Elisabeth Larsson
Division of Scientific Computing, IT Department, Uppsala University
Radial basis function (RBF) approximation methods are attractive for solving PDEs due to their flexibility with respect to geometry, the potential for high-order accuracy, and their ease of use. Since global approximations come with a high computational cost, the trend has been towards localized approximations. The two main classes are stencil-based methods (RBF-FD) and partition of unity methods (RBF-PUM). These are cost efficient and work well. However, it has been difficult to derive complete convergence proofs for the collocation methods.
Recently, several authors have investigated how to introduce oversampling into the PDE solution procedures. This improves the approximation stability, and the least-squares versions of the methods can be computationally competitive compared with their collocation counterparts. Furthermore, for the least-squares methods we are able to derive convergence proofs using approaches based on the continuous approximation problem. In this talk, we present recent algorithmic and theoretical developments as well as numerical results for PDE problems.