Multiscale Finite Element approach for “weakly” random problems and related issues

Claude Le Bris; Frédéric Legoll; Florian Thomines

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2014)

  • Volume: 48, Issue: 3, page 815-858
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Le Bris, Claude, Legoll, Frédéric, and Thomines, Florian. "Multiscale Finite Element approach for “weakly” random problems and related issues." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.3 (2014): 815-858. <http://eudml.org/doc/273339>.

@article{LeBris2014,
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.},
author = {Le Bris, Claude, Legoll, Frédéric, Thomines, Florian},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {weakly stochastic homogenization; finite elements; Galerkin methods; highly oscillatory PDE; numerical experiment},
language = {eng},
number = {3},
pages = {815-858},
publisher = {EDP-Sciences},
title = {Multiscale Finite Element approach for “weakly” random problems and related issues},
url = {http://eudml.org/doc/273339},
volume = {48},
year = {2014},
}

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 - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 3
SP - 815
EP - 858
AB - 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.
LA - eng
KW - weakly stochastic homogenization; finite elements; Galerkin methods; highly oscillatory PDE; numerical experiment
UR - http://eudml.org/doc/273339
ER -

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