Bloch Approximation in Homogenization and Applications

Abstract

The classical problem of homogenization of elliptic operators with periodically oscillating coefficients is revisited in this paper. As is well known, the homogenization process in a classical framework is concerned with the study of asymptotic behavior of solutions $u^\varepsilon$ of boundary value problems associated with such operators when the period $\varepsilon>0$ of the coefficients is small. In a previous work by C. Conca and M. Vanninathan [SIAM J. Appl. Math., 57 (1997), pp. 1639--1659], a new proof of weak convergence as $\varepsilon\to 0$ towards the homogenized solution was furnished using Bloch wave decomposition. Following the same approach here, we go further and introduce what we call Bloch approximation, which will provide energy norm approximation for the solution $u^\varepsilon$. We develop several of its main features. As a simple application of this new object, we show that it contains both the first and second order correctors. Necessarily, the Bloch approximation will have to capture the oscillations of the solution in a sharper way. The present approach sheds new light and offers an alternative for viewing classical results.