Numerical solution of elliptic partial differential equations by Bloch waves method
Carlos Conca, Srinivasan Natesan, M. Vanninathan · Dialnet (Universidad de la Rioja) · 2001
This paper deals with a numerical study of classical homogenization of elliptic linear operators with periodic oscillating coefficients (period eY ). A method introduced by Conca and Vanninathan [7] based on Bloch waves that homogenize this kind of operators is used for the numerical approximation of their solution u e . The novelty of their approach consists of using the spectral decomposition of the operator on all R N to obtain a new approximation of u e –the so-called Bloch approximation θ e– which provides an alternative to the classical two-scale expansion u e (x) = u ∗ (x) + Σe kuk(x, x e ), and therefore, θ e contains implicitly at least the homogenized solution u ∗ and the first order corrector term eu1. Besides, it is defined through an elegant oscillating integral. The Bloch approximation θ e is obtained by computing, for every value of the Bloch variable η in the reciprocal cell Y 0 (Brillouin zone), the components of u ∗ on the first Bloch mode associated with the periodic structure of the medium. The main goal of this paper is to report exhaustive numerical experiments including a comparative study between both the classical and Bloch approaches. The important issue emerging from the numerical results states that θ e is more close to u e , i.e., is a better approximation of u e than the first order corrector term in the case of smooth coefficients whereas in the discontinuous coefficients case, both perform almost in the same way.