[en] We prove an $L^2$-estimate for the homogenization of an elliptic operator $A_{\mathrm{\epsilon <!--\; ??\; -->}}$ in a domain $\mathrm{\Omega <!--\; ??\; -->}$ with a Neumann boundary condition on the boundary $\mathrm{\partial <!--\; ???\; -->}\mathrm{\Omega <!--\; ??\; -->}$. The coefficients of the operator $A_{\mathrm{\epsilon <!--\; ??\; -->}}$ are rapidly oscillating over different groups of variables with periods of different orders of smallness as $\mathrm{\epsilon <!--\; ??\; -->}\to <!--\; ???\; -->0$. We assume minimal regularity of the data, which makes it possible to impart to the result the meaning of an estimate in the operator $(L^2(\mathrm{\Omega <!--\; ??\; -->})\to <!--\; ???\; -->L^2(\mathrm{\Omega <!--\; ??\; -->}))$-norm for the difference of the resolvents of the original and homogenized problems. We also find an approximation to the resolvent of the original problem in the operator $(L^2(\mathrm{\Omega <!--\; ??\; -->})\to <!--\; ???\; -->H^1(\mathrm{\Omega <!--\; ??\; -->}))$-norm.

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