Local Error Estimates For Radial Basis Function Interpolation

This paper includes three parts. The first part gives some basic introduction to the meshless methods of Partial Differential equation. In this part, the RBF collocation method based on Radial Basis Function interpolation is described in detail, especially the Kansa's method and symmetric method. The second part includes the definition of positive definite function and conditional positive definite function, the basic form of Radial Basis Function interpolation, the correctness of the inversion of the interpolation matrix, the error estimate based on Kringing function. Two numerical examples is used to observe the boundary error of the Radial Basis Function interpolation. These observations show that the boundary error have a great effect on the overall error estimate. The third part gives a local error estimate for the Radial Basis Function on manifold by constructing a admissible vector. This method avoid satisfying the cone condition, which is required for the classical discussion of the radial basis interpolation, and is useful for obtaining the theoretical error estimate of the meshless method of numerical solution of PDE.