A numerical study of 2D integrated RBFNs incorporating Cartesian grids for solving 2D elliptic differential problems

Abstract

This paper reports a numerical discretisation scheme, based on two-dimensional integrated radial-basis-function networks (2D-IRBFNs) and rectangular grids, for solving second-order elliptic partial differential equations defined on 2D non-rectangular domains. Unlike finite-difference and 1D-IRBFN Cartesian-grid techniques, the present discretisation method is based on an approximation scheme that allows the field variable and its derivatives to be evaluated anywhere within the domain and on the boundaries, regardless of the shape of the problem domain. We discuss the following two particular strengths, which the proposed Cartesian-grid-based procedure possesses, namely (i) the implementation of Neumann boundary conditions on irregular boundaries and (ii) the use of high-order integration schemes to evaluate flux integrals arising from a control-volume discretisation on irregular domains. A new preconditioning scheme is suggested to improve the 2D-IRBFN matrix condition number. Good accuracy and high-order convergence solutions are obtained.