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SeMA Journal

, Volume 67, Issue 1, pp 1–37 | Cite as

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

  • Franck OuakiEmail author
  • Grégoire Allaire
  • Sylvain Desroziers
  • Guillaume Enchéry
Article

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.

Keywords

Convection-diffusion Periodic homogenization  Multiscale finite element method 

Mathematics Subject Classification

35B27 74Q10 65M55 

Notes

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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Copyright information

© Sociedad Española de Matemática Aplicada 2014

Authors and Affiliations

  • Franck Ouaki
    • 1
    Email author
  • Grégoire Allaire
    • 2
  • Sylvain Desroziers
    • 3
  • Guillaume Enchéry
    • 3
  1. 1.CEA Saclay, DEN, DMN, SRMAGif/YvetteFrance
  2. 2.CMAP UMR-CNRS 7641, École PolytechniquePalaiseau CedexFrance
  3. 3.IFP Energies nouvellesRueil-MalmaisonFrance

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