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A priori error estimate of a multiscale finite element method for transport modeling

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Abstract

This work proposes an a priori error estimate of a multiscale finite element method to solve convection-diffusion problems where both velocity and diffusion coefficient exhibit strong variations at a scale which is much smaller than the domain of resolution. In that case, classical discretization methods, used at the scale of the heterogeneities, turn out to be too costly. Our method, introduced in Allaire et al. (A Multiscale Finite Element Method for Transport Modeling. September 10–14, 2012), aims at solving this kind of problems on coarser grids with respect to the size of the heterogeneities by means of particular basis functions. These basis functions are defined using cell problems and are designed to reproduce the variations of the solution on an underlying fine grid. Since all cell problems are independent from each other, these problems can be solved in parallel, which makes the method very efficient when used on parallel architectures. This article focuses on the proof of an a priori error estimate of this method.

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Acknowledgments

The results presented in this paper are part of F. Ouaki’s PhD work which was supported by IFP Energies nouvelles.

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Authors and Affiliations

  1. CEA Saclay, DEN, DMN, SRMA, Bat 453 p26D, 91191, Gif/Yvette, France

    Franck Ouaki

  2. CMAP UMR-CNRS 7641, École Polytechnique, 91128, Palaiseau Cedex, France

    Grégoire Allaire

  3. IFP Energies nouvelles, 1 et 4 avenue de Bois-Préau, 92852, Rueil-Malmaison, France

    Sylvain Desroziers & Guillaume Enchéry

Authors
  1. Franck Ouaki
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  2. Grégoire Allaire
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  3. Sylvain Desroziers
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  4. Guillaume Enchéry
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Corresponding author

Correspondence to Franck Ouaki.

Additional information

G. Allaire is a member of the DEFI project at INRIA Saclay Ile-de-France.

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Ouaki, F., Allaire, G., Desroziers, S. et al. A priori error estimate of a multiscale finite element method for transport modeling. SeMA 67, 1–37 (2015). https://doi.org/10.1007/s40324-014-0023-8

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  • DOI: https://doi.org/10.1007/s40324-014-0023-8

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