  In this paper, we extend the idea of "geometric reconstruction" to couple a
nonlocal diffusion model directly with the classical local diffusion in one
dimensional space. This new coupling framework removes interfacial
inconsistency, ensures the flux balance, and satisfies energy conservation as
well as the maximum principle, whereas none of existing coupling methods for
nonlocal-to-local coupling satisfies all of these properties. We establish the
well-posedness and provide the stability analysis of the coupling method. We
investigate the difference to the local limiting problem in terms of the
nonlocal interaction range. Furthermore, we propose a first order finite
difference numerical discretization and perform several numerical tests to
confirm the theoretical findings. In particular, we show that the resulting
numerical result is free of artifacts near the boundary of the domain where a
classical local boundary condition is used, together with a coupled fully
nonlocal model in the interior of the domain.
