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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$O(h^k + \delta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mi>k</mi> </msup> <mo>+</mo> <mi>δ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$h$</EquationSource> <EquationSource Format="MATHML"><math> <mi>h</mi> </math></EquationSource> </InlineEquation> is the mesh size, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation> is the nonlocal horizon, and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$k$</EquationSource> <EquationSource Format="MATHML"><math> <mi>k</mi> </math></EquationSource> </InlineEquation> is the order of the FEM basis. Without shape regularity, the bound becomes <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$O(h^{k+1}/\delta + \delta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">/</mo> <mi>δ</mi> <mo>+</mo> <mi>δ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. 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 reduces the computational cost tremendously. Numerical experiments verify the theoretical results and demonstrate the outstanding performance of the proposed numerical approach.

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

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发表日期：

2026年4月10日

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Yanzun Meng,
Zuoqiang Shi

摘要

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 $O(h^k + \delta )$ $O (h^{k} + δ)$ , where $h$ $h$ is the mesh size, $\delta $ $δ$ is the nonlocal horizon, and $k$ $k$ is the order of the FEM basis. Without shape regularity, the bound becomes $O(h^{k+1}/\delta + \delta )$ $O (h^{k + 1} / δ + δ)$ . 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 reduces the computational cost tremendously. Numerical experiments verify the theoretical results and demonstrate the outstanding performance of the proposed numerical approach.