MsFEM à la Crouzeix-Raviart for Highly Oscillatory Elliptic Problems



We introduce and analyze a multiscale finite element type method (MsFEM) in the vein of the classical Crouzeix-Raviart finite element method that is specifically adapted for highly oscillatory elliptic problems. We illustrate numerically the efficiency of the approach and compare it with several variants of MsFEM.


Homogenization Finite elements Galerkin methods Highly oscillatory PDE 

Mathematics Subject Classification

35B27 65M60 65M12 



The third author acknowledges the hospitality of INRIA. We thank William Minvielle for his remarks on a preliminary version of this article.


  1. 1.
    Aarnes, J.: On the use of a mixed multiscale finite element method for greater flexibility and increased speed or improved accuracy in reservoir simulation. Multiscale Model. Simul. 2(3), 421–439 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aarnes, J., Heimsund, B.O.: Multiscale discontinuous Galerkin methods for elliptic problems with multiple scales. In: Engquist, B., Lötstedt, P., Runborg, O. (eds.) Multiscale Methods in Science and Engineering. Lecture Notes in Computational Science and Engineering, vol. 44, pp. 1–20. Springer, Berlin (2005) CrossRefGoogle Scholar
  3. 3.
    Abdulle, A.: Multiscale method based on discontinuous Galerkin methods for homogenization problems. C. R. Math. Acad. Sci. Paris 346(1–2), 97–102 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Abdulle, A.: Discontinuous Galerkin finite element heterogeneous multiscale method for elliptic problems with multiple scales. Math. Comput. 81(278), 687–713 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Arbogast, T.: Implementation of a locally conservative numerical subgrid upscaling scheme for two-phase Darcy flow. Comput. Geosci. 6(3–4), 453–481 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Arbogast, T.: Mixed multiscale methods for heterogeneous elliptic problems. In: Graham, I.G., Hou, T.Y., Lakkis, O., Scheichl, R. (eds.) Numerical Analysis of Multiscale Problems. Lecture Notes in Computational Science and Engineering, vol. 83, pp. 243–283. Springer, Berlin (2011) CrossRefGoogle Scholar
  7. 7.
    Arbogast, T., Boyd, K.J.: Subgrid upscaling and mixed multiscale finite elements. SIAM J. Numer. Anal. 44(3), 1150–1171 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bensoussan, A., Lions, J.L., Papanicolaou, G.: Asymptotic analysis for periodic structures. Studies in Mathematics and Its Applications, vol. 5. North-Holland, Amsterdam (1978) zbMATHGoogle Scholar
  9. 9.
    Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction. Grundlehren der Mathematischen Wissenschaften, vol. 223. Springer, Berlin (1976) CrossRefzbMATHGoogle Scholar
  10. 10.
    Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York (2008) CrossRefzbMATHGoogle Scholar
  11. 11.
    Chen, Z., Cui, M., Savchuk, T.Y.: The multiscale finite element method with nonconforming elements for elliptic homogenization problems. Multiscale Model. Simul. 7(2), 517–538 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chen, Z., Hou, T.Y.: A mixed multiscale finite element method for elliptic problems with oscillating coefficients. Math. Comput. 72(242), 541–576 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cioranescu, D., Donato, P.: An Introduction to Homogenization. Oxford Lecture Series in Mathematics and Its Applications, vol. 17. Clarendon, Oxford (1999) zbMATHGoogle Scholar
  14. 14.
    Crouzeix, M., Raviart, P.A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I. RAIRO. Anal. Numér. 7(3), 33–75 (1973) MathSciNetGoogle Scholar
  15. 15.
    Efendiev, Y., Hou, T.Y.: Multiscale Finite Element Method: Theory and Applications. Surveys and Tutorials in the Applied Mathematical Sciences, vol. 4. Springer, New York (2009) Google Scholar
  16. 16.
    Efendiev, Y.R., Hou, T.Y., Wu, X.H.: Convergence of a nonconforming multiscale finite element method. SIAM J. Numer. Anal. 37(3), 888–910 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Engquist, B., Souganidis, P.: Asymptotic and Numerical Homogenization. Acta Numerica, vol. 17. Cambridge University Press, Cambridge (2008) Google Scholar
  18. 18.
    Ern, A., Guermond, J.L.: Theory and Practice of Finite Elements. Applied Mathematical Sciences, vol. 159. Springer, Berlin (2004) CrossRefzbMATHGoogle Scholar
  19. 19.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001) zbMATHGoogle Scholar
  20. 20.
    Gloria, A.: An analytical framework for numerical homogenization. Part II: Windowing and oversampling. Multiscale Model. Simul. 7(1), 274–293 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hou, T.Y., Wu, X.H.: A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134(1), 169–189 (1997) MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hou, T.Y., Wu, X.H., Cai, Z.: Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comput. 68(227), 913–943 (1999) MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hou, T.Y., Wu, X.H., Zhang, Y.: Removing the cell resonance error in the multiscale finite element method via a Petrov-Galerkin formulation. Commun. Math. Sci. 2(2), 185–205 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Jikov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, Berlin (1994) CrossRefGoogle Scholar
  25. 25.
    Le Bris, C., Legoll, F., Lozinski, A.: A MsFEM type approach for perforated domains. (2013, in preparation) Google Scholar
  26. 26.
    Le Bris, C., Legoll, F., Thomines, F.: Multiscale Finite Element approach for “weakly” random problems and related issues. arXiv:1111.1524 [math.NA]
  27. 27.
    Malqvist, A., Peterseim, D.: Localization of elliptic multiscale problems. arXiv:1110.0692 [math.NA]
  28. 28.
    Owhadi, H., Zhang, L.: Localized bases for finite dimensional homogenization approximations with non-separated scales and high-contrast. Multiscale Model. Simul. 9, 1373–1398 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Rannacher, R., Turek, S.: Simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differ. Equ. 8, 97–111 (1982) MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.École Nationale des Ponts et ChausséesMarne-La-Vallée Cedex 2France
  2. 2.MICMAC project-teamINRIA RocquencourtLe Chesnay CedexFrance
  3. 3.Institut de Mathématiques de ToulouseUniversité Paul SabatierToulouse Cedex 9France
  4. 4.Laboratoire de Mathématiques CNRS UMR 6623Université de Franche-ComtéBesançon CedexFrance

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