  We develop two approximation schemes for solving the cell equation and the
discounted cell equation using Aubry-Mather-Fathi theory. The Hamiltonian is
supposed to be Tonelli, time-independent , and periodic in space. By Legendre
transform it is equivalent to find a fixed point of some nonlinear operator,
called Lax-Oleinik operator, which may be discounted or not. By discretizing in
time, we are led to solve an additive eigenvalue problem involving a discrete
Lax-Oleinik operator. We show how to approximate the effective Hamiltonian and
some weak KAM solutions by letting the time step in the discrete model tend to
zero. We also obtain a selected discrete weak KAM solution as in [Davini et al
2014] and show it converges to a particular solution of the cell equation. In
order to unify the two settings, continuous and discrete , we develop a more
general formalism of short-range interactions.
