H -adaptive RBF-FD method for the high-dimensional convection-diffusion equation

Abstract

Radial basis function-generated finite difference method (RBF-FD) has been a popular method for simulating the derivatives of a function and has been successfully applied for the partial differential equations (PDEs). In this paper we introduce an effective h-adaptive RBF-FD method to the convection-diffusion equation in high-dimension space including two dimensions and three dimensions. The derivative of the solution is represented on overlapping a new influence domain through RBF-FD by using Thin Plane Spline Radial Basis Functions (TPS) augmented with additional polynomial functions. The number of the nodes added in the domain by the h-adaptive RBF-FD method is triggered by an error indicator, which very simply depends on the local residual norm. Several numerical examples are given to demonstrate the validity of h-adaptive RBF-FD method for the convection-diffusion equation in high-dimension space.