Bloch-wave homogenization for spectral asymptotic analysis of the periodic Maxwell operator
Kirill Cherednichenko, Sébastien R. L. Guenneau · Waves in Random and Complex Media · 2007
This paper is devoted to the asymptotic behavior of the spectrum of the three-dimensional Maxwell operator in a bounded periodic heterogeneous dielectric medium T = [−T,T]3, T > 0, as the structure period η, such that η−1 T is a positive integer, tends to 0. The domain T is extended periodically to the whole of ℝ 3, so that the original operator is understood as acting in a space of T-periodic functions. We use the so-called Bloch-wave homogenization technique which, unlike the classical homogenization method, is capable of characterizing a renormalized limit of the spectrum (called the Bloch spectrum) [6 Allaire, G. and Conca, C. 1998. Bloch wave homogenization and spectral asymptotic analysis. Journal de Mathematiques Pures et Appliquees, 77: 153–208. [Crossref], [Web of Science ®] , [Google Scholar]]. The related procedure is concerned with sequences of eigenvalues Λη of the resolvent of the order of the square of the medium period, which correspond to the oscillations of high-frequencies of order η−1. The Bloch-wave description is obtained via the notion of two-scale convergence for bounded self-adjoint operators, and a proof of the ‘completeness’ of the limiting spectrum is provided. The results obtained theoretically are illustrated by finite element computations.