We consider homogenization problems for linear elliptic equations in divergence form. The coefficients are assumed to be a local perturbation of some periodic background. We prove and Lipschitz convergence of the two-scale expansion, with explicit rates. For this purpose, we use a corrector adapted to this particular setting, and defined in (Comm. Partial Differential Equations40 (2015) 2173–2236; Comm. Partial Differential Equations43 (2018) 965–997), and apply the same strategy of proof as Avellaneda and Lin in (Comm. Pure Appl. Math.40 (1987) 803–847). We also propose an abstract setting generalizing our particular assumptions for which the same estimates hold.
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