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

Illinois Institute Of Technology, Chicago IL

Investigators

Abstract

ABSTRACT An extension of a meshless method for the numerical solution of partial differential equations based on radial basis functions is proposed. In particular, the combination of (1) radial basis functions, (2) compact support, and (3) multilevel algorithms is suggested for the collocation solution of nonlinear partial differential equations. In this manner accurate and computationally efficient algorithms can be designed. The research focuses on the implementation of these algorithms, as well as the investigation of some related theoretical issues such as convergence rates and well-posedness. Partial differential equations play an important role in many areas of science and engineering. They are at the core of many mathematical models used, e.g., meteorological models, molecular simulations in chemistry and physics, simulations in such areas as semiconductor modeling, study of materials, fluid dynamics, etc.. In this project we focus on the design of algorithms potentially applicable to any of these areas. Algorithms for high-performance parallel hardware are also considered. The tools employed (radial basis functions) are of fairly recent origin (1980s), but are slowly being accepted by a growing number of scientists and engineers. Their main advantage is that no complicated (and expensive) underlying mesh structure is required as is for the standard finite element or finite volume techniques.