Nonlocal diffusion and peridynamic models with Neumann type constraints and their numerical approximations

This paper studies nonlocal diffusion models associated with a finite nonlocal horizon parameter δ that characterizes the range of nonlocal interactions. The focus is on the variational formulation associated with Neumann type constraints and its numerical approximations. We establish the well-posedness for some variational problems associated and study their local limit as δ → 0. A main contribution is to derive a second order convergence to the local limit. We then discuss the numerical approximations including standard finite element methods and quadrature based finite difference methods. We study their convergence in the nonlocal setting and in the local limit.