An improved localized method of approximate particular solutions for solving elliptic PDEs

In this paper we improve the localized method of approximate particular solutions (LMAPS) in Yao et al. (2011) by utilizing the polyharmonic splines (PS) radial basis function (RBF) for solving elliptic partial differential equations (PDEs). LMAPS has been widely circulated since it is published in 2010. The multiquadric (MQ) has been considered as the most popular choice among all RBFs. However, adjusting the shape parameter is a critical issue when utilizing the original LMAPS. In this paper, we modified LMAPS by combining conditionally positive definite RBF-PS and an additional low degree of polynomial basis in the localization process. The accuracy of the proposed LMAPS is significantly improved. We can simply increase the order of PS to achieve even higher accuracy. Other than the unexpected high accuracy, there is no need to deal with the difficult issues of choosing optimal shape parameter. This is a huge advantage in the RBF simulations of PDEs. In the numerical experiments, we will present the pros and cons of improved LMAPS (ILMAPS) using PS and some commonly used RBFs (MQ, Matérn, and Gaussian) versus the original LMAPS (OLMAPS).