Modeling, Structure and Discretization of Hierarchical Mixed-Dimensional Partial Differential Equations

  • J. M. NordbottenEmail author
  • W. M. Boon
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 125)


Mixed-dimensional partial differential equations arise in several physical applications, wherein parts of the domain have extreme aspect ratios. In this case, it is often appealing to model these features as lower-dimensional manifolds embedded into the full domain. Examples are fractured and composite materials, but also wells (in geological applications), plant roots, or arteries and veins.

In this manuscript, we survey the structure of mixed-dimensional PDEs in the context where the sub-manifolds are a single dimension lower than the full domain, including the important aspect of intersecting sub-manifolds, leading to a hierarchy of successively lower-dimensional sub-manifolds. We are particularly interested in partial differential equations arising from conservation laws. Our aim is to provide an introduction to such problems, including the mathematical modeling, differential geometry, and discretization.



The authors wish to thank Gunnar Fløystad, Eirik Keilegavlen, Jon Eivind Vatne and Ivan Yotov for valuable comments and discussions on this topic. The authors also with to thank the two anonymous reviewers who provided helpful comments on the initial version of this manuscript. This research is funded in part by the Norwegian Research Council grants: 233736 and 250223.


  1. 1.
    J.M. Nordbotten, M.A. Celia, Geological Storage of CO2: Modeling Approaches for Large-Scale Simulation (Wiley, Hoboken, 2012)Google Scholar
  2. 2.
    J. Bear, Hydraulics of Groundwater (McGraw-Hill, New York City, 1979)Google Scholar
  3. 3.
    P.G. Ciarlet, Mathematical Elasticity Volume II: Theory of Plates (Elsevier, Amsterdam, 1997)zbMATHGoogle Scholar
  4. 4.
    C. Alboin, J. Jaffré, J.E. Roberts, C. Serres, Domain decomposition for flow in porous media with fractures, in 14th Conference on Domain Decomposition Methods in Sciences and Engineering, Cocoyoc, Mexico, 1999Google Scholar
  5. 5.
    T.H. Sandve, I. Berre, J.M. Nordbotten, An efficient multi-point flux approximation method for discrete fracture-matrix simulations. J. Comput. Phys. 231, 3784–3800 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    M. Karimi-Fard, L.J. Durlofsky, K. Aziz, An effcient discrete-fracture model applicable for general-purpose reservoir simulations. SPE J. 9, 227–236 (2004)CrossRefGoogle Scholar
  7. 7.
    V. Martin, J. Jaffré, J.E. Roberts, Modeling fractures and barriers as interfaces for flow in porous media. SIAM J. Sci. Comput. 26, 1557–1691 (2005)MathSciNetCrossRefGoogle Scholar
  8. 8.
    M.W. Licht, Complexes of discrete distributional differential forms and their homology theory. Found. Comput. Math. (2016).
  9. 9.
    Y.C. Yortsos, A theoretical analysis of vertical flow equilibrium. Transp. Porous Media 18, 107–129 (1995)CrossRefGoogle Scholar
  10. 10.
    N. Schwenk, B. Flemisch, R. Helmig, B.I. Wohlmuth, Dimensionally reduced flow models in fractured porous media. Comput. Geosci. 16, 277–296 (2012)CrossRefGoogle Scholar
  11. 11.
    L. Formaggia, A. Fumagalli, A. Scotti, P. Ruffo, A reduced model for Darcy’s problem in networks of fractures. ESAIM: Math. Model. Numer. Anal. 48, 1089–1116 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    W.M. Boon, J.M. Nordbotten, I. Yotov, Robust discretization of flow in fractured porous media. SIAM J. Numer. Anal. 56(4), 2203–2233Google Scholar
  13. 13.
    W.M. Boon, J.M. Nordbotten, J.E. Vatne, Exterior calculus for mixed-dimensional partial differential equations. arXiv:1710.00556Google Scholar
  14. 14.
    M. Spivak, Calculus on Manifolds (Addison-Wesley, Reading, 1965)zbMATHGoogle Scholar
  15. 15.
    D.N. Arnold, R.S. Falk, R. Winther, Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1–155 (2006)MathSciNetCrossRefGoogle Scholar
  16. 16.
    R. Hiptmair, Finite elements in computational electromagnetism. Acta Numer. 11, 237–339 (2002)MathSciNetCrossRefGoogle Scholar
  17. 17.
    C. D’Angelo, A. Scotti, A mixed finite element method for Darcy flow in fractured porous media with non-matching grids. ESAIM: Math. Model. Numer. Anal., 465–489 (2012)Google Scholar
  18. 18.
    N. Frih, V. Martin, J.E. Roberts, A. Saada, Modeling fractures as interfaces with nonmatching grids. Comput. Geosci. 16, 1043–10060 (2012)CrossRefGoogle Scholar
  19. 19.
    R. Helmig, C. Braun, M. Emmert, MUFTE: A Numerical Model for Simulation of Multiphase Flow Processes in Porous and Fractured Porous Media (Universität Stuttgart, Stuttgart, 1994)Google Scholar
  20. 20.
    A. Logg, K.-A. Mardal, G.N. Wells, et al., Automated Solution of Differential Equations by the Finite Element Method (Springer, Berlin, 2012)CrossRefGoogle Scholar
  21. 21.
    X. Tunc, F. I, T. Gallouët, M.C. Cacas, P. Havé, A model for conductive faults with non-matching grids. Comput. Geosci. 16, 277–296 (2012)CrossRefGoogle Scholar
  22. 22.
    X. Claeys, R. Hiptmair, Integral equations on multi-screens. Integr. Equ. Oper. Theory 77(2), 167–197 (2013)MathSciNetCrossRefGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BergenBergenNorway

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