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 - <p style="text-align: justify;">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.</p>