TY - JOUR
T1 - Discrete Maximum Principle and Convergence of Poisson Problem for the Finite Point Method
AU - Lv , Guixia
AU - Sun , Shunkai
AU - Shen , Longjun
JO - Journal of Computational Mathematics
VL - 3
SP - 245
EP - 264
PY - 2017
DA - 2017/06
SN - 35
DO - http://doi.org/10.4208/jcm.1605-m2015-0397
UR - https://global-sci.org/intro/article_detail/jcm/9772.html
KW - Finite point method, Directional difference, Meshless, Discrete maximum principle, Convergence analysis.
AB - <p style="text-align: justify;">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.</p>