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MsFEM à la Crouzeix-Raviart for Highly Oscillatory Elliptic Problems

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Book cover Partial Differential Equations: Theory, Control and Approximation

Abstract

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.

The work of the first two authors is partially supported by ONR under Grant (No. N00014-12-1-0383) and EOARD under Grant (No. FA8655-10-C-4002).

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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Abdulle, A.: Discontinuous Galerkin finite element heterogeneous multiscale method for elliptic problems with multiple scales. Math. Comput. 81(278), 687–713 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  7. Arbogast, T., Boyd, K.J.: Subgrid upscaling and mixed multiscale finite elements. SIAM J. Numer. Anal. 44(3), 1150–1171 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  9. Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction. Grundlehren der Mathematischen Wissenschaften, vol. 223. Springer, Berlin (1976)

    Book  MATH  Google Scholar 

  10. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York (2008)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cioranescu, D., Donato, P.: An Introduction to Homogenization. Oxford Lecture Series in Mathematics and Its Applications, vol. 17. Clarendon, Oxford (1999)

    MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  17. Engquist, B., Souganidis, P.: Asymptotic and Numerical Homogenization. Acta Numerica, vol. 17. Cambridge University Press, Cambridge (2008)

    Google Scholar 

  18. Ern, A., Guermond, J.L.: Theory and Practice of Finite Elements. Applied Mathematical Sciences, vol. 159. Springer, Berlin (2004)

    Book  MATH  Google Scholar 

  19. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001)

    MATH  Google Scholar 

  20. Gloria, A.: An analytical framework for numerical homogenization. Part II: Windowing and oversampling. Multiscale Model. Simul. 7(1), 274–293 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  24. Jikov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, Berlin (1994)

    Book  Google Scholar 

  25. Le Bris, C., Legoll, F., Lozinski, A.: A MsFEM type approach for perforated domains. (2013, in preparation)

    Google Scholar 

  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. Malqvist, A., Peterseim, D.: Localization of elliptic multiscale problems. arXiv:1110.0692 [math.NA]

  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)

    Article  MathSciNet  MATH  Google Scholar 

  29. Rannacher, R., Turek, S.: Simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differ. Equ. 8, 97–111 (1982)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

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

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Author notes

  1. Alexei Lozinski

    Present address: Laboratoire de Mathématiques CNRS UMR 6623, Université de Franche-Comté, 16 route de Gray, 25030, Besançon Cedex, France

Authors and Affiliations

  1. École Nationale des Ponts et Chaussées, 6 et 8 avenue Blaise Pascal, 77455, Marne-La-Vallée Cedex 2, France

    Claude Le Bris & Frédéric Legoll

  2. MICMAC project-team, INRIA Rocquencourt, 78153, Le Chesnay Cedex, France

    Claude Le Bris & Frédéric Legoll

  3. Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse Cedex 9, France

    Alexei Lozinski

Authors
  1. Claude Le Bris
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  2. Frédéric Legoll
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  3. Alexei Lozinski
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Corresponding author

Correspondence to Claude Le Bris .

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Editors and Affiliations

  1. Dept. Mathematics, City University of Hong Kong, Hong Kong, Hong Kong SAR

    Philippe G. Ciarlet

  2. School of Mathematical Sciences, Fudan University, Shanghai, People's Republic of China

    Tatsien Li

  3. Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie(Paris VI), Paris, France

    Yvon Maday

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Le Bris, C., Legoll, F., Lozinski, A. (2014). MsFEM à la Crouzeix-Raviart for Highly Oscillatory Elliptic Problems. In: Ciarlet, P., Li, T., Maday, Y. (eds) Partial Differential Equations: Theory, Control and Approximation. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41401-5_11

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