Homogenization of first order equations with u/ε-periodic Hamiltonian: Rate of convergence as εto 0 and numerical approximation of the effective Hamiltonian

We consider homogenization problems for first order Hamilton-Jacobi equations with $u^\epsilon/\epsilon$ periodic dependence, recently introduced by C. Imbert and R. Monneau, and also studied by G. Barles: this unusual dependence leads to a nonstandard cell problems. We study the rate of convergence of the solution to the solution of the homogenized problem when the parameter $\epsilon$ tends to 0. We obtain the same rates as those obtained by I. Capuzzo Dolcetta and H. Ishii for the more usual homogenization problems without the dependence in $u^\epsilon/\epsilon$. In a second part, we study Eulerian schemes for the approximation of the cell problems. We prove that when the grid steps tend to zero, the approximation of the effective Hamiltonian converges to the effective Hamiltonian.