  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.
