We address multiscale elliptic problems with random coefficients that are a perturbation of multiscale deterministic problems. Our approach consists in taking benefit of the perturbative context to suitably modify the classical Finite Element basis into a deterministic multiscale Finite Element basis. The latter essentially shares the same approximation properties as a multiscale Finite Element basis directly generated on the random problem. The specific reference method that we use is the Multiscale Finite Element Method. Using numerical experiments, we demonstrate the efficiency of our approach and the computational speed-up with respect to a more standard approach. In the stationary setting, we provide a complete analysis of the approach, extending that available for the deterministic periodic setting.
Keywords: weakly stochastic homogenization, finite elements, Galerkin methods, highly oscillatory PDE
@article{M2AN_2014__48_3_815_0,
author = {Le Bris, Claude and Legoll, Fr\'ed\'eric and Thomines, Florian},
title = {Multiscale {Finite} {Element} approach for {\textquotedblleft}weakly{\textquotedblright} random problems and related issues},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {815--858},
year = {2014},
publisher = {EDP-Sciences},
volume = {48},
number = {3},
doi = {10.1051/m2an/2013122},
mrnumber = {3264336},
zbl = {1300.65007},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2013122/}
}
TY - JOUR AU - Le Bris, Claude AU - Legoll, Frédéric AU - Thomines, Florian TI - Multiscale Finite Element approach for “weakly” random problems and related issues JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2014 SP - 815 EP - 858 VL - 48 IS - 3 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2013122/ DO - 10.1051/m2an/2013122 LA - en ID - M2AN_2014__48_3_815_0 ER -
%0 Journal Article %A Le Bris, Claude %A Legoll, Frédéric %A Thomines, Florian %T Multiscale Finite Element approach for “weakly” random problems and related issues %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2014 %P 815-858 %V 48 %N 3 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2013122/ %R 10.1051/m2an/2013122 %G en %F M2AN_2014__48_3_815_0
Le Bris, Claude; Legoll, Frédéric; Thomines, Florian. Multiscale Finite Element approach for “weakly” random problems and related issues. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 3, pp. 815-858. doi: 10.1051/m2an/2013122
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