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].
Accepté le :
DOI : 10.1051/cocv/2017061
Keywords: Elliptic PDEs, Oscillatory coefficients, Homogenization, Coarse-graining
Le Bris, Claude 1 ; Legoll, Frédéric 1 ; Lemaire, Simon 1
@article{COCV_2018__24_4_1345_0,
author = {Le Bris, Claude and Legoll, Fr\'ed\'eric and Lemaire, Simon},
title = {On the best constant matrix approximating an oscillatory matrix-valued coefficient in divergence-form operators},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1345--1380},
year = {2018},
publisher = {EDP Sciences},
volume = {24},
number = {4},
doi = {10.1051/cocv/2017061},
zbl = {1419.35020},
mrnumber = {3922435},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2017061/}
}
TY - JOUR AU - Le Bris, Claude AU - Legoll, Frédéric AU - Lemaire, Simon TI - On the best constant matrix approximating an oscillatory matrix-valued coefficient in divergence-form operators JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 1345 EP - 1380 VL - 24 IS - 4 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2017061/ DO - 10.1051/cocv/2017061 LA - en ID - COCV_2018__24_4_1345_0 ER -
%0 Journal Article %A Le Bris, Claude %A Legoll, Frédéric %A Lemaire, Simon %T On the best constant matrix approximating an oscillatory matrix-valued coefficient in divergence-form operators %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 1345-1380 %V 24 %N 4 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2017061/ %R 10.1051/cocv/2017061 %G en %F COCV_2018__24_4_1345_0
Le Bris, Claude; Legoll, Frédéric; Lemaire, Simon. On the best constant matrix approximating an oscillatory matrix-valued coefficient in divergence-form operators. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1345-1380. doi: 10.1051/cocv/2017061
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