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

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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.