  In this paper, we propose nonlocal diffusion models with Dirichlet boundary.
These nonlocal diffusion models preserve the maximum principle and also have
corresponding variational form. With these good properties, It is relatively
easy to prove the well-posedness and the vanishing nonlocality convergence.
Furthermore, by specifically designed weight function, we can get a nonlocal
diffusion model with second order convergence which is optimal for nonlocal
diffusion models.
