Type of Culminating Activity
Master of Science in Mathematics
Grady Wright, Ph.D.
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.
Lohmeier, Joseph, "A Fictitious Point Method for Handling Boundary Conditions in the RBF-FD Method" (2011). Boise State University Theses and Dissertations. Paper 246.