Recently, several types of nonlocal discrete differential operators have emerged either from meshfree particle methods or from nonlocal continuum mechanics, such as peridynamics. In this article, we discuss the mathematical formulation as well as construction of the nonlocal discrete differential operators. Based on a least-square minimization procedure and the associated Moore–Penrose inverse, we have found a general form of the shape tensor and a unified expression for the first type nonlocal differential operators. We then conduct a convergence study, which provides the interpolation error estimate for the first type discrete nonlocal different operators. We have shown that as the radius of the horizon approaches to zero, the first type nonlocal differential operators will converge to the local differential operators. Moreover, we have demonstrated the computational performance of the first type nonlocal differential operators in several numerical examples.

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非局部微分算子的逼近论

最近，从无网格粒子方法或非局部连续介质力学（例如近场动力学）中出现了几种类型的非局部离散微分算子。在本文中，我们讨论了非局部离散微分算子的数学公式和构造。基于最小二乘最小化程序和相关的 Moore-Penrose 逆，我们找到了形状张量的一般形式和第一类非局部微分算子的统一表达式。然后我们进行收敛研究，它为第一类离散非局部不同算子提供插值误差估计。我们已经证明，随着地平线半径接近零，第一类非局部微分算子将收敛到局部微分算子。而且，