On the equivalence between least-squares and kernel approximations in meshless methods

Meshless methods using least-squares approximations and kernel approximatio ns are based on non-shifted and shifted polynomial basis, respectively. We show that, mathematically, the shifted and nonshifted polynomial basis give rise to identical interpolation functions when the nodal volumes are set t o unity in kernel approximations. This result indicates that mathematically the least-squares and kernel approximations are equivalent. However, for l arge point distributions or for higher-order polynomial basis the numerical errors with a non-shifted approach grow quickly compared to a shifted appr oach, resulting in violation of consistency conditions. Hence, a shifted po lynomial basis is better suited from a numerical implementation point of vi ew. Finally, we introduce an improved finite cloud method which uses a shif ted polynomial basis and a fixed-kernel approximation for construction of i nterpolation functions and a collocation technique for discretization of th e governing equations. Numerical results indicate that the improved finite cloud method exhibits superior convergence characteristics compared to our original implementation [Aluru and Li (2001)] of the finite cloud method.