Using Radial Basis Function Method Solving Elliptic Partial Differential Equation

Many mathematical models of physics and engineering practical problem can be described by elliptic partial differential equations, for example the elasticity geometry electromagnetics in physics and some engineering models. But the exact solution of elliptic partial differential equation only can be obtained in some especial conditions, therefore for some general elliptic partial differential equations, using numerical methods to solve is a kind of more effective method. Traditional numerical method mainly includes finite element method and finite difference method, therefore these methods have been very matured for years of development. Finite element method must generate mesh in advance for solution, so the requirements more and more become a problem that must to be considered. The pretreatment of the generating mesh in the process of realizing calculation uses much time. While solved region is irregular and the dimension of solved region is very high, using finite element method to solve is very difficult. The finite difference method needs grid nodes to be regular distribution, thus when physical district is irregular the treatment of the solved region is difficult. Radial basis function method what is a effectively method to solve partial differential equations appeared in 1970*s and has got great development in the past thirty years, and it has stable convergence order and approximation order.By constantly improving and development, the theory of radial bases function method already has a sound basis. Radial bases meshless method mainly contains two forms: Collocation Method and Galerkin Method. In this paper we use these two methods to solve elliptic partial differential equations and give the error estimate. In the end using numerical example verifies prefatory conclusion.