Abstract

This paper deals with a numerical study of classical homogenization of elliptic linear operators with periodic oscillating coefficients (period εY). The importance of such problems in engineering applications is quite well-known. A method introduced by Conca and Vanninathan [SIAM J. Appl. Math. 1997; 57:1639-1659] based on Bloch waves that homogenize this kind of operators is used for the numerical approximation of their solution uε. The novelty of their approach consists of using the spectral decomposition of the operator on ℝN to obtain a new approximation of uε - the socalled Bloch approximation θε - which provides an alternative to the classical two-scale expansion uε(x) = u* (x) + ∑εk uk(x, x/ε), and therefore, θε contains implicitly at least the homogenized solution u* and the first- and second-order corrector terms. The Bloch approximation θε is obtained by computing, for every value of the Bloch variable η in the reciprocal cell Y′ (Brillottin zone), the components of u* on the first Bloch mode associated with the periodic structure of the medium. Though theoretical basis of the method already exists, there is no evidence of its numerical performance. The main goal of this paper is to report on some numerical experiments including a comparative study between both the classical and Bloch approaches. The important conclusion emerging from the numerical results states that θε is closer to uε', i.e. is a better approximation of uε than the first- and second-order corrector terms, specifically in the case of high-contrast materials. Copyright © 2005 John Wiley & Sons, Ltd.