  We proceed here with our systematic study, initiated in [3], of multiscale
problems with defects, within the context of homogenization theory. The case
under consideration here is that of a diffusion equation with a diffusion
coefficient of the form of a periodic function perturbed by an $L^r (R^d ) , 1
< r < +$\infty$$ , function modeling a localized defect. We outline the proof
of the following approximation result: the corrector function, the existence of
which has been established in [3,4], allows to approximate the solution of the
original multiscale equation with essentially the same accuracy as in the
purely periodic case. The rates of convergence may however vary, and are made
precise, depending upon the $L^r$ integrability of the defect. The
generalization to an abstract setting is mentioned. Our proof exactly follows,
step by step, the pattern of the original proof of Avellaneda and Lin in [1] in
the periodic case, extended in the works of Kenig and collaborators [13], and
borrows a lot from it. The details of the results announced in this Note are
given in our forthcoming publications [2,12].
