Sobolev Space In The Radial Basis Function Interpolation And Its Applications

In this paper we study the method of interpolation by radial basis functions in H~k(Ω)(k ≥ 1) and give some error estimates. By means of such interpolation with a special kind of radial basis function, we construct a basis in H~k(Ω)(k ≥ 1). Combined with the Galerkin method, this theory can be applied to solve boundary value problems for elliptic partial differential equations (such as the third boundary value problem for Poisson equation and the corresponding problem for the biharmonic equation), and some numerical experiments are also given.We can not apply Galerkin methods with radial basis functions to the first boundary value problem (the Dirichlet boundary value problem) of the second-order elliptic equation, because it requires that the approximate functions should fit the essential boundary condition, while the spaces spanned by the radial basis functions fails to satisfy this kind of boundary condition, and it is the same for the higher-order elliptic equation. To ovecome this difficulty, we use the natural boundary value problem(as the third boundary value problem for the second-order elliptic equation) to approximte the corresponding Dirichlet boundary value problem. We deal with the biharmonic equation and proved that under our assumptions, the week solution of the natural boundary value problem strongly converges (in H~2) to that of the first boundary value problem. Some numerical experiments are presented to show the effectiveness of our method.We also present a way to compute the global data density.