Efficient quadrature rules for finite element discretizations of nonlocal equations

TL;DR

This paper introduces efficient quadrature rules for finite element discretizations of nonlocal diffusion equations, improving computational efficiency and implementation in 3D through mollification, adaptive integration, and parallelization.

Contribution

It presents a novel mollification-based approach combined with adaptive and parallel techniques to enhance efficiency and accuracy in nonlocal finite element discretizations.

Findings

Mollified solutions converge to exact solutions as mollification parameter decreases.

The method demonstrates high accuracy on 2D and 3D test cases.

Parallel implementation shows good scalability and efficiency.

Abstract

In this paper we design efficient quadrature rules for finite element discretizations of nonlocal diffusion problems with compactly supported kernel functions. Two of the main challenges in nonlocal modeling and simulations are the prohibitive computational cost and the nontrivial implementation of discretization schemes, especially in three-dimensional settings. In this work we circumvent both challenges by introducing a parametrized mollifying function that improves the regularity of the integrand, utilizing an adaptive integration technique, and exploiting parallelization. We first show that the "mollified" solution converges to the exact one as the mollifying parameter vanishes, then we illustrate the consistency and accuracy of the proposed method on several two- and three-dimensional test cases. Furthermore, we demonstrate the good scaling properties of the parallel implementation of the adaptive algorithm and we compare the proposed method with recently developed techniques for efficient finite element assembly.