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Volume 36, Issue 5
Efficient Implementation of 3D FEM for Nonlocal Poisson Problem with Different Ball Approximation Strategies

Gengjian Chen, Yuheng Ma & Jiwei Zhang

DOI: 10.4208/cicp.OA-2023-0210

Commun. Comput. Phys., 36 (2024), pp. 1378-1410.

Published online: 2024-12

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  • Abstract

Nonlocality brings many challenges to the implementation of finite element methods (FEM) for nonlocal problems, such as a large number of neighborhood query operations being invoked on the meshes. Besides, the interactions are usually limited to Euclidean balls, so direct numerical integrals often introduce numerical errors. The issues of interactions between the ball and finite elements have to be carefully dealt with, such as using ball approximation strategies. In this paper, an efficient representation and construction methods for approximate balls are presented based on the combinatorial map, and an efficient parallel algorithm is also designed for the assembly of nonlocal linear systems. Specifically, a new ball approximation method based on Monte Carlo integrals, i.e., the fullcaps method, is also proposed to compute numerical integrals over the intersection region of an element with the ball.

  • Keywords

Nonlocal problem, finite element method, combinatorial map, approximate ball, Monte Carlo integration, parallel computing.

  • AMS Subject Headings

65Y10, 65D30, 65R99, 34K28, 45A99

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-36-1378, author = {Chen , GengjianMa , Yuheng and Zhang , Jiwei}, title = {Efficient Implementation of 3D FEM for Nonlocal Poisson Problem with Different Ball Approximation Strategies}, journal = {Communications in Computational Physics}, year = {2024}, volume = {36}, number = {5}, pages = {1378--1410}, abstract = {

Nonlocality brings many challenges to the implementation of finite element methods (FEM) for nonlocal problems, such as a large number of neighborhood query operations being invoked on the meshes. Besides, the interactions are usually limited to Euclidean balls, so direct numerical integrals often introduce numerical errors. The issues of interactions between the ball and finite elements have to be carefully dealt with, such as using ball approximation strategies. In this paper, an efficient representation and construction methods for approximate balls are presented based on the combinatorial map, and an efficient parallel algorithm is also designed for the assembly of nonlocal linear systems. Specifically, a new ball approximation method based on Monte Carlo integrals, i.e., the fullcaps method, is also proposed to compute numerical integrals over the intersection region of an element with the ball.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2023-0210}, url = {http://global-sci.org/intro/article_detail/cicp/23612.html} }
TY - JOUR T1 - Efficient Implementation of 3D FEM for Nonlocal Poisson Problem with Different Ball Approximation Strategies AU - Chen , Gengjian AU - Ma , Yuheng AU - Zhang , Jiwei JO - Communications in Computational Physics VL - 5 SP - 1378 EP - 1410 PY - 2024 DA - 2024/12 SN - 36 DO - http://doi.org/10.4208/cicp.OA-2023-0210 UR - https://global-sci.org/intro/article_detail/cicp/23612.html KW - Nonlocal problem, finite element method, combinatorial map, approximate ball, Monte Carlo integration, parallel computing. AB -

Nonlocality brings many challenges to the implementation of finite element methods (FEM) for nonlocal problems, such as a large number of neighborhood query operations being invoked on the meshes. Besides, the interactions are usually limited to Euclidean balls, so direct numerical integrals often introduce numerical errors. The issues of interactions between the ball and finite elements have to be carefully dealt with, such as using ball approximation strategies. In this paper, an efficient representation and construction methods for approximate balls are presented based on the combinatorial map, and an efficient parallel algorithm is also designed for the assembly of nonlocal linear systems. Specifically, a new ball approximation method based on Monte Carlo integrals, i.e., the fullcaps method, is also proposed to compute numerical integrals over the intersection region of an element with the ball.

Chen , GengjianMa , Yuheng and Zhang , Jiwei. (2024). Efficient Implementation of 3D FEM for Nonlocal Poisson Problem with Different Ball Approximation Strategies. Communications in Computational Physics. 36 (5). 1378-1410. doi:10.4208/cicp.OA-2023-0210
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