On the best constant matrix approximating an oscillatory matrix-valued coefficient in divergence-form operators

¹ École des Ponts ParisTech, CERMICS, 6 et 8 avenue Blaise Pascal, 77455 Marne-la-Vallée Cedex 2, France
² Inria Paris, MATHERIALS project-team, 2 rue Simone Iff, CS 42112, 75589 Paris Cedex 12, France
³ École des Ponts ParisTech, Laboratoire Navier – UMR 8205, 6 et 8 avenue Blaise Pascal, 77455 Marne-la-Vallée Cedex 2, France

^* Corresponding author: frederic.legoll@enpc.fr

Received: 17 December 2016
Accepted: 5 September 2017

Abstract

We approximate an elliptic problem with oscillatory coefficients using a problem of the same type, but with constant coefficients. We deliberately take an engineering perspective, where the information on the oscillatory coefficients in the equation can be incomplete. A theoretical foundation of the approach in the limit of infinitely small oscillations of the coefficients is provided, using the classical theory of homogenization. We present a comprehensive study of the implementation aspects of our method, and a set of numerical tests and comparisons that show the potential practical interest of the approach. The approach detailed in this article improves on an earlier version briefly presented in [C. Le Bris, F. Legoll and K. Li, C.R. Acad. Sci. Paris, Série I  351 (2013) 265–270].