Convergence of Stationary RBF-schemes for the numerical solution of evolution equations

In this paper we establish convergence rates for semi-discrete stationary RBF schemes for the classical heat equation and, more generally, for a large class of translation invariant pseudo-differenti al evolution equations which include the fractional heat equation and the Kolmogorov-Fokker-Planck equations of Levy processes (under natural conditions on the Levy measure), but also hyperbolic equations such as the half-wave equation

Radial Basis Function Methods in Fluid-Structure Interaction

This thesis focuses on the use of a number of radial basis function (RBF) methods in computational fluid-structure interaction (FSI). RBF provide a general interpolation framework and have been applied in a number of different ways in various areas of FSI, including bi-directional fluid-structure coupling, mesh or point cloud motion, and in the numerical solution of PDE themselves. This thesis presents novel techniques in all three of these areas. Firstly, an efficiency improvement to the state-of-the-art in RBF mesh motion. Secondly, improvement and simplification of the handing of both moving and static boundaries in RBF-Finite Difference (RBF-FD) methods for numerically solving PDE. Lastly, a partitioned FSI solver is developed using a new coupling method, which extends the current applicability of RBF-FD to a more general class of FSI problems

Meshfree Methods Using Localized Kernel Bases

Radial basis functions have been used to construct meshfree numerical methods for interpolation and for solving partial differential equations. Recently, a localized basis of radial basis functions has been developed on the sphere. In this dissertation, we investigate applying localized kernel bases for interpolation, approximation, and for novel discretization methods for numerically solving partial differential equations and integral equations. We investigate methods for partial differential equations on spheres using newly explored bases constructed from radial basis functions and associated quadrature methods. We explore applications of radial basis functions to anisotropic nonlocal diffusion problems and we develop theoretical frameworks for these methods