Solving Nonlinear Elliptic PDEs in 2D and 3D Using Polyharmonic Splines and Low-Degree of Polynomials

In this paper, the improved localized method of approximated particular solutions (ILMAPS) using polyharmonic splines (PHS) together with a low-degree of polynomial basis is used to approximate solutions of various nonlinear elliptic Partial Differential Equations (PDEs). The method is completely meshfree, and it uses a radial basis function (RBF) that has no shape parameters. The discretization process is done through a simple collocation technique on a set of points in the local domain of influence. Resulted system of nonlinear algebraic equations is solved by the Picard method. The performance of the proposed method is tested on various nonlinear elliptical problems, including the Poisson-type problems in 2D and 3D with constant or variable coefficients on rectangular or irregular domains and the Poisson–Boltzmann equation with Dirichlet boundary conditions or mixed boundary conditions. The effect of domain shapes in 2D and 3D, types of boundary conditions, and degrees of PHS, and order of polynomial basis are examined. The performance of the method is compared with other bases such as multiquadrics (MQ) basis functions, and with results reported in the literature (method of particular solutions using polynomials). The numerical experiments suggest that ILMAPS with polyharmonic splines yields considerably superior accuracy than other RBFs as well as other approaches reported in the literature for solving nonlinear elliptic PDEs.