Igor Tominec: Stability Aspects of Radial Basis Function Methods for Solving PDEs

Abstract

Radial basis function (RBF) methods belong to a class of methods that discretize a PDE on nodes scattered over a computational domain. In contrast, other methods, such as the finite element method, require a connected set of nodes forming a mesh. Computing meshes can be more challenging in certain cases. However, RBF discretizations can encounter various numerical instabilities. For instance, when dealing with an elliptic PDE problem, instabilities may arise due to the imposition of Neumann boundary conditions. Similarly, discretizing a linear hyperbolic problem can also lead to instabilities. Until recently, there have been limited theoretical investigations on these instabilities.

In this seminar, I will outline an approach to analyze RBF methods and utilize this analysis to enhance their stability properties when solving problems in solid mechanics and fluid mechanics. Furthermore, at the end of the seminar, I will provide an introduction to my postdoctoral project focused on the numerics of ice sheet models.