How to approximate the heat equation with Neumann boundary conditions by   nonlocal diffusion problems
  We present a model for nonlocal diffusion with Neumann boundary conditions in a bounded smooth domain prescribing the flux through the boundary. We study the limit of this family of nonlocal diffusion operators when a rescaling parameter related to the kernel of the nonlocal operator goes to zero. We prove that the solutions of this family of problems converge to a solution of the heat equation with Neumann boundary conditions. 
C. Cortazar, M. Elgueta, J.D. Rossi and N. Wolanski