Nonlocality brings many challenges to the implementation of finite element methods (FEM) for nonlocal problems, such as large number of queries and invoke operations 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 combinatorial map, and an efficient parallel algorithm is also designed for 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.

翻译：非局部性给非局部问题的有限元法实现带来了诸多挑战，例如网格上的大量查询和调用操作。此外，相互作用通常局限于欧几里得球，因此直接数值积分常引入数值误差。球与有限元之间的相互作用问题必须谨慎处理，例如采用球近似策略。本文基于组合图提出了一种高效的近似球表示与构造方法，并设计了一种用于非局部线性系统组装的并行算法。具体而言，本文还提出了一种基于蒙特卡洛积分的球近似方法，即全帽法（fullcaps method），用于计算单元与球相交区域上的数值积分。