Efficient Implementation of 3D FEM for Nonlocal Poisson Problem with Different Ball Approximation Strategies (Communications in Computational Physics, 2024)
Nonlocality brings many challenges to the implementation of finite elementmethods (FEM) for nonlocal problems, such as a large number of neighborhood queryoperations being invoked on the meshes. Besides, the interactions are usually limitedto Euclidean balls, so direct numerical integrals often introduce numerical errors. Theissues of interactions between the ball and finite elements have to be carefully dealtwith, such as using ball approximation strategies. In this paper, an efficient representation and construction methods for approximate balls are presented based on thecombinatorial map, and an efficient parallel algorithm is also designed for the assembly of nonlocal linear systems. Specifically, a new ball approximation method based onMonte Carlo integrals, i.e., the fullcaps method, is also proposed to compute numericalintegrals over the intersection region of an element with the ball.