On the Equivalence Between Least-Squares and Kernel Approximations in Meshless Methods

Abstract

Meshless methods using least-squares ap- proximations and kernel approximations are based on non-shifted and shifted polynomial basis, respectively. We show that, mathematically, the shifted and non- shifted polynomial basis give rise to identical interpola- tion functions when the nodal volumes are set to unity in kernel approximations. This result indicates that math- ematically the least-squares and kernel approximations are equivalent. However, for large point distributions or for higher-order polynomial basis the numerical errors with a non-shifted approach grow quickly compared to a shifted approach, resulting in violation of consistency conditions. Hence, a shifted polynomial basis is better suited from a numerical implementation point of view. Finally, we introduce an improved nite cloud method which uses a shifted polynomial basis and a x ed-kernel approximation for construction of interpolation functions and a collocation technique for discretization of the gov- erning equations. Numerical results indicate that the im- proved nite cloud method exhibits superior convergence characteristics compared to our original implementation (Aluru and Li (2001)) of the nite cloud method.