On approximation theory of nonlocal differential operators

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.