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We develop two approximation schemes for solving the cell equation and thediscounted cell equation using Aubry-Mather-Fathi theory. The Hamiltonian issupposed to be Tonelli, time-independent , and periodic in space. By Legendretransform it is equivalent to find a fixed point of some nonlinear operator,called Lax-Oleinik operator, which may be discounted or not. By discretizing intime, we are led to solve an additive eigenvalue problem involving a discreteLax-Oleinik operator. We show how to approximate the effective Hamiltonian andsome weak KAM solutions by letting the time step in the discrete model tend tozero. We also obtain a selected discrete weak KAM solution as in [Davini et al2014] and show it converges to a particular solution of the cell equation. Inorder to unify the two settings, continuous and discrete , we develop a moregeneral formalism of short-range interactions.

Convergence of discrete Aubry-Mather model in the continuous limit