Volume 24, Number 4, October–December 2018
|Page(s)||1345 - 1380|
|Published online||26 October 2018|
On the best constant matrix approximating an oscillatory matrix-valued coefficient in divergence-form operators
École des Ponts ParisTech, CERMICS, 6 et 8 avenue Blaise Pascal,
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: email@example.com
Accepted: 5 September 2017
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].
Mathematics Subject Classification: 35J / 35B27 / 74Q15
Key words: Elliptic PDEs / Oscillatory coefficients / Homogenization / Coarse-graining
© EDP Sciences, SMAI 2018
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