For a homogenization problem associated to a linear elliptic operator, we
prove the existence of a distributional corrector and we find an approximation
scheme for the homogenized coefficients. We also study the convergence rates in
the asymptotic almost periodic setting, and we show that the rates of
convergence for the zero order approximation, are near optimal. The results
obtained constitute a step towards the numerical implementation of results from
the deterministic homogenization theory beyond the periodic setting. To
illustrate this, numerical simulations based on finite volume method are
provided to sustain our theoretical results.Comment: 49 pages, 10 figure