Reproducing kernel enhanced local radial basis collocation method

Abstract

Abstract Standard radial basis functions (RBFs) offer exponential convergence, however, the method is suffered from the large condition numbers due to their ‘nonlocal’ approximation. The nonlocality of RBFs also limits their applications to small‐scale problems. The reproducing kernel functions, on the other hand, provide polynomial reproducibility in a ‘local’ approximation, and the corresponding discrete systems exhibit relatively small condition numbers. Nonetheless, reproducing kernel functions produce only algebraic convergence. This work intends to combine the advantages of RBFs and reproducing kernel functions to yield a local approximation that is better conditioned than that of the RBFs, while at the same time offers a higher rate of convergence than that of reproducing kernel functions. Further, the locality in the proposed approximation allows its application to large‐scale problems. Error analysis of the proposed method is also provided. Numerical examples are given to demonstrate the improved conditioning and accuracy of the proposed method. Copyright © 2007 John Wiley & Sons, Ltd.