Localized radial basis functions with partition of unity properties

Abstract

In this paper we introduce a localization of radial basis function (RBF) under the general framework of partition of unity. A reproducing kernel that reproduces polynomials is used as the localizing function of RBF. It is shown that the proposed approach yields a similar convergence to that of the non localized RBF, while a better conditioned discrete system than that of the radial basis collocation method is achieved. Analyses of error and stability of the proposed method for solving boundary value problems are presented. Numerical examples are given to demonstrate the effectiveness of the proposed method.