Hybrid Finite Volume Discretization of Linear Elasticity Models on General Meshes

  • Daniele A. Di Pietro
  • Robert Eymard
  • Simon Lemaire
  • Roland Masson
Conference paper
Part of the Springer Proceedings in Mathematics book series (PROM, volume 4)


This paper presents a new discretization scheme for linear elasticity models using only one degree of freedom per face corresponding to the normal component of the displacement. The scheme is based on a piecewise constant gradient construction and a discrete variational formulation for the displacement field. The tangential components of the displacement field are eliminated using a second order linear interpolation. Our main motivation is the coupling of geomechanical models and porous media flows arising in reservoir or CO2 storage simulations. Our scheme guarantees by construction the compatibility condition between the displacement discretization and the usual cell centered finite volume discretization of the Darcy flow model. In addition it applies on general meshes possibly non conforming such as Corner Point Geometries commonly used in reservoir and CO2 storage simulations.


Hybrid finite volumes linear elasticity general meshes 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M.A. Murad and A.F.D. Loula. On Stability and Convergence of Finite Element Approximations of Biot’s Consolidation Problem. Int. Jour. Numer. Eng., 37:645–667, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Methods. Springer-Verlag, New-York, 1991.Google Scholar
  3. 3.
    A. Settari and F.M. Mourits. Coupling of Geomechanics and Reservoir simulation models. Comp. Meth. Adv. Geomech., Siriwardane and Zaman and Balkema, Rotterdam, 2151–2158, 1994.Google Scholar
  4. 4.
    J. Droniou, R. Eymard, T. Gallouët and R. Herbin. A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods. Math. Models Methods Appl. Sci., 20(2):265–295, 2010.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    R. Eymard, T. Gallouët and R. Herbin. Discretisation of heterogeneous and anisotropic diffusion problems on general non-conforming meshes, SUSHI: a scheme using stabilisation and hybrid interfaces. IMA J. Numer. Anal., 30(4):1009–1043, 2010. See also
  6. 6.
    F. Brezzi, K. Lipnikov and V. Simoncini. A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci., 15:1533–1553, 2005.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    D.K. Ponting. Corner Point Geometry in reservoir simulation. In Clarendon Press, editor, Proc. ECMOR I, 45–65, Cambridge, 1989.Google Scholar
  8. 8.
    D.A. Di Pietro and J.M. Gratien. Lowest order methods for diffusive problems on general meshes: A unified approach to definition and implementation. These proceedings, 2011.Google Scholar
  9. 9.
    R. Eymard, T. Gallouët and R. Herbin. Finite Volume Methods. Handbook of Numerical Analysis, 7:713–1020, 2000.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Daniele A. Di Pietro
    • 1
  • Robert Eymard
    • 2
  • Simon Lemaire
    • 1
  • Roland Masson
    • 1
  1. 1.IFP Énergies nouvellesRueil-MalmaisonFRANCE
  2. 2.Université Paris-Est Marne-la-ValléeChamps-sur-MarneFRANCE

Personalised recommendations