),\\nand the dual-space multilevel kernel-splitting (DMK) framework. It is\\ninsensitive to flaws in the triangulation, permitting not just nonconforming\\nelements, but arbitrary aspect ratio triangles, gaps and various other\\ndegeneracies. On a single CPU core, the scheme computes the solution at a rate\\ncomparable to that of the fast Fourier transform (FFT) in work per gridpoint.\",\"PeriodicalId\":501162,\"journal\":{\"name\":\"arXiv - MATH - Numerical Analysis\",\"volume\":\"32 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-18\",\"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-2409.11998\",\"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-2409.11998","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
在过去的二十年里,已经开发出几种快速、稳健和高阶精确的方法,用于利用势理论求解复杂几何中的泊松方程。在这种方法中,我们不是将边际微分方程本身离散化,而是首先求体积积分来计算域内的源分布,然后求解边界积分方程来施加指定的边界条件。在这里,我们提出了一种新的快速算法,这种算法易于实现,而且几乎与任何离散化技术兼容,包括非结构化域三角测量,如标准有限元或有限体积方法中使用的算法。我们的方法结合了早先在氦方程势理论、渐近分析、非均匀快速傅立叶变换(NUFFT)和双空间多级内核拆分(DMK)框架方面的工作。它对三角剖分中的缺陷很敏感,不仅允许不规则的元素,还允许任意长宽比的三角形、间隙和其他各种退行性。在单个 CPU 内核上,该方案计算解的速度可与快速傅立叶变换 (FFT) 计算每个网格点的工作量相媲美。
A Lightweight, Geometrically Flexible Fast Algorithm for the Evaluation of Layer and Volume Potentials
Over the last two decades, several fast, robust, and high-order accurate
methods have been developed for solving the Poisson equation in complicated
geometry using potential theory. In this approach, rather than discretizing the
partial differential equation itself, one first evaluates a volume integral to
account for the source distribution within the domain, followed by solving a
boundary integral equation to impose the specified boundary conditions. Here,
we present a new fast algorithm which is easy to implement and compatible with
virtually any discretization technique, including unstructured domain
triangulations, such as those used in standard finite element or finite volume
methods. Our approach combines earlier work on potential theory for the heat
equation, asymptotic analysis, the nonuniform fast Fourier transform (NUFFT),
and the dual-space multilevel kernel-splitting (DMK) framework. It is
insensitive to flaws in the triangulation, permitting not just nonconforming
elements, but arbitrary aspect ratio triangles, gaps and various other
degeneracies. On a single CPU core, the scheme computes the solution at a rate
comparable to that of the fast Fourier transform (FFT) in work per gridpoint.