Circulant matrix and conformal mapping for solving partial differential equations

In recent years, a fast radial basis function (RBF) solver for surface interpolation has been developed by Karageorghis et al. (2007). In this paper, we follow up their work by extending the idea of fast evaluation of the surface interpolation to efficiently solve various types of partial differential equations (PDEs). We look into the possibility of solving PDEs with conformal mapping to map the RBF circular points on a disk to the interior of an irregular domain. We also compare the pros and cons for these two approaches. In both approaches, the circulant matrix formulation and the matrix decomposition method are the central ideas of fast computation. In numerical experiments, we test these two approaches for solving PDEs using up to 160 000 RBF interpolation points with greater accuracy and efficiency.