TAILIEUCHUNG - RBF stencils for poisson equation

In the paper we present a method for finding the weight vector called stencil with the help of RBF interpolation. This stencil is the foundation for constructing meshless finite difference scheme for boundary value problems. The results of numerical experiments show that the numerical solution obtained by RBF-FD with the stencils generated by Gauss RBF interpolation is much more accurate then the solution obtained by FEM. | Đặng Thị Oanh Tạp chí KHOA HỌC & CÔNG NGHỆ 78(02): 63 - 66 RBF STENCILS FOR POISSON EQUATION Dang Thi Oanh* Faculty of Information Technology - TNU ABSTRACT In the paper we present a method for finding the weight vector called stencil with the help of RBF interpolation. This stencil is the foundation for constructing meshless finite difference scheme for boundary value problems. The results of numerical experiments show that the numerical solution obtained by RBF-FD with the stencils generated by Gauss RBF interpolation is much more accurate then the solution obtained by FEM. Keyword: Radial Basis Function (RBF), meshless method, shape parameter, stencil INTRODUCTION* Because of the difficulties to create, maintain and update complex meshes needed for the standard finite difference, finite element or finite volume discretisations of the partial differential equations, meshless methods attract growing attention. In particular, strong form methods such as collocation or generalised finite differences are attractive because they avoid costly numerical integration of the non-polynomial shape functions on non-standard domains often encountered in those meshless methods that are based on the weak formulation of PDE. Thanks to their excellent local approximation power, radial basis functions are an ideal tool to produce numerical differentiation stencils for the Laplacian and other partial differential operators on irregular centres, without any need for a mesh. This leads to exceptionally promising RBF based generalised finite difference method. The essense of the method is to find a weight vector called stencil corresponding a row of stiffness matrix in FEM. In this paper we give formulas for finding the stencil based on RBF, and then we use them for discretising the Poisson equation on a nonuniform set of centres. In [1], [2] for the above purpose we used RBF interpolation with additional polynomial term and performed many numerical examples in complicated .

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