On homogenization estimates in Neuman boundary value problem for an elliptic equation with multiscale coefficients

Abstract

Homogenization of a scalar elliptic equation in a bounded domain with Neuman boundary condition is studied. Coefficients of the operator are oscillating over two different groups of variables with different small periods $\varepsilon$ and $\delta=\delta(\varepsilon)$. We assume that ${\delta}/{\varepsilon}$ tends to zero as $\varepsilon$ tends to zero. It is known that the limit problem is obtained through reiterated homogenization procedure and corresponds to an elliptic equation with constant coefficients. The difference for resolvents of the initial and the limit problems is estimated in operator $(L^2\to L^2)$-norm. This estimate is of order $\tau=\tau(\varepsilon)$ which is the maximum of $\ve$ and ${\delta}/{\varepsilon}$. We find also the approximation of the initial resolvent in operator $(L^2\to H^1)$-norm, it is of order $\sqrt{\tau}$.