Asymptotically Compatible Error Bound of Finite Element Method for Nonlocal Diffusion Model with An Efficient Implementation

TL;DR

This paper establishes an asymptotically compatible error bound for FEM applied to nonlocal diffusion models, covering different mesh regularities, and introduces an efficient implementation that reduces computational costs significantly.

Contribution

It provides the first comprehensive error analysis for FEM in nonlocal diffusion models across various mesh types and proposes a decoupling technique for efficient computation.

Findings

Error bounds of O(h^k + δ) for shape-regular meshes

Error bounds of O(h^{k+1}/δ + δ) without shape regularity

Numerical experiments confirm theoretical accuracy and efficiency

Abstract

This paper presents an asymptotically compatible error bound for the finite element method (FEM) applied to a nonlocal diffusion model. The analysis covers two scenarios: meshes with and without shape regularity. For shape-regular meshes, the error is bounded by \(O(h^k + \delta)\), where \(h\) is the mesh size, \(\delta\) is the nonlocal horizon, and \(k\) is the order of the FEM basis. Without shape regularity, the bound becomes \(O(h^{k+1}/\delta + \delta)\). In addition, we present an efficient implementation of the finite element method of nonlocal model. The direct implementation of the finite element method of nonlocal model requires computation of $2n$-dimensional integrals which are very expensive. For the nonlocal model with Gaussian kernel function, we can decouple the $2n$-dimensional integral to 2-dimensional integrals which reduce the computational cost tremendously. Numerical experiments verify the theoretical results and demonstrate the outstanding performance of the proposed numerical approach.