Improved numerical solver for Kansa's method based on affine space decomposition

Radial Basis functions (RBFs) have been successfully developed as a truly mesh-free method to find the numerical solutions of partial differential equations (PDEs). In particular, the asymmetric RBF collocation method (Kansa's method) is one of the most frequently used methods due to its ease of implementation. To achieve high accuracy, the resultant system of RBF-PDE problem usually becomes badly conditioned. We propose in this paper an improved solution method based on an affine space decomposition that decouples the influence between the interior and boundary collocations. Numerical examples are given to compare the proposed method with several direct methods.