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：非局部性对非本地性问题采用限定元素方法(FEM)带来许多挑战,例如大量查询和援引网外线性操作。此外,互动通常仅限于欧洲球,因此直接数字组合往往会引入数字错误。球与有限元素之间的互动问题必须谨慎处理,例如使用球近似战略。本文根据组合图提出了近似球的有效代表性和构造方法,还为非本地线性系统组装设计了高效的平行算法。具体地说,还提议采用基于蒙特卡洛组合的新的球近似法,即全盖法,在与球相交的区域计算数字组合。