Fredrik Fryklund: Function extension with radial basis functions for solving non-homogeneous PDEs

Abstract:

In many applications it is desirable to solve partial differential equations on irregular domains with high accuracy and speed. Integral equation based methods offer this ability for homogeneous elliptic PDEs. For the non-homogeneous elliptic PDEs the solution would involve evaluating a volume potential over the irregular domain, which is difficult to do accurately. Instead the problem can be split in two
parts: In the first the given right hand side function is extended to a rectangular domain, discretized with a uniform grid; a problem well suited for fast spectral solvers. The second part consists of solving the homogeneous PDE on the irregular domain with modified boundary conditions. For this, a boundary integral method with special techniques for highly accurate numerical integration of singular and nearly singular integrands is employed.

The success of our method relies on a technique to efficiently compute a high-regularity extension of a function to a domain embedding the given irregular domain. Function extension is an essential component in increasing the applicatility of boundary integral methods to non-homogeneous PDEs, which is needed e.g. in extension from Stokes to Navier-Stokes equations.

Joint work with Anna-Karin Tornberg (KTH Royal Institute of Technology, Sweden) and Erik Lehto (KTH Royal Institute of Technology, Sweden).