TY  - EJOU
AU  - Jin, Xiaozhong  
AU  - Li, Gang  
AU  - Aluru, N. R.  

TI  - On the Equivalence Between Least-Squares and Kernel Approximations in Meshless Methods
T2  - Computer Modeling in Engineering \& Sciences

PY  - 2001
VL  - 2
IS  - 4
SN  - 1526-1506

AB  - Meshless methods using least-squares approximations 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 interpolation functions when the nodal volumes are set to unity in kernel approximations. This result indicates that mathematically 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 finite cloud method which uses a shifted polynomial basis and a fixed-kernel approximation for construction of interpolation functions and a collocation technique for discretization of the 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.
KW  - 

DO  - 10.3970/cmes.2001.002.447