Advertisement

The nonconforming virtual element method for the Navier-Stokes equations

  • Xin Liu
  • Zhangxin Chen
Article
  • 91 Downloads

Abstract

In this paper a unified nonconforming virtual element scheme for the Navier-Stokes equations with different dimensions and different polynomial degrees is described. Its key feature is the treatment of general elements including non-convex and degenerate elements. According to the properties of an enhanced nonconforming virtual element space, the stability of this scheme is proved based on the choice of a proper velocity and pressure pair. Furthermore, we establish optimal error estimates in the discrete energy norm for velocity and the L2 norm for both velocity and pressure. Finally, we test some numerical examples to validate the theoretical results.

Keywords

Nonconforming virtual element method Navier-Stokes equations General elements Stability Energy and L2 optimal error estimates 

Mathematics Subject Classification (2010)

65N30 65N12 65N15 76D05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

The authors thank Rui Li for his valuable discussions during the stages of this research.

References

  1. 1.
    Crouzeix, M., Raviart, P. A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Serie Rouge 7, 33–75 (1973)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Chen, Z, Douglas, J.: Approximation of coefficients in hybrid and mixed methods for nonlinear parabolic problems. Mat. Aplic. Comp. 10, 137–160 (1991)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Crouzeix, M., Falk, R. S.: Nonconforming finite elements for the Stokes problem. Math. Comp. 52, 437–456 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Fortin, M., Soulie, M.: A non-conforming piecewise quadratic finite element on triangles. Int. J. Numer. Meth. Eng. 19, 505–520 (1983)CrossRefzbMATHGoogle Scholar
  5. 5.
    Matthies, G., Tobiska, L.: Inf-sup stable non-conforming finite elements of arbitrary order on triangles. Numer. Math. 102, 293–309 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Baran, A. E., Stoyan, G.: Gauss-Legendre elements: a stable, higher order non-conforming finite element family. Computing 79, 1–21 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Stoyan, G., Baran, A. E.: Crouzeix-Velte decompositions for higher-order finite elements. Comput. Math. Appl. 51, 967–986 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chen, Z.: BDM mixed methods for a nonlinear elliptic problem. J. Comput. Appl. Math. 53, 207–223 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cai, Z., Douglas, J., Ye, X.: A stable nonconforming quadrilateral finite element method for the stationary Stokes and Navier-Stokes equations. Calcolo 36, 215–232 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Matthies, G.: Inf-sup stable nonconforming finite elements of higher order on quadrilaterals and hexahedra. ESAIM-Math. Model Num. 41, 855–874 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Rannacher, R., Turek, S.: Simple nonconforming quadrilateral Stokes element. Numer. Meth. Part. D E. 8, 97–111 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fortin, M.: A three-dimensional quadratic nonconforming element. Numer. Math. 46, 269–279 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Karakashian, O. A., Jureidini, W. N.: Nonconforming finite element method for the stationary Navier-Stokes equations. SIAM J. Numer. Anal. 35, 93–120 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Beirão da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L. D., Russo, A.: Basic principles of virtual element methods. Math. Mod. Meth. Appl. Sci. 23, 199–214 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ayuso de Dios, B., Lipnikov, K., Manzini, G.: The nonconforming virtual element method. ESAIM-Math. Model Num. 50, 879–904 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Cangiani, A., Manzini, G., Sutton, O. J.: Conforming and nonconforming virtual element methods for elliptic problems. IMA J. Numer. Anal. 37, 1317–1354 (2016)MathSciNetGoogle Scholar
  17. 17.
    Brezzi, F., Marini, L. D.: Virtual element method and discontinuous Galerkin methods. In: Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations (Feng, X. B., Karakashian, Ohannes, Xing, Y. L., Editors), Lecture Notes in The IMA Volumes in Mathematics and its Applications, vol. 157, pp. 209–221. Springer (2014)Google Scholar
  18. 18.
    Brezzi, F., Falk, R. S., Marini, L. D.: Basic principles of mixed virtual element method. ESAIM-Math. Model Num. 48, 1227–1240 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Brezzi, F., Lipnikov, K., Shashkov, M., Simoncini, V.: A new discretization methodology for diffusion problems on generalized polyhedral meshes. Comput. Methods Appl. Mech. Eng. 196, 3682–3692 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Brezzi, F., Lipnikov, K., Shashkov, M.: Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. 43, 1872–1896 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Cangiani, A., Manzini, G., Russo, A.: Convergence analysis of a mimetic finite difference method for general second-order elliptic problems. SIAM J. Numer. Anal. 47, 2612–2637 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hyman, J., Shashkov, M.: Mimetic discretizations for Maxwells equations and the equations of magnetic diffusion. Prog Electromagn. Res. 32, 89–121 (2001)CrossRefGoogle Scholar
  23. 23.
    Campbell, J., Shashkov, M.: A tensor artificial viscosity using a mimetic finite difference algorithm. J. Comput. Phys. 172, 739–765 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Beirão da Veiga, L., Lipnikov, K., Manzini, G.: Convergence analysis of the high-order mimetic finite difference method. Numer. Math. 113, 325–356 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lipnikov, K., Morel, J., Shashkov, M.: Mimetic finite difference methods for diffusion equations on non-orthogonal non-conformal meshes. J. Comput. Phys. 199, 589–597 (2004)CrossRefzbMATHGoogle Scholar
  26. 26.
    Ahmed, B., Alsaedi, A., Brezzi, F., Marini, L. D., Russo, A.: Equivalent projectors for virtual element methods. Comput. Math. Appl. 66, 376–391 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Beirão da Veiga, L., Dassi, F., Russo, A.: High-order Virtual Element Method on polyhedral meshes. Comput. Math. with App. 74, 1110–1122 (2017)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Beirão da Veiga, L., Brezzi, F., Marini, L. D., Russo, A.: The hitchhiker’s guide to the virtual element method. Math. Mod. Meth. Appl. Sci. 24, 1541–1573 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Vacca, G., Beirão da Veiga, L.: Virtual element methods for parabolic problems on polygonal meshes. Numer. Meth. Part. D E. 31, 2110–2134 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Brezzi, F., Marini, L. D.: Virtual element method for plate bending problems. Comput. Methods Appl. Mech. Engrg. 253, 455–462 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Antonietti, P. F., Beirão da Veiga, L., Mora, D., Verani, M.: A stream virtual element formulation of the Stokes problem on polygonal meshes. SIAM J. Numer. Anal. 52, 386–404 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Beirão da Veiga, L., Lovadina, C., Vacca, G.: Divergence free virtual elements for the Stokes problem on polygonal meshes. ESAIM-Math. Model Num. 51, 509–535 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Cangiani, A., Gyrya, V., Manzini, G.: The nonconforming virtual element method for the Stokes equations. SIAM J. Numer. Anal. 54, 3411–3435 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Liu, X., Li, J., Chen, Z.: A nonconforming virtual element method for the Stokes problem on general meshes. Comput. Methods Appl. Mech. Engrg. 320, 694–711 (2017)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Beirão da Veiga, L., Lovadina, C., Vacca, G.: Virtual Elements for the Navier-Stokes problem on polygonal meshes, arXiv:1703.00437
  36. 36.
    Benedetto, M. F., Berrone, S., Pieraccini, S., Scialo, S.: The virtual element method for discrete fracture network simulations. Comput. Methods Appl. Mech. Engrg. 280, 135–156 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Beirão da Veiga, L., Brezzi, F., Marini, L. D.: Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51, 794–812 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Gain, A. L., Talischi, C., Paulino, G. H.: On the virtual element method for three-dimensional elasticity problems on arbitrary polyhedral meshes. Comput. Methods Appl. Mech. Engrg. 282, 132–160 (2014)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Girault, V., Raviart, P.: Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics. Springer-Verlag, Berlin (1986)CrossRefzbMATHGoogle Scholar
  40. 40.
    Chen, Z.: Finite element methods and their applications, Scientific Computation. Springer, Berlin (2005)Google Scholar
  41. 41.
    Brenner, S. C., Scott, L. R.: The mathematical theory of finite element methods, Texts in Applied Mathematics. Springer, Berlin (2008)CrossRefGoogle Scholar
  42. 42.
    Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations, Springer Series in Computational Mathematics. Springer-Verlag, Berlin (1994)zbMATHGoogle Scholar
  43. 43.
    Anderson, J. D.: Computational fluid dynamics: the basics with applications. McGraw Hill, New York (1995)Google Scholar
  44. 44.
    Beirão da Veiga, L., Brezzi, F., Marini, L. D., Russo, A.: Virtual element methods for general second order elliptic problems on polygonal meshes. Math. Mod. Meth. Appl. Sci. 26, 729–750 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Mora, D., Rivera, G., Rodriguez, R.: A virtual element method for the steklov eigenvalue problem. Math. Mod. Meth. Appl. Sci. 25, 1421–1445 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Mascotto, L.: A therapy for the ill-conditioning in the Virtual Element Method, arXiv:1705.10581
  47. 47.
    Manzini, G., Cangiani, A., Sutton, O.: The conforming virtual element method for the convection-diffusion-reaction equation with variable coeffcients, Los Alamos National Laboratory, LA-UR-14-27710.  https://doi.org/10.2172/1159207 (2014)
  48. 48.
    Adak, D., Natarajan, E.: A unified analysis of nonconforming virtual element methods for convection diffusion reaction problem, arXiv:1601.01077
  49. 49.
    Roos, H. G., Stynes, M., Tobiska, L.: Robust numerical methods for singularly perturbed differential equations: convection-diffusion-reaction and flow problems, Springer Science Business Media. Springer, Berlin (2008)zbMATHGoogle Scholar
  50. 50.
    Knobloch, P., Tobiska, L.: The \(P_{1}^{mod}\) element: A new nonconforming finite element for convection-diffusion problems. SIAM J. Numer. Anal. 41, 436–456 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    John, V., Maubach, J. M., Tobiska, L.: Nonconforming streamline-diffusion-finite-element-methods for convection-diffusion problems. Numer. Math. 78, 165–188 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsXi’an Jiaotong UniversityXi’anChina
  2. 2.Istituto di Matematica Applicata e Tecnologie Informatiche del C.N.RPaviaItaly
  3. 3.College of Petroleum EngineeringChina University of PetroleumBeijingChina
  4. 4.Department of Chemical and Petroleum Engineering, Schulich School of EngineeringUniversity of CalgaryCalgaryCanada

Personalised recommendations