Numerical solution of elliptic partial differential equations by Bloch waves method

• Carlos Conca ^[1] ; Srinivasan Natesan ^[2] ; Muthusamy Vanninathan ^[3]
  1. [1] Universidad de Chile

     Universidad de Chile

     Santiago, Chile

     
  2. [2] Centro de Modelamiento Matemático, UMR 2071 CNRS-UChile
  3. [3] IISc-TIFR Mathematics Programme, TIFR Center, Bangalore, India

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• Localización: XVII Congreso de Ecuaciones Diferenciales y Aplicaciones ; VII Congreso de Matemática Aplicada: Salamanca, 14-28 septiembre 2001 / coord. por Luis Ferragut Canals, Anastasio Pedro Santos Yanguas, 2001, ISBN 8469961446, págs. 63-83
• Idioma: inglés
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• Resumen

  • This paper deals with a numerical study of classical homogenization of elliptic linear operators with periodic oscillating coefficients (period εY ). 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 ε . 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 ε –the so-called Bloch approximation θ ε– which provides an alternative to the classical two-scale expansion u ε (x) = u ∗ (x) + Σε kuk(x, x ε ), and therefore, θ ε contains implicitly at least the homogenized solution u ∗ and the first order corrector term εu1. Besides, it is defined through an elegant oscillating integral. The Bloch approximation θ ε 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 ε is more close to u ε , i.e., is a better approximation of u ε 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.