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

摘要提出了一种基于径向基函数的无网格偏微分方程数值求解方法的推广。特别提出了(1)径向基函数、(2)紧支持和(3)多层算法的组合，用于非线性偏微分方程的配置解。通过这种方式，可以设计出精确且计算效率高的算法。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研究。偏微分方程在科学和工程的许多领域起着重要的作用。它们是许多数学模型的核心，例如，气象模型，化学和物理中的分子模拟，半导体建模，材料研究，流体动力学等领域的模拟。在这个项目中，我们专注于算法的设计，这些算法可能适用于任何这些领域。高性能并行硬件的算法也被考虑。所使用的工具（径向基函数）是相当近的起源（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。

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.