Radial basis function methods for PDE problems
Our research focuses on RBF methods for solving time-dependent/independent PDE problems. We utilize RBF methods in collocation, Galerkin, and hybrid schemes.
Collocation:
The idea is essentially following Pseudospectral collocation methods where differentiation matrices are utilized to solve boundary value or initial boundary value problems. The differentiation matrices are formed using either global and/or local RBF approximations instead of polynomials. For time-dependent cases, method of lines is used where RBF approximation is used in space and common time-stepping methods are used to advance solutions in time. Current projects are:
- Adaptive Local RBF Methods. (A. Heryudono, E. Larsson). The figure below shows interpolation of 2 dimensional Franke function where RBF points are refined/coarsened automatically based on residuals.
- RBF Methods for Solving Helmholtz Problems: General theory, convergence rates, a posteriori error estimates. (U. Pettersson, E. Larsson). The figure below shows a sound wave coming in from the left and propagating through an M-shaped region. The computation was performed using about 800 scattered RBF node points.
Galerkin:
RBF approximation offers high order approximations in non-trivial geometries and Galerkin methods have a well developed underlying mathematical theory. The prospect of combining the two is attractive. One of the issues to deal with is quadrature accurate enough not to ruin the expected high order convergence. The following MSc thesis is a pilot project.
- RBF Methods in Discontinuous Galerkin (DG) scheme. (B. Rodhe, S-E. Ekström, E. Larsson). The figure below shows a discontinuous Galerkin solution to a simple convective problem, where the underlying approximation is RBF based.
Hybrid Method:
RBF methods are coupled with other legacy methods such as Pseudospectral, Finite-Difference, and Finite-Element. Current projects are:
- FEM-RBF. (E. Larsson, A. Heryudono, A. Målqvist). The goal is to couple FEM and RBF for problems with mixed regularity. As an example, solid mechanics problems on irregular geometries with cracks. Solutions near cracks are approximated with FEM whereas RBF methods are used in regions with smooth solutions. This 17 month project is funded by the Marie Curie FP7 program (123.694 euro).
- RBF-PUM. RBF Partition of Unity Method for Collocation Problems. (E. Larsson, A. Heryudono).