The modified localized method of approximated particular solutions for solving elliptic equations with mixed boundary conditions on scattered data

Abstract

The localized method of approximated particular solutions (LMAPS) was first introduced in 2011 [31]. The method is then modified by employing integrated polyharmonic splines with polynomial basis. However, the current reported results on LMAPS still limited to the evenly distributed data points and Dirichlet boundary conditions. On the other hand, the traditional point-wise moving least square method is improved by piece-wise moving least squares (PMLS) in [19] for scattered data approximation. The paper proved that the PMLS is is an optimal design for data approximation. In this paper, the modified LMAPS is further improved by involving the Hermite-type PMLS to construct shape functions. The improved LMAPS is called piece-wise smoothed LMAPS (PS-LMAPS). Together with the original LMAPS, PS-LMAPS is used to solve elliptic partial differential equations with Dirichlet and Neumann mixed boundary conditions on the scattered data points. Performance of PS-LMAPS in comparison with LMAPS is tested on two Poisson equation with mixed boundary conditions using evenly-spaced nodes and scattered nodes, modified Helmholtz equations and a non-smooth problem. Particularly, PS-LMAPS can avoid some of the ill-conditioning issues of the system as shown in Example 3. The computational complexity ratio and relative error ratio indicate that the PS-LMAPS is much more efficient than that original LMAPS. The conclusion is supported by theoretical analysis of computational complexity and numerical experiments on the error analysis.