A Note on Explicit Convergence Rates of Nonlocal Peridynamic Operators in $L^q$-Norm

Abstract

This note investigates the explicit convergence rates of nonlocal peridynamic operators to their classical (local) counterparts in $L^q$-norm. Previous results used Fourier series and hence were restricted to showing convergence in $L^2$. Moreover, convergence rates were not explicit due to the use of the Lebesgue Dominated Convergence Theorem. Some previous results have also used the Taylor Remainder Theorem in differential form, but this often required an assumption of bounded fifth-order derivatives. We do not use these tools, but instead use the Hardy-Littlewood Maximal function, and combine it with the integral form of the Taylor Remainder Theorem. This approach allows us to establish convergence in the $L^q$-norm ($1 \leq q \leq \infty$) for nonlocal peridynamic partial derivatives, which immediately yields convergence rates for the corresponding nonlocal peridynamic divergence, gradient, and curl operators to their local counterparts as the radius (a.k.a., ``horizon'') of the nonlocal interaction $\delta \to 0$. Moreover, we obtain an explicit rate of order $\mathcal{O}(\delta^2)$. This result contributes to the understanding of the relationship between nonlocal and local models, which is essential for applications in multiscale modeling and simulations.