A Cartesian-grid collocation method based on radial-basis-function networks for solving PDEs in irregular domains

Abstract

[Abstract]: This paper reports a new Cartesian-grid collocation method based on radial-basis-function networks (RBFNs) for numerically solving elliptic partial differential equations (PDEs) in irregular domains. The domain of interest is embedded in a Cartesian grid, and the governing equation is discretized by using a collocation approach. The new features here are (a) One-dimensional integrated RBFNs are employed to represent the variable along each line of the grid, resulting in a significant improvement of computational efficiency, (b) The present method does not require complicated interpolation techniques for the treatment of Dirichlet boundary conditions in order to achieve a high level of accuracy, and (c) Normal derivative boundary conditions are imposed by means of integration constants. The method is verified through the solution of second- and fourth-order PDEs; accurate results and fast convergence rates are obtained.