Computational Geosciences

, Volume 22, Issue 1, pp 233–260 | Cite as

Comparison between cell-centered and nodal-based discretization schemes for linear elasticity

  • Halvor M. Nilsen
  • Jan Nordbotten
  • Xavier Raynaud
Original Paper


In this paper, we study newly developed methods for linear elasticity on polyhedral meshes. Our emphasis is on applications of the methods to geological models. Models of subsurface, and in particular sedimentary rocks, naturally lead to general polyhedral meshes. Numerical methods which can directly handle such representation are highly desirable. Many of the numerical challenges in simulation of subsurface applications come from the lack of robustness and accuracy of numerical methods in the case of highly distorted grids. In this paper, we investigate and compare the Multi-Point Stress Approximation (MPSA) and the Virtual Element Method (VEM) with regard to grid features that are frequently seen in geological models and likely to lead to a lack of accuracy of the methods. In particular, we look at how the methods perform near the incompressible limit. This work shows that both methods are promising for flexible modeling of subsurface mechanics.


Multi-point stress approximation Virtual element method Mimetic finite difference Geomechanics Linear elasticity Polyhedral grids 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



We thank Odd Anderson from SINTEF ICT in Oslo for his help to set up the numerical test case for the Biot model.


  1. 1.
    Aavatsmark, I.: An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci. 6, 405–432 (2002)CrossRefGoogle Scholar
  2. 2.
    Agélas, L., Guichard, C., Masson, R.: Convergence of finite volume MPFA 0 type schemes for heterogeneous anisotropic diffusion problems on general meshes. International Journal on Finite Volumes, 7 (2010)Google Scholar
  3. 3.
    Arnold, D.N., Brezzi, F., Douglas, J.: PEERS: a new mixed finite element for plane elasticity. Japan Journal of Applied Mathematics 1(2), 347–367 (1984)CrossRefGoogle Scholar
  4. 4.
    Beirão da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D., Russo, A.: Basic principles of virtual element methods. Mathematical Models and Methods in Applied Sciences 23(01), 199–214 (2013)CrossRefGoogle Scholar
  5. 5.
    Beirão da Veiga, L., Brezzi, F., Donatella Marini, L.: Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51(2), 794–812 (2013)CrossRefGoogle Scholar
  6. 6.
    Beirão da Veiga, L., Lipnikov, K., Manzini, G.: Mimetic Finite Difference Method for Elliptic Problems, volume 11 Springer (2014)Google Scholar
  7. 7.
    Biot, M.A.: General theory of three-dimensional consolidation. J. Appl. Phys. 12(2), 155–164 (1941)CrossRefGoogle Scholar
  8. 8.
    Brezzi, F., Lipnikov, K., Shashkov, M.: Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. 43(5), 1872–1896 (2005)CrossRefGoogle Scholar
  9. 9.
    Brezzi, F., Lipnikov, K., Simoncini, V.: A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 15(10), 1533–1551 (2005)CrossRefGoogle Scholar
  10. 10.
    Gain, A.L., Talischi, C., Paulino, G.H.: On the virtual element method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes. Comput. Methods Appl. Mech. Eng. 282, 132–160 (2014)CrossRefGoogle Scholar
  11. 11.
    Haga, J.B., Osnes, H., Langtangen, H.P.: On the causes of pressure oscillations in low-permeable and low-compressible porous media. Int. J. Numer. Anal. Methods Geomech. 36(12), 1507–1522 (2012)CrossRefGoogle Scholar
  12. 12.
    Keilegavlen, E., Nordbotten, J.M.: Finite volume methods for elasticity with weak symmetry. Int. J. Numer. Methods Eng. NME-Jul-16-0496.R2 (2017)Google Scholar
  13. 13.
    Klausen, R.A., Winther, R.: Convergence of multipoint flux approximations on quadrilateral grids. Numerical Methods for Partial Differential Equations 22(6), 1438–1454 (2006)CrossRefGoogle Scholar
  14. 14.
    Klausen, R.A., Winther, R.: Robust convergence of multi point flux approximation on rough grids. Numer. Math. 104(3), 317–337 (2006)CrossRefGoogle Scholar
  15. 15.
    Lie, K.-A., Krogstad, S., Ligaarden, I.S., Natvig, J.R., Nilsen, H.M., Skaflestad, B.: Open source MATLAB implementation of consistent discretisations on complex grids. Comput. Geosci. 16, 297–322 (2012)CrossRefGoogle Scholar
  16. 16.
    Lipnikov, K., Manzini, G., Shashkov, M.: Mimetic finite difference method. J. Comput. Phys. 257, 1163–1227 (2014)CrossRefGoogle Scholar
  17. 17.
    Lipnikov, K., Shashkov, M., Yotov, I.: Local flux mimetic finite difference methods. Numer. Math. 112(1), 115–152 (2009)CrossRefGoogle Scholar
  18. 18.
    The MATLAB reservoir simulation toolbox, version 2016a, 7 (2016)Google Scholar
  19. 19.
    Nagtegaal, J.C., Parks, D.M., Rice, J.R.: On numerically accurate finite element solutions in the fully plastic range. Comput. Methods Appl. Mech. Eng. 4(2), 153–177 (1974)CrossRefGoogle Scholar
  20. 20.
    Nilsen, H.M., Lie, K.-A., Natvig, J.R.: Accurate modelling of faults by multipoint, mimetic, and mixed methods. SPE J. 17(2), 568–579 (2012)CrossRefGoogle Scholar
  21. 21.
    Nordbotten, J.M.: Cell-centered finite volume discretizations for deformable porous media. Int. J. Numer. Methods Eng. 100(6), 399–418 (2014)CrossRefGoogle Scholar
  22. 22.
    Nordbotten, J.M.: Convergence of a cell-centered finite volume discretization for linear elasticity. SIAM J. Numer. Anal. 53(6), 2605–2625 (2015)CrossRefGoogle Scholar
  23. 23.
    Nordbotten, J.M.: Stable cell-centered finite volume discretization for biot equations. SIAM J. Numer. Anal. 54(2), 942–968 (2016)CrossRefGoogle Scholar
  24. 24.
    Raynaud, X., Nilsen, H.M., Andersen, O.: Virtual element method for geomechanical simulations of reservoir models. In: ECMOR XV–15th European Conference on the Mathematics of Oil Recovery, Amsterdam, Netherlands (2016)Google Scholar
  25. 25.
    Raynaud, X., Nilsen, H.M., Andersen, O.: Virtual Element Method for Geomechanics on Reservoir Grids. arXiv:1606.09508v2 (2017)
  26. 26.
    Zienkiewicz, O.C., Zhu, J.Z.: The superconvergent patch recovery and a posteriori error estimates. part 1: the recovery technique. Int. J. Numer. Methods Eng. 33(7), 1331–1364 (1992)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Applied Mathematics and CyberneticsSINTEFOsloNorway
  2. 2.Department of MathematicsUniversity of BergenBergenNorway
  3. 3.Department of Civil and Environmental EngineeringPrinceton UniversityPrincetonUSA

Personalised recommendations