Mixed Mimetic Spectral Element Method Applied to Darcy’s Problem

  • Pedro Pinto RebeloEmail author
  • Artur Palha
  • Marc Gerritsma
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 95)


We present a discretization for Darcy’s problem using the recently developed Mimetic Spectral Element Method (Kreeft et al. (2011) Mimetic framework on curvilinear quadrilaterals of arbitrary order. Submitted to FoCM, Arxiv preprint arXiv:1111.4304). The gist lies in the exact discrete representation of integral relations. In this paper, an anisotropic flow through a porous medium is considered and a discretization of a full permeability tensor is presented. The performance of the method is evaluated on standard test problems, converging at the same rate as the best possible approximation.


Edge Function Saddle Point Problem Spectral Element Method Anisotropic Permeability Positive Definite Tensor 
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The authors gratefully acknowledge the funding received by FCT – Foundation for science and technology Portugal through SRF/BD/36093/2007 and SFRH/BD/79866/2011 and the anonymous reviewers for their helpful comments.


  1. 1.
    D. Arnold, R. Falk, and R. Winther. Finite element exterior calculus: from Hodge theory to numerical stability. American Mathematical Society, 47(2):281–354, 2010.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    J. Bonelle and A. Ern. Analysis of compatible discrete operator schemes for elliptic problems on polyhedral meshes. arXiv preprint arXiv:1211.3354, 2012.Google Scholar
  3. 3.
    A Bossavit. Discretization of electromagnetic problems. Handbook of numerical analysis, 13:105–197, 2005.Google Scholar
  4. 4.
    F Brezzi, A Buffa, and K Lipnikov. Mimetic finite differences for elliptic problems. Mathematical Modelling and Numerical Analysis, 43(2):277–296, 2009.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    W. L. Burke. Applied differential geometry. Cambridge Univ Pr, 1985.Google Scholar
  6. 6.
    M. Desbrun, A. Hirani, M. Leok, and J. Marsden. Discrete exterior calculus. Arxiv preprint math/0508341, 2005.Google Scholar
  7. 7.
    T. Frankel. The Geometry of Physics. Cambridge University Press, 2nd edition, 2004.Google Scholar
  8. 8.
    M Gerritsma. Edge functions for spectral element methods. Spectral and High Order Methods for Partial differential equations, Eds J.S. Hesthaven & E.M. Rønquist, Lecture Notes in Computational Science and Engineering, 76.Google Scholar
  9. 9.
    M. Gerritsma, R. Hiemstra, J. Kreeft, A. Palha, P. Pinto Rebelo, and D. Toshniwal. The geometric basis of numerical methods. Proceedings ICOSAHOM 2012 (this issue), 2012.Google Scholar
  10. 10.
    R. Herbin and F. Hubert. Benchmark on discretization schemes for anisotropic diffusion problems on general grids. Finite volumes for complex applications V, pages 659–692, 2008.Google Scholar
  11. 11.
    R. Hiemstra and M. Gerritsma. High order methods with exact conservation properties. Proceedings ICOSAHOM 2012 (this issue), 2012.Google Scholar
  12. 12.
    R. Hiemstra, R. Huijsmans, and M. Gerritsma. High order gradient, curl and divergence conforming spaces, with an application to compatible isogeometric analysis. Submitted to J. Comp Phys., arXiv preprint arXiv:1209.1793, 2012.Google Scholar
  13. 13.
    A. Hirani. Discrete Exterior Calculus. PhD thesis, California Institute of Technology, 2003.Google Scholar
  14. 14.
    A. Hirani, K. Nakshatrala, and J. Chaudhry. Numerical method for Darcy flow derived using discrete exterior calculus. arXiv preprint arXiv:0810.3434, 2008.Google Scholar
  15. 15.
    J. Hyman, M. Shashkov, and S. Steinberg. The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials. Journal of Computational Physics, 132(1):130–148, 1997.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    J. Kreeft and M. Gerritsma. Higher-order compatible discretization on hexahedrals. Proceedings ICOSAHOM 2012 (this issue), 2012.Google Scholar
  17. 17.
    J. Kreeft and M. Gerritsma. Mixed mimetic spectral element method for stokes flow: a pointwise divergence-free solution. Journal of Computational Physics, 2012.Google Scholar
  18. 18.
    J. Kreeft and M. Gerritsma. A priori error estimates for compatible spectral discretization of the stokes problem for all admissible boundary conditions. arXiv preprint arXiv:1206.2812, 2012.Google Scholar
  19. 19.
    J. Kreeft, A. Palha, and M. Gerritsma. Mimetic framework on curvilinear quadrilaterals of arbitrary order. Submitted to FoCM, Arxiv preprint arXiv:1111.4304, 2011.Google Scholar
  20. 20.
    A. Masud and T.J.R. Hughes. A stabilized mixed finite element method for darcy flow. Computer Methods in Applied Mechanics and Engineering, 191(39):4341–4370, 2002.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    A. Palha, P. Pinto Rebelo, and M. Gerritsma. Mimetic spectral element solution for conservative advection. Proceedings ICOSAHOM 2012 (this issue), 2012.Google Scholar
  22. 22.
    A. Palha, P. Pinto Rebelo, R. Hiemstra, J. Kreeft, and M. Gerritsma. Physics-compatible discretization techniques on single and dual grids, with application to the Poisson equation of volume forms. Submitted to J. Comp Phys., 2012.Google Scholar
  23. 23.
    E Tonti. On the formal structure of physical theories. preprint of the Italian National Research Council, 1975.Google Scholar
  24. 24.
    D. Toshniwal, R.H.M. Huijsmans, and M. Gerritsma. A geometric approach towards momentum conservation. Proceedings ICOSAHOM 2012 (this issue), 2012.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Pedro Pinto Rebelo
    • 1
    Email author
  • Artur Palha
    • 1
  • Marc Gerritsma
    • 1
  1. 1.Faculty of Aerospace Engineering, Aerodynamics GroupDelft University of TechnologyDelftThe Netherlands

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