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Multiscale Finite Element approach for “weakly” random problems and 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 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 homogenizationfinite elementsGalerkin methodshighly oscillatory PDE
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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