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

TL;DR

This paper presents a high-performance finite element implementation for nonlocal Poisson problems, introducing novel ball approximation strategies and parallel algorithms to improve accuracy and computational efficiency.

Contribution

It introduces a new Monte Carlo-based ball approximation method and efficient parallel assembly techniques for nonlocal FEM implementations.

Findings

Enhanced accuracy with the fullcaps Monte Carlo method.

Significant reduction in computation time through parallel algorithms.

Improved handling of nonlocal interactions in FEM.

Abstract

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.