Localized Hermite method of approximate particular solutions for solving the Poisson equation

Abstract

In this paper, we propose a localized Hermite method of approximate particular solutions (LHMAPS) for solving the Poisson equation. Unlike the localized method of approximate particular solutions (LMAPS) that approximates only function values of the solution in different local neighborhoods of collocation nodes by using particular solutions of radial basis functions, the proposed method employs mixed basis functions, combining radial basis functions and their particular solutions for the Laplace operator within local stencils to simultaneously approximate both the solution and its Laplacian. Numerical experiments show that significantly improves the accuracy of LMAPS.