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A Nonlocal Biharmonic Model with $Γ$-Convergence to Local Model and an Efficient Numerical Method

Weiye Gan, Tangjun Wang, Qiang Du, Zuoqiang Shi

TL;DR

This work addresses a nonlocal biharmonic equation with clamped boundary data by formulating a nonlocal energy $F_n$ that combines a nonlocal Laplacian term with a boundary penalty, and it proves $F_n$ converges in the $\Gamma$-sense to the local biharmonic energy $F$ as the horizon $\delta\to0$. It establishes well-posedness (existence and uniqueness) of minimizers for $F_n$, verifies $F_n\stackrel{\Gamma}{\longrightarrow}F$ in $L^2(\Omega)$, and proves convergence of minimizers, under kernels $K,R,\tilde{R}$ and horizon/penalty parameters with $\delta_n\to0$, $\xi_n\to0$, and $\delta_n^2/\xi_n\to0$. An efficient finite element method exploiting Gaussian kernel separability is developed, achieving first-order convergence to the local model, and numerical experiments in 1D and 2D validate the theoretical results and reveal boundary-penalty effects. Collectively, the results provide a rigorous bridge between nonlocal higher-order models and local biharmonic theory, with practical, scalable computation for applied elasticity and related areas.

Abstract

Nonlocal models and their associated theories have been extensively investigated in recent years. Among these, nonlocal versions of the classical Laplace operator have attracted the most attention, while higher-order nonlocal operators have been studied far less. In this work, we focus on the nonlocal counterpart of the classical biharmonic operator together with the clamped boundary condition ($u$ and $\frac{\partial u}{\partial n}$ are given on the boundary). We develop the variational formulation of a nonlocal biharmonic model, establish the existence and uniqueness of its solution, and analyze its convergence as the nonlocal horizon approaches zero. In addition, we propose an efficient finite element method to solve the nonlocal model and the numerical results verify the analytical properties of the nonlocal model and its solution.

A Nonlocal Biharmonic Model with $Γ$-Convergence to Local Model and an Efficient Numerical Method

TL;DR

This work addresses a nonlocal biharmonic equation with clamped boundary data by formulating a nonlocal energy that combines a nonlocal Laplacian term with a boundary penalty, and it proves converges in the -sense to the local biharmonic energy as the horizon . It establishes well-posedness (existence and uniqueness) of minimizers for , verifies in , and proves convergence of minimizers, under kernels and horizon/penalty parameters with , , and . An efficient finite element method exploiting Gaussian kernel separability is developed, achieving first-order convergence to the local model, and numerical experiments in 1D and 2D validate the theoretical results and reveal boundary-penalty effects. Collectively, the results provide a rigorous bridge between nonlocal higher-order models and local biharmonic theory, with practical, scalable computation for applied elasticity and related areas.

Abstract

Nonlocal models and their associated theories have been extensively investigated in recent years. Among these, nonlocal versions of the classical Laplace operator have attracted the most attention, while higher-order nonlocal operators have been studied far less. In this work, we focus on the nonlocal counterpart of the classical biharmonic operator together with the clamped boundary condition ( and are given on the boundary). We develop the variational formulation of a nonlocal biharmonic model, establish the existence and uniqueness of its solution, and analyze its convergence as the nonlocal horizon approaches zero. In addition, we propose an efficient finite element method to solve the nonlocal model and the numerical results verify the analytical properties of the nonlocal model and its solution.

Paper Structure

This paper contains 9 sections, 15 theorems, 111 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Assumptions and main results
  3. Well-posedness of nonlocal functionals
  4. $\Gamma$-Convergence of nonlocal functionals
  5. Convergence of minimizers
  6. Numerical Validation
  7. Numerical method
  8. Numerical results
  9. Conclusion

Key Result

Theorem 2.3

Suppose that $\Omega$ is a bounded Lipschitz domain in $\mathbb{R}^d$. $K,R,\tilde{R}$ are kernels satisfying Assumption 2.1. Then, for arbitrary fixed $\delta_n$ and $\xi_n$, there exists a unique minimizer of $F_n:L^2(\Omega)\rightarrow\mathbb{R}$ defined as eq:nonlocal-functional.

Figures (3)

  • Figure 1: Error plot with different $\delta$ in log-log scale. Lines in different colors and markers represent different $N$. A zoomed-in view of the boxed region is provided as an inset to better visualize the differences among the curves.
  • Figure 2: Error plot with $\delta$ in log-log scale. Lines in different colors and markers represent different ground-truth functions $u_{\text{gt}}$. $N=20$ is fixed.
  • Figure 3: Error plot with $c$ as in $\xi=\delta/c$, in log-log scale. Lines in different colors and markers represent different $\delta$. $N=20$ is fixed.

Theorems & Definitions (28)

  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3: Well-posedness
  • Theorem 2.4: $\Gamma$-convergence
  • Theorem 2.5: Convergence of minimizers
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Definition 3.3: Ellipticity
  • ...and 18 more