Solve partial differential equations by meshless subdomains method combined with RBFs

In this paper, we will introduce a new meshless method for the numerical solutions of the partial differential equations. It can be regarded as one of the Petrov–Galerkin methods (see [Zhang yun-xin, Tan yong-ji, Solve PDEs using meshless Petrov–Galerkin Methods with RBFs, submitted for publication]), in which the test space is generated by the radial basis functions (RBFs), and the trial space is generated by some characteristic functions. But it is more efficient than the usual Petrov–Galerkin methods because the complicated integrals in the usual Petrov–Galerkin methods are replaced by the simple local boundary integrals. The numerical examples indicate it is really an accurate and efficient meshless method. Certainly our method also can be regarded as one of the weighted residual methods, or the generalization of the collocation methods.