  We developed a new self-adjoint, consistent, and stable coupling strategy for
nonlocal diffusion models, inspired by the quasinonlocal atomistic-to-continuum
method for crystalline solids. The proposed coupling model is coercive with
respect to the energy norms induced by the nonlocal diffusion kernels as well
as the $L^2$ norm, and it satisfies the maximum principle. A finite difference
approximation is used to discretize the coupled system, which inherits the
property from the continuous formulation. Furthermore, we design a numerical
example which shows the discrepancy between the fully nonlocal and fully local
diffusions, whereas the result of the coupled diffusion agrees with that of the
fully nonlocal diffusion.
