The localized method of approximated particular solutions for near-singular two- and three-dimensional problems

In this paper, the localized method of approximate particular solutions (LMAPS) using radial basis functions (RBFs) has been simplified and applied to near-singular elliptic problems in two- and three-dimensional spaces. The leave-one-out cross validation (LOOCV) is used in LMAPS to search for a good shape parameter of multiquadric RBF. The main advantage of the method is that a small number of neighboring nodes can be chosen for each influence domain in the discretization to achieve high accuracy. This is especially efficient for three-dimension problems. There is no need to apply adaptivity on node distribution near the region containing spikes of the forcing terms. To examine the performance and limitations of the method, we deliberately push the spike of the forcing term to be extremely large and still obtain excellent results. LMAPS is far superior than the compactly supported RBF (Chen et al. 2003) for such elliptic boundary value problems.