Skip to main content
SpringerLink
Log in

Equal Order Discontinuous Finite Volume Element Methods for the Stokes Problem

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

The aim of this paper is to develop and analyze a family of stabilized discontinuous finite volume element methods for the Stokes equations in two and three spatial dimensions. The proposed scheme is constructed using a baseline finite element approximation of velocity and pressure by discontinuous piecewise linear elements, where an interior penalty stabilization is applied. A priori error estimates are derived for the velocity and pressure in the energy norm, and convergence rates are predicted for velocity in the \(L^2\)-norm under the assumption that the source term is locally in \( H^1\). Several numerical experiments in two and three spatial dimensions are presented to validate our theoretical findings.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Agmon, S.: Lectures on Elliptic Boundary Value Problems. AMS Providence, Rhode Island (2010)

    MATH  Google Scholar 

  2. Anaya, V., Mora, D., Ruiz-Baier, R.: An augmented mixed finite element method for the vorticity–velocity–pressure formulation of the Stokes equations. Comput. Methods Appl. Mech. Eng. 267, 261–274 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  3. Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Badia, S., Codina, R., Gudi, T., Guzmán, J.: Error analysis of discontinuous Galerkin methods for the Stokes problem under minimal regularity. IMA J. Numer. Anal. (2013). doi:10.1093/imanum/drt023

  5. Baroli, D., Quarteroni, A., Ruiz-Baier, R.: Convergence of a stabilized discontinuous Galerkin method for incompressible nonlinear elasticity. Adv. Comput. Math. 39, 425–443 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  6. Becker, R., Capatina, D., Joie, J.: Connections between discontinuous Galerkin and nonconforming finite element methods for the Stokes equations. Numer. Methods Partial Differ. Equ. 28, 1013–1041 (2012)

    Article  MathSciNet  Google Scholar 

  7. Bercovier, M., Engelman, M.: A finite element for the numerical solution of viscous incompressible flows. J. Comput. Phys. 30, 181–201 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  8. Berggren, M., Ekström, S.E., Nordström, J.: A discontinuous Galerkin extension of the vertex-centered edge-based finite volume method. Commun. Comput. Phys. 5, 456–468 (2009)

    MathSciNet  Google Scholar 

  9. Bi, C., Geng, J.: A discontinuous finite volume element method for second-order elliptic problems. Numer. Methods Partial Differ. Equ. 28, 425–440 (2012)

    Article  MATH  Google Scholar 

  10. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods, vol. 15 of Springer Series in Computational Mathematics. Springer, New York (1991)

    Book  Google Scholar 

  11. Bürger, R., Ruiz-Baier, R., Torres, H.: A stabilized finite volume element formulation for sedimentation–consolidation processes. SIAM J. Sci. Comput. 34, B265–B289 (2012)

    Article  MATH  Google Scholar 

  12. Cai, Z.: On the finite volume element method. Numer. Math. 58, 713–735 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  13. Chatzipantelidis, P., Makridakis, C., Plexousakis, M.: A-posteriori error estimates for a finite volume method for the Stokes problem in two dimensions. Appl. Numer. Math. 46, 45–58 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Chou, S.C.: Analysis and convergence of a covolume method for the generalized Stokes problem. Math. Comput. 66, 85–104 (1997)

    Article  MATH  Google Scholar 

  15. Chung, E.T., Kim, H.H., Widlund, O.B.: Two-level overlapping Schwarz algorithms for a staggered discontinuous Galerkin method. SIAM J. Numer. Anal. 51(1), 47–67 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  16. Cockburn, B., Kanschat, G., Schötzau, D.: An equal-order DG method for the incompressible Navier–Stokes equations. J. Sci. Comput. 40, 188–210 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. Cockburn, B., Kanschat, G., Schötzau, D., Schwab, C.: Local discontinuous Galerkin methods for the Stokes system. SIAM J. Numer. Anal. 40, 319–343 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  18. Cui, M., Ye, X.: Superconvergence of finite volume methods for the Stokes equations. Numer. Methods PDEs 25, 1212–1230 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  19. Cui, M., Ye, X.: Unified analysis of finite volume methods for the Stokes equations. SIAM J. Numer. Anal. 48, 824–839 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  20. Ewing, R.E., Lin, T., Lin, Y.: On the accuracy of the finite volume element method based on piecewise linear polynomials. SIAM J. Numer. Anal. 39, 1865–1888 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  21. Feistauer, M., Felcman, J., Lukác̆ová-Medvid’ová, M.: Combined finite element-finite volume solution of compressible flow. J. Comput. Appl. Math. 63, 179–199 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  22. Fortin, M.: Finite element solution of the Navier–Stokes equations. Acta Numer. 5, 239–284 (1993)

    Article  MathSciNet  Google Scholar 

  23. Ganesan, S., Matthies, G., Tobiska, L.: Local projection stabilization of equal order interpolation applied to the Stokes problem. Math. Comput. 77, 2039–2060 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  24. Gatica, G.N., Márquez, A., Sánchez, M.A.: Analysis of a velocity–pressure-pseudostress formulation for the stationary Stokes equations. Comput. Methods Appl. Mech. Eng. 199, 1064–1079 (2010)

    Article  MATH  Google Scholar 

  25. Girault, V., Rivière, B., Wheeler, M.F.: A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier–Stokes problems. Math. Comput. 74, 53–84 (2005)

    Article  MATH  Google Scholar 

  26. Girault, V., Wheeler, M.F.: Discontinuous Galerkin methods. In: Partial Differential Equations, Computational Methods in Applied Sciences, vol. 16, pp. 3–26 (2008)

  27. Hansbo, P., Larson, M.G.: Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche’s method. Comput. Methods Appl. Mech. Eng. 191, 1895–1908 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  28. Hansbo, P., Larson, M.G.: Piecewise divergence-free discontinuous Galerkin methods for Stokes flow. Commun. Numer. Meth. Eng. 24, 355–366 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  29. Kim, H.H., Chung, E.T., Lee, C.S.: A staggered discontinuous Galerkin method for the Stokes system. SIAM J. Numer. Anal. 51, 3327–3350 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  30. Kumar, S.: A mixed and discontinuous Galerkin finite volume element method for incompressible miscible displacement problems in porous media. Numer. Methods Partial Differ. Equ. 28, 1354–1381 (2012)

    Article  MATH  Google Scholar 

  31. Kumar, S., Nataraj, N., Pani, A.K.: Discontinuous Galerkin finite volume element methods for second order linear elliptic problems. Numer. Meth. Partial Differ. Equ. 25, 1402–1424 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  32. Lazarov, R., Ye, X.: Stabilized discontinuous finite element approximations for Stokes equations. J. Comput. Appl. Math. 198, 236–252 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  33. Li, J., Chen, Z.: A new stabilized finite volume method for the stationary Stokes equations. Adv. Comput. Math. 30, 141–152 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  34. Lin, Y., Liu, J., Yang, M.: Finite volume element methods: an overview of recent developments. Int. J. Numer. Anal. Model. B 4, 14–24 (2013)

    MathSciNet  Google Scholar 

  35. Liu, J., Mu, L., Ye, X.: An adaptive discontinuous finite volume element method for elliptic problems. J. Comput. Appl. Math. 235, 5422–5431 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  36. Liu, J., Mu, L., Ye, X., Jari, R.: Convergence of the discontinuous finite volume method for elliptic problems with minimal regularity. J. Comput. Appl. Math. 236, 4537–4546 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  37. Nicaise, S., Djadel, K.: Convergence analysis of a finite volume method for the Stokes system using non-conforming arguments. IMA J. Numer. Anal. 25, 523–548 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  38. Notsu, H.: Numerical computations of cavity flow problems by a pressure stabilized characteristic-curve finite element scheme. Trans. Jpn. Soc. Comput. Eng. Sci. 08, 20080032–20080051 (2008)

    Google Scholar 

  39. Quarteroni, A., Ruiz-Baier, R.: Analysis of a finite volume element method for the Stokes problem. Numer. Math. 118, 737–764 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  40. Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer, Berlin (1997)

    Google Scholar 

  41. Ruiz-Baier, R., Torres, H.: Numerical solution of a multidimensional sedimentation problem using finite volume-elements. Appl. Numer. Math. (2014). doi:10.1016/j.apnum.2013.12.006

  42. Schötzau, D., Schwab, C., Toselli, A.: Mixed hp-DGFEM for incompressible flows. II. Geometric edge meshes. IMA J. Numer. Anal. 24, 273–308 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  43. Shahbazi, K., Fischer, P.F., Ethier, C.R.: A high-order discontinuous Galerkin method for the unsteady incompressible NavierStokes equations. J. Comput. Phys. 222, 391–407 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  44. Toselli, A.: Hp discontinuous Galerkin approximations for the Stokes problem. Math. Models Methods Appl. Sci. 12, 1565–1597 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  45. Yang, Q., Jiang, Z.: A discontinuous mixed covolume method for elliptic problems. J. Comput. Appl. Math. 235, 2467–2476 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  46. Ye, X.: On the relationship between finite volume and finite element methods applied to the Stokes equations. Numer. Methods Partial Differ. Equ. 17, 440–453 (2001)

    Article  MATH  Google Scholar 

  47. Ye, X.: A new discontinuous finite volume method for elliptic problems. SIAM J. Numer. Anal. 42, 1062–1072 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  48. Ye, X.: A discontinuous finite volume method for the Stokes problem. SIAM J. Numer. Anal. 44, 183–198 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  49. Yin, Z., Jiang, Z., Xu, Q.: A discontinuous finite volume method for the Darcy–Stokes equations. J. Appl. Math. 2012, 761242–761258 (2012)

    MathSciNet  Google Scholar 

Download references

Acknowledgments

We thank Dr. Thirupathi Gudi (IISc, Bangalore) for his valuable suggestions during the early stage of this work, and we gratefully acknowledge the support by the University of Lausanne.

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Space Science and Technology, Thiruvananthapuram, 695 547, Kerala, India

    Sarvesh Kumar

  2. Institut des Sciences de la Terre, FGSE, Université de Lausanne, Géopolis Quartier Unil-Mouline, 1015, Lausanne, Switzerland

    Ricardo Ruiz-Baier

Authors
  1. Sarvesh Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ricardo Ruiz-Baier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ricardo Ruiz-Baier.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kumar, S., Ruiz-Baier, R. Equal Order Discontinuous Finite Volume Element Methods for the Stokes Problem. J Sci Comput 65, 956–978 (2015). https://doi.org/10.1007/s10915-015-9993-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-015-9993-7

Keywords

Mathematics Subject Classification

Search

Navigation