Abstract: A new approximate method based on coupled radial basis functions and polynomial basis functions is presented.Only nodes information is needed when this method is used for interpolation over a set of discrete nodes,the background mesh is totally unnecessary.The establishment,properties,interpolation behaviors of approximate functions,the shape functions and their derivatives are described in detail.The modified variation principle and partition of unity quadrature are employed with coupled approximate function to solve boundary value problems. Numerical examples show excellent agreement with exact solutions.

Key words: radial basis functions, polynomial basis functions, meshless method, Galerkin method, partition of unity quadrature