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A Mimetic Spectral Element Method for Non-Isotropic Diffusion Problems

  • B. GervangEmail author
  • K. Olesen
  • M. Gerritsma
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 119)

Abstract

We present a mimetic spectral element method for the solution of the stationary Darcy’s problem. We show that the divergence constraint is satisfied exactly for both heterogeneous, non-isotropic, and deformed mesh problems.

References

  1. 1.
    R. Abraham, J. Marsden, T. Ratiu, Manifolds, Tensors Analysis, and Applications. Applied Mathematical Sciences, vol. 75 (Springer, Berlin, 2001)Google Scholar
  2. 2.
    M. Bouman, A. Palha, J. Kreeft, M. Gerritsma, in A Conservative Spectral Element Method for Curvilinear Domains, ed. by J.S. Hesthaven, E.M. Rønquist. Lecture Notes in Computational Sciences and Engineering, vol. 76 (Springer, Heidelberg, 2011), pp. 111–119Google Scholar
  3. 3.
    C. Canuto, M. Hussaini, A. Quarteroni, T. Zang, Spectral Methods, Fundamentals in Single Domains (Springer, Berlin, 2006)zbMATHGoogle Scholar
  4. 4.
    T. Frankel, The Geometry of Physics (Cambridge University Press, Cambridge, 2012)Google Scholar
  5. 5.
    M. Gerritsma, Edge Functions for Spectral Element Methods, in Spectral and High Order Methods for Partial Differential Equations, ed. by J.S. Hesthaven, E.M. Rønquist. Lecture Notes in Computational Science and Engineering, vol. 76 (Springer, Heidelberg, 2011), pp. 199–207Google Scholar
  6. 6.
    J.M. Hyman, M. Shashkov, S. Steinberg, The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials. J. Comput. Phys. 132, 130–148 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    J. Kreeft, A. Palha, M. Gerritsma, Mimetic framework on curvilinear quadrilaterals of arbitrary order. Arxiv preprint (2011)Google Scholar
  8. 8.
    A. Palha, P.P. Rebelo, R. Hiemstra, J. Kreeft, M. Gerritsma, Physics-compatible discretization techniques on single and dual grids, with application to the Poisson equation of volume forms. J. Comput. Phys. 257, 1394–1422 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    J.N. Reddy, An Introduction to Continuum Mechanics, 2nd edn. (Cambridge University Press, Cambridge, 2013)Google Scholar
  10. 10.
    A. Samii, C. Michoski, C. Dawson, A parallel and adaptive hybridized discontinuous Galerkin method for anisotropic nonhomogeneous diffusion. Comput. Methods Appl. Math. Eng. 134, 118–139 (2016)CrossRefMathSciNetGoogle Scholar
  11. 11.
    E. Tonti, Why starting from differential equations for computational physics? J. Comput. Phys. 257, 1260–1290 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    M. Wheeler, G. Xue, I. Yoto, A multiscale mortar multipoint flux mixed finite element method. ESIAM: M2AN 46, 759–796 (2012)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of EngineeringAarhus UniversityAarhusDenmark
  2. 2.Faculty of Aerospace EngineeringDelft University of TechnologyDelftThe Netherlands

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