Oversampled radial basis function methods for solving partial differential equations

Sammanfattning: Partial differential equations (PDEs) describe complex real-world phenomena such as weather dynamics,  object deformations, financial trading prices, and fluid-structure interaction.  The solutions of PDEs are commonly used to enhance the understanding of these phenomena and also as leverage to make technological improvements to consumer products. In the present thesis, we  develop numerical methods for solving PDEs using computers. The focus is on radial basis  function (RBF) methods that are appreciated for their high-order accuracy and ease of implementation in higher dimensions, but can  sometimes face numerical stability challenges.  To circumvent the stability issues, we use an oversampled approach to discretize PDEs as opposed to the more commonly used collocated approach.  Throughout the thesis, we mainly use the RBF-generated finite difference (RBF-FD) method, but we also use the RBF partition  of unity method (RBF-PUM) and Kansa's global RBF method in one part of the thesis.  The first two methods are local in the sense that the underlying discretization matrices are sparse,  while the third method is global, leading to dense discretization matrices. In Paper I we improve the stability properties of the RBF-FD method through an oversampling approach  when solving an elliptic model problem with derivative-type boundary conditions, and provide a theoretical analysis.  In Paper II we develop an unfitted RBF-FD method and by that simplify the handling of complex  computational domains by relaxing the requirement that  the set of nodes has to conform to the boundary of the domain. We make the first steps toward a simulation of the thoracic diaphragm in Paper III,  where we use an unfitted RBF-FD method to solve a linear elastic PDE and employ data smoothing to leverage high-order convergence of the numerical solution. In Paper IV we explore the stability properties behind the RBF-FD method, Kansa's method, and RBF-PUM  when they are applied to a  linear time-dependent hyperbolic PDE. We find that Kansa's method and RBF-PUM can become stable under sufficient oversampling of the system of equations.  On the other hand, the insufficient regularity of the numerical solution prevents the RBF-FD method from being stable in time, no matter the oversampling.  In Paper V we use the residual viscosity stabilization framework to locally stabilize the Gibbs phenomenon present in the RBF-FD solutions  to shock-inducing nonlinear time-dependent conservation laws such as the compressible Euler system of equations.