@Article{cmes.2001.002.447,
AUTHOR = {Xiaozhong  Jin, Gang  Li, N. R.  Aluru},
TITLE = {On the Equivalence Between Least-Squares and Kernel Approximations in Meshless Methods},
JOURNAL = {Computer Modeling in Engineering \& Sciences},
VOLUME = {2},
YEAR = {2001},
NUMBER = {4},
PAGES = {447--462},
URL = {http://www.techscience.com/CMES/v2n4/24745},
ISSN = {1526-1506},
ABSTRACT = {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.},
DOI = {10.3970/cmes.2001.002.447}
}