Publication Date

8-20-2011

Type of Culminating Activity

Thesis

Degree Title

Master of Science in Mathematics

Department

Mathematics

Major Advisor

Grady Wright, Ph.D.

Abstract

The main attraction of using radial basis functions (RBFs) for generating finite difference type approximations (RBF-FD) is that they naturally work for unstructured or scattered nodes. Therefore, a geometrically complex domain can be efficiently discretized using scattered nodes and continuous differential operators such as the Laplacian can be effectively approximated locally using RBF-FD formulas on these nodes. This RBF-FD method is becoming more and more popular as an alternative to the finite-element since it avoids the sometimes complex and expensive step of mesh generation and the RBF-FD method can achieve much higher orders of accuracy. One of the issues with the RBF-FD method is how to properly handle non-Dirichlet boundary conditions. In this thesis, we describe an effective method for handling Neumann conditions in the case of Poisson's equation. The method uses fictitious points and generalized Hermite-Birkhoff interpolation to enforce the boundary conditions and to improve the accuracy of the RBF-FD method near boundaries. We present several numerical experiments using the method and investigate its convergence and accuracy.

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