Numerical approximation of the effective Hamiltonian and of the Aubry set for first order Hamilton-Jacobi equations

We introduce an approximation for rst order Hamilton-Jacobi equations with a convex Hamil- tonian periodic in the space variable. We use a rst order semi-Lagrangian scheme to compute a solution of the so called cell problem which allows us to compute the effective Hamiltonian. We exploit the information included in the solutions of the cell problem and in the effective Hamil- tonian to obtain an approximation of the Aubry set. These objects are relevant from a PDE as well as from a dynamical point of view. Two numerical tests illustrate the effectiveness of our approximation.