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Error estimates of finite element methods for nonlocal problems using exact or approximated interaction neighborhoods

Qiang Du, Hehu Xie, Xiaobo Yin, Jiwei Zhang

TL;DR

The paper analyzes the asymptotic error between nonlocal diffusion solutions with a bounded interaction neighborhood and the local diffusion limit as the horizon $\delta$ and mesh size $h$ vanish, for both exact spherical neighborhoods and polygonal approximations. It develops a conforming discontinuous Galerkin (CDG) framework to obtain explicit error bounds in the energy and $L^2$ norms, detailing how the locality limit is preserved under diverse neighborhood geometries and kernel types. The key contributions include rigorous $\delta$- and $h$-dependent error estimates, conditions under which polygonal neighbourhoods preserve asymptotic compatibility, and a suite of numerical experiments validating the theory and illustrating practical AC behavior for peridynamics-like models. The results provide quantitative guidance for choosing discretization parameters and neighborhood approximations in multiscale nonlocal simulations with guaranteed convergence to the local limit.

Abstract

We study the asymptotic error between the finite element solutions of nonlocal models with a bounded interaction neighborhood and the exact solution of the limiting local model. The limit corresponds to the case when the horizon parameter, the radius of the spherical nonlocal interaction neighborhood of the nonlocal model, and the mesh size simultaneously approach zero. Two important cases are discussed: one involving the original nonlocal models and the other for nonlocal models with polygonal approximations of the nonlocal interaction neighborhood. Results of numerical experiments are also reported to substantiate the theoretical studies.

Error estimates of finite element methods for nonlocal problems using exact or approximated interaction neighborhoods

TL;DR

The paper analyzes the asymptotic error between nonlocal diffusion solutions with a bounded interaction neighborhood and the local diffusion limit as the horizon and mesh size vanish, for both exact spherical neighborhoods and polygonal approximations. It develops a conforming discontinuous Galerkin (CDG) framework to obtain explicit error bounds in the energy and norms, detailing how the locality limit is preserved under diverse neighborhood geometries and kernel types. The key contributions include rigorous - and -dependent error estimates, conditions under which polygonal neighbourhoods preserve asymptotic compatibility, and a suite of numerical experiments validating the theory and illustrating practical AC behavior for peridynamics-like models. The results provide quantitative guidance for choosing discretization parameters and neighborhood approximations in multiscale nonlocal simulations with guaranteed convergence to the local limit.

Abstract

We study the asymptotic error between the finite element solutions of nonlocal models with a bounded interaction neighborhood and the exact solution of the limiting local model. The limit corresponds to the case when the horizon parameter, the radius of the spherical nonlocal interaction neighborhood of the nonlocal model, and the mesh size simultaneously approach zero. Two important cases are discussed: one involving the original nonlocal models and the other for nonlocal models with polygonal approximations of the nonlocal interaction neighborhood. Results of numerical experiments are also reported to substantiate the theoretical studies.
Paper Structure (19 sections, 13 theorems, 146 equations, 7 figures, 10 tables)

This paper contains 19 sections, 13 theorems, 146 equations, 7 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Convergence of the nonlocal solutions to the local limit
  3. Convergence of the nonlocal solutions to the local limit
  4. Convergence of the nonlocal solutions with polygonal approximations of the spherical interaction neighborhoods
  5. Examples of the kernel function
  6. Error estimate of the linear CDG method for the nonlocal problems with respect to the horizon parameter and the mesh size
  7. Brief review of the linear CDG method for nonlocal problems
  8. The CDG approximation for the nonlocal problem with spherical interaction neighborhoods
  9. The CDG approximation for the nonlocal problem with polygonal interaction neighborhoods
  10. Error estimates between the nonlocal discrete solutions and the local exact solution
  11. Numerical experiment
  12. Nonlocal problems with an exact RHS function and VC
  13. Numerical results for a fixed horizon parameter
  14. Numerical results for the case of a fixed ratio between the horizon parameter and the mesh size
  15. Numerical results for power law between the horizon parameter and the mesh size
  16. ...and 4 more sections

Key Result

Theorem 2.2

Suppose $u_{0}\in C^{4}_{b}(\Omega)$ is the solution of the local problem local_diffusion, the family of kernels $\{\gamma_{\delta}\}$ satisfies kernel_finite, General_kernel, and Let $u_\delta$ be the solution of the nonlocal problem nonlocal_diffusion. If $\widetilde{u}_{0}$ is a $C^{4}$ extension of $u_{0}$ and then it holds that

Figures (7)

  • Figure 1: Polygonally approximated balls, taken from du2022convergence
  • Figure 2: Inscribed polygon in $B_{\delta}({\bf x})$ and its image by \ref{['Affine_map']}
  • Figure 3: \ref{['Example:Continuous']}: the consistent mesh and corresponding exactcaps solution $u_{\delta}^{h}$, $\delta=0.4$, $h=0.05$
  • Figure 4: \ref{['Example:Continuous']}: the non-consistent mesh and corresponding solution $u_{\delta}^{h,H}$, $\delta=0.4$, $h=0.05\sqrt{2}$, $H=3h/8$
  • Figure 5: \ref{['Example:Continuous']}: zoom around point $(1,1)$ of $u_{\delta}^{h}$ and $u_{\delta,n_{\delta}}^{h}$, $h=\delta=0.4$
  • ...and 2 more figures

Theorems & Definitions (28)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Definition 2.4
  • Theorem 2.5
  • Theorem 2.6
  • proof
  • Theorem 2.7
  • proof
  • ...and 18 more