Multilevel Algorithms for the Numerical Solution of Partial Differential Equations using Compactly Supported Radial Basis Functions

使用紧支持径向基函数的偏微分方程数值解的多级算法

• 批准号: 0073636
• 负责人: Gregory Fasshauer
• 金额: $ 8.8万
• 依托单位: Illinois Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-08-01 至 2003-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0073636&HistoricalAwards=false
• 关键词: Multilevel; Algorithms; Numerical; Solution; Partial

ABSTRACTAn extension of a meshless method for the numerical solution ofpartial differential equations based on radial basis functions isproposed. In particular, the combination of (1) radialbasis functions, (2) compact support, and (3) multilevel algorithms issuggested for the collocation solution of nonlinear partialdifferential equations. In this manner accurate and computationally efficient algorithms can be designed. The research focuses on theimplementation of these algorithms, as well as the investigation ofsome related theoretical issues such as convergence rates andwell-posedness. Partial differential equations play an important role in many areas ofscience and engineering. They are at the core of many mathematicalmodels used, e.g., meteorological models, molecular simulations inchemistry and physics, simulations in such areas as semiconductormodeling, study of materials, fluid dynamics, etc.. In this project wefocus on the design of algorithms potentially applicable to any ofthese areas. Algorithms for high-performance parallel hardware arealso considered. The tools employed (radial basis functions) are offairly recent origin (1980s), but are slowly being accepted by agrowing number of scientists and engineers. Their main advantage isthat no complicated (and expensive) underlying mesh structure isrequired as is for the standard finite element or finite volumetechniques.

摘要提出了一种基于径向基函数的偏微分方程数值解无网格方法的推广。特别地，（1）径向基函数，（2）紧支撑，（3）多层算法的组合被提出用于非线性偏微分方程的配置解。以这种方式，可以设计精确且计算高效的算法。本文的研究重点是这些算法的实现，以及一些相关的理论问题，如收敛速度和适定性的调查。偏微分方程在科学和工程的许多领域中起着重要的作用。它们是许多实用模型的核心，例如，气象模型、化学和物理学中的分子模拟、电子模型、材料研究、流体动力学等领域的模拟。在这个项目中，我们专注于设计可能适用于任何这些领域的算法。高性能并行硬件的算法也被认为是。所使用的工具（径向基函数）是最近的起源（20世纪80年代），但正在慢慢地被越来越多的科学家和工程师所接受。它们的主要优点是不需要像标准有限元或有限体积技术那样复杂（和昂贵）的底层网格结构。