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.

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.

DOI : 10.1051/m2an/2013122
Classification : 35B27, 65M60, 65M12, 35R60, 60H
Mots-clés : weakly stochastic homogenization, finite elements, Galerkin methods, highly oscillatory PDE 
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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. http://www.numdam.org/articles/10.1051/m2an/2013122/

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