High order RBF-FD approximations with application to a scattering problem

A recently suggested technique for high order meshless approximations is described and analyzed in this paper. It involves constructing ordinary Radial Basis Function-generated finite difference approximations augmented with monomials up to a given order to ensure higher convergence rates. These approximations are used to solve the Poisson's equation on an annulus to demonstrate the predicted convergence rates. The presented methodology is then applied to a scattering problem, which is described by a coupled system of two complex-valued PDEs on two domains, sharing a common boundary.

I. Introduction

Radial basis function-generated finite differences (RBF-FD) were suggested among others by Tolstykh and Shirobokov [1]. They combine good approximation properties of radial basis functions (RBFs) with their ability to adapt to scattered grids, as the Lagrange interpolation problem can be shown to be nonsingular for many commonly used RBFs [2]. This comes contrary to other suggested generalizations of finite differences to scattered grids, such as the Finite Point method [3], where no such guarantee is given.