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Multiscale Finite Element approach for “weakly” random problemsand related issues

Published online by Cambridge University Press:  08 April 2014

Claude Le Bris ,
Frédéric Legoll  and
Florian Thomines
Claude Le Bris
Affiliation:
CERMICS, École Nationale des Ponts et Chaussées, Université Paris-Est, 6 et 8 avenue Blaise Pascal, 77455 Marne-La-Vallée Cedex 2, France.. lebris@cermics.enpc.fr INRIA Rocquencourt, MICMAC team-project, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France.
Frédéric Legoll
Affiliation:
INRIA Rocquencourt, MICMAC team-project, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France. Laboratoire Navier, École Nationale des Ponts et Chaussées, Université Paris-Est, 6 et 8 avenue Blaise Pascal, 77455 Marne-La-Vallée Cedex 2, France.; legoll@lami.enpc.fr; thominef@lami.enpc.fr
Florian Thomines
Affiliation:
INRIA Rocquencourt, MICMAC team-project, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France. Laboratoire Navier, École Nationale des Ponts et Chaussées, Université Paris-Est, 6 et 8 avenue Blaise Pascal, 77455 Marne-La-Vallée Cedex 2, France.; legoll@lami.enpc.fr; thominef@lami.enpc.fr
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Abstract

We address multiscale elliptic problems with random coefficients that are a perturbationof multiscale deterministic problems. Our approach consists in taking benefit of theperturbative context to suitably modify the classical Finite Element basis into adeterministic multiscale Finite Element basis. The latter essentially shares the sameapproximation properties as a multiscale Finite Element basis directly generated on therandom problem. The specific reference method that we use is the Multiscale Finite ElementMethod. Using numerical experiments, we demonstrate the efficiency of our approach and thecomputational speed-up with respect to a more standard approach. In the stationarysetting, we provide a complete analysis of the approach, extending that available for thedeterministic periodic setting.

Keywords

Weakly stochastic homogenizationfinite elementsGalerkin methodshighly oscillatory PDE

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Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 3 , May 2014 , pp. 815 - 858
Copyright
© EDP Sciences, SMAI 2014

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