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Nonlocality brings many challenges to the implementation of finite elementmethods (FEM) for nonlocal problems, such as large number of queries and invokeoperations on the meshes. Besides, the interactions are usually limited toEuclidean balls, so direct numerical integrals often introduce numericalerrors. The issues of interactions between the ball and finite elements have tobe carefully dealt with, such as using ball approximation strategies. In thispaper, an efficient representation and construction methods for approximateballs are presented based on combinatorial map, and an efficient parallelalgorithm 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 overthe intersection region of an element with the ball.

High performance implementation of 3D FEM for nonlocal Poisson problem with different ball approximation strategies