On the implementation of linear finite element method for nonlocal diffusion model over 2D domain

TL;DR

The paper addresses the computational challenge of nonlocal diffusion modeled by integral operators in 2D by proposing a linear FEM discretization that reduces the intrinsic $4$D integration to a tractable $2$D form. It derives a rigorous reduction of the stiffness matrix from $4$D to $2$D through integration by parts and triangle-based kernel evaluations, leveraging the assumption that the kernel $R$ is polynomial on $[0,1]$ to obtain explicit closed-form integrals over triangles. The approach yields explicit formulas for intra- and inter-element contributions and boundary terms, enabling efficient assembly of the nonlocal system. The method extends naturally to Dirichlet boundaries (via MP-dirichlet) and is positioned for 3D generalization, offering a practical pathway for implementing nonlocal diffusion models with standard FEM tools and improved computational efficiency.

Abstract

We propose an implementation of linear finite element method for nonlocal diffusion problem in 2D space. In the implementation, we reduce the integral from 4D to 2D which would simplify the computation significantly.

Figures (1)

• Figure 1: Illustration of the integral over triangle. Left: decomposition of the integral domain; Right: integral over triangle.