An integrated-RBF technique based on Galerkin formulation for elliptic differential equations

This paper presents a new radial-basis-function (RBF) technique for solving elliptic differential equations (DEs). The RBF solutions are sought to satisfy (a) the boundary conditions in a local sense using the point-collocation formulation, (b) the governing equation in a global sense using the Galerkin formulation. In contrast to Galerkin finite-element techniques, the present Neumann boundary conditions are imposed in an exact manner. Unlike conventional RBF techniques, the present RBF approximations are constructed “locally” on grid lines through integration and they are expressed in terms of nodal variable values. The proposed technique can provide an approximate solution that is a CpCp function across the subdomain interfaces (p—the order of the DE). Several numerical examples are presented to demonstrate the attractiveness of the present implementation.