Adaptive RBF-FD method

Abstract

Radial-basis-function-generated finite differences (RBF-FD) is a method for solving par- tial differential equations (PDEs), which is developed into a fully automatic adaptive me- thod during the course of this work. RBF-FD is a strong form meshless method, which means that it does not require a mesh of the problem domain, but uses only a set of nodes as the basis for the discretization. A large part of this PhD is dedicated to algorithms for meshless node generation. A new algorithm for construction of variable density mesh- less discretizations in arbitrary spatial dimensions is developed. It can generate points in the interior and on the boundary, has provable minimal spacing requirements, can generate N points in O(N log N ) time and the resulting node sets are compatible with RBF-FD. This algorithm is used as the basis of a newly proposed h-adaptive procedure for elliptic problems. The behavior of the procedure is analyzed on classical 2D and 3D adaptive Poisson problems. Furthermore, several contact problems from linear elastic- ity are solved, demonstrating successful adaptive derefinement and refinement, with the densest parts of the discretization being more than a million times denser than the coars- est. Finally, the software developed for this work and broader research is presented and published online as an open source library for solving PDEs with strong form methods.