Homogenization of an Elliptic System as the Cells of Periodicity are Refined in One Direction

We homogenize a second-order elliptic system with anisotropic fractal structure characteristic of many real objects: the cells of periodicity are refined in one direction. This problem is considered in the rectangle with Dirichlet conditions given on two sides and periodicity conditions on two other sides. An explicit formula for the homogenized operator is established, and an asymptotic estimate of the remainder is obtained. The accuracy of approximation depends on the exponent $\varkappa\in(0,1/2]$ of smoothness of the right-hand side with respect to slow variables (the Sobolev–Slobodetskii space) and is estimated by $O(h^\varkappa)$ for $\varkappa\in(0,1/2)$ and by $O(h^{1/2}(1+|\log h|))$ for $\varkappa=1/2$.