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

Qiang Du, Hehu Xie, Xiaobo Yin, Jiwei Zhang
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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.
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使用精确或近似交互邻域的非局部问题有限元方法的误差估计
我们研究了有界相互作用邻域的非局部模型有限元解与极限局部模型精确解之间的渐近误差。该极限对应的情况是:地平线参数、非局部模型的球形非局部相互作用邻域半径和网格尺寸同时趋近于零。本文讨论了两种重要情况:一种是原始非局部模型,另一种是非局部模型的非局部相互作用邻域的多边形近似值。还报告了数值实验结果,以证实理论研究。
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