Homogenization of periodic structures via Bloch decomposition

Abstract

In this paper, the classical problem of homogenization of elliptic operators in arbitrary domains with periodically oscillating coefficients is considered. Using Bloch wave decomposition, a new proof of convergence is furnished. It sheds new light and offers an alternate way to view the classical results. In a natural way, this method leads us to work in the Fourier space and thus in a framework dual to the one used by L. Tartar [Problémes d'Homogénéisation dans les Equations aux Dérivées Partielles, Cours Peccot au Collége de France, 1977] in his method of homogenization. Further, this technique offers a nontraditional way of calculating the homogenized coefficients which is easy to implement in the computer.