ERROR ESTIMATES FOR THE APPROXIMATION OF THE EFFECTIVE HAMILTONIAN FABIO CAMILLI, ITALO CAPUZZO DOLCETTA AND DIOGO A.GOMES Abstract. We study approximation schemes for the cell problem arising in homogeniza- tion of Hamilton-Jacobi equations. We prove several error estimates concerning the rate of convergence of the approximation scheme to the effective Hamiltonian, both in the optimal control setting and as well as in the calculus of variations setting. 1. Introduction Starting from [23], where the basic approach to periodic homogenization via viscosity solutions was outlined, the homogenization theory for Hamilton-Jacobi equations has re- ceived a considerable interest both from theoretical, as well as from an applied viewpoint (see, for example, [3], [7], [9], [15], [24], [25]). An essential step in the homogenization procedure is the identification of the structure of the Hamiltonian of the limit problem, the so-called effective Hamiltonian H (P ). The function H (P ) is the unique value of the parameter λ for which the cell problem H (x, Du + P )= λ x ∈ T N (T N is the N -dimensional torus) admits a (periodic) viscosity solution u, called the cor- rector. Since this equation, in general, cannot be explicitly solved except in special cases, see [8], it is important to design numerical schemes to approximate the solution. From the numerical point of view, this is a very difficult task since it requires the approximation of a first order Hamilton-Jacobi equation in which the unknowns are both the viscosity solution u and the constant H (P ). Moreover, while H (P ) is uniquely identified by the cell problem, the corresponding solution u is, in general, not unique. A numerical scheme for the computation of the effective Hamiltonian H (P ) was proposed in [20], where a convergence theorem and some error estimates were proved. This scheme relies on an inf-sup representation formula for H (P ) and therefore it does not require the solution of the cell problem. On the other hand, since the computation of the approximate value of H (P ), for any fixed P , requires the solution of a minimax optimization problem, this scheme is computationally very expensive. Furthermore, the scheme in [20] does not approximate the solution. Date : May 12, 2006. 1