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Volume 35, Issue 3
Discrete Maximum Principle and Convergence of Poisson Problem for the Finite Point Method

Guixia Lv, Shunkai Sun & Longjun Shen

DOI: 10.4208/jcm.1605-m2015-0397

J. Comp. Math., 35 (2017), pp. 245-264.

Published online: 2017-06

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  • 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.

  • Keywords

Finite point method, Directional difference, Meshless, Discrete maximum principle, Convergence analysis.

  • AMS Subject Headings

65D25, 65N06, 65N12, 65N15.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

lv_guixia@iapcm.ac.cn (Guixia Lv)

sun_shunkai@iapcm.ac.cn (Shunkai Sun)

shenlj@iapcm.ac.cn (Longjun Shen)

  • BibTex
  • RIS
  • TXT
@Article{JCM-35-245, author = {Lv , GuixiaSun , Shunkai and Shen , Longjun}, title = {Discrete Maximum Principle and Convergence of Poisson Problem for the Finite Point Method}, journal = {Journal of Computational Mathematics}, year = {2017}, volume = {35}, number = {3}, pages = {245--264}, 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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1605-m2015-0397}, url = {http://global-sci.org/intro/article_detail/jcm/9772.html} }
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 -

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.

Lv , GuixiaSun , Shunkai and Shen , Longjun. (2017). Discrete Maximum Principle and Convergence of Poisson Problem for the Finite Point Method. Journal of Computational Mathematics. 35 (3). 245-264. doi:10.4208/jcm.1605-m2015-0397
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