Discrete Maximum Principle and Convergence of Poisson Problem for the Finite Point Method

Abstract

This paper makes some mathematical analyses for the finite point method based on directional difference. By virtue of the explicit expressions of numerical formulae using only five neighboring points for computing first-order and second-order directional differentials, a new methodology is presented to discretize the Laplacian operator defined on 2D scattered point distributions. Some sufficient conditions with very weak limitations are obtained, under which the resulted schemes are positive schemes. As a consequence, the discrete maximum principle is proved, and the first order convergent result of $O(h)$ is achieved for the nodal solutions defined on scattered point distributions, which can be raised up to $O(h^2)$ on uniform point distributions.