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On the semi-discrete stabilized finite volume method for the transient Navier–Stokes equations

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Abstract

A stabilized finite volume method for solving the transient Navier–Stokes equations is developed and studied in this paper. This method maintains conservation property associated with the Navier–Stokes equations. An error analysis based on the variational formulation of the corresponding finite volume method is first introduced to obtain optimal error estimates for velocity and pressure. This error analysis shows that the present stabilized finite volume method provides an approximate solution with the same convergence rate as that provided by the stabilized linear finite element method for the Navier–Stokes equations under the same regularity assumption on the exact solution and a slightly additional regularity on the source term. The stability and convergence results of the proposed method are also demonstrated by the numerical experiments presented.

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References

  1. Bank, R.E., Rose, D.J.: Some error estimates for the box method. SIAM J. Numer. Anal. 24, 777–787 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bochev, P., Dohrmann, C.R., Gunzburger, M.D.: A computational study of stabilized, low 581 order C 0 finite element approximations of Darcy equations. Comput. Mech. 38, 323–333 (2006)

    Google Scholar 

  3. Bochev, P., Dohrmann, C.R., Gunzburger, M.D.: Stabilization of low-order mixed finite elements for the Stokes equations. SIAM J. Numer. Anal. 44, 82–101 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, Z.: The control volume finite element methods and their applications to multiphase flow. Netw. Heterog. Media 1, 689–706 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, Z., Li, R., Zhou, A.: A note on the optimal L 2-estimate of finite volume element method. Adv. Comput. Math. 16, 291–303 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chou, S.H., Li, Q.: Error estimates in L 2, H 1 and L  ∞  in co-volume methods for elliptic and parabolic problems: a unified approach. Math. Comput. 69, 103–120 (2000)

    MathSciNet  MATH  Google Scholar 

  8. Chatzipantelidis, P.: A finite volume method based on the Crouzeix–Raviart element for elliptic PDEs in two dimensions. Numer. Math. 82, 409–432 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chatzipantelidis, P., Lazarov, R.D., Thomée, V.: Error estimates for a finite volume method for parabolic equations in convex polygonal domains. Numer. Methods Partial Differ. Equ. 20, 650–674 (2004)

    Article  MATH  Google Scholar 

  10. Chen, Z.: Finite Element Methods and Their Applications. Spring-Verlag, Heidelberg (2005)

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  12. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

  13. Dohrmann, C.R., Bochev, P.: A stabilized finite element method for the Stokes problem based on polynomial pressure projections. Int. J. Numer. Methods Fluids 46, 183–201 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ewing, R.E., Lazarov, R.D., Lin, Y.: Finite volume element approximations of nonlocal reactive flows in porous media. Numer. Methods Partial Differ. Equ. 16, 285–311 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. 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  MathSciNet  MATH  Google Scholar 

  16. Eymard, R., Gallouét, T., Herbin, R.: Finite volume methods. Handbook of Numerical Analysis VII, vol. 46, pp. 713–1020. North Holland (2000)

  17. Eymard, R., Herbin, R., Latché, J.C.: On a stabilized colocated finite volume scheme for the Stokes problem. ESAIM: M2AN 40, 501–528 (2006)

    Article  MATH  Google Scholar 

  18. Eymard, R., Herbin, R., Latché, J.C., Piar, B.: Convergence analysis of a locally stabilized collocated finite volume scheme for incompressible flows. ESAIM: M2AN 43, 889–927 (2009)

    Article  MATH  Google Scholar 

  19. Girault, V., Raviart, P.A.: Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms. Springer-Verlag, Berlin (1986)

    Book  MATH  Google Scholar 

  20. He, Y., Li, J.: A stabilized finite element method based on local polynomial pressure projection for the stationary Navier–Stokes equation. Appl. Numer. Math. 58, 1503–1514 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. He, Y., Lin, Y., Sun, W.: Stabilized finite element method for the non-stationary Navier–Stokes problem. Discrete Continuous Dyn. Syst. Ser. B 6, 41–68 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier–Stokes problem I: Regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal. 19, 275–311 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  23. Huang, J., Xi, S.: On the finite volume element method for general self-adjoint elliptic problems. SIAM J. Numer. Anal. 35, 1762–1774 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hill, A.T., Süli, E.: Approximation of the global attractor for the incompressible Navier–Stokes problem. IMA J. Numer. Anal. 20, 633–667 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Li, J., He, Y.: A new stabilized finite element method based on two local Gauss integration for the Stokes equations. J. Comput. Appl. Math. 214, 58–65 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Li, J., He, Y., Chen, Z.: A new stabilized finite element method for the transient Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 197, 22–35 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Li, J., He, Y., Chen, Z.: Performance of several stabilized finite element methods for the Stokes equations based on the lowest equal-order pairs. Computing 86, 37–51 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Li, J., Shen, L., Chen, Z.: Convergence and stability of a stabilized finite volume method for the stationary Navier–Stokes equations. BIT Numer. Math. 50, 823–842 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Li, R.: Generalized difference methods for a nonlinear Dirichlet problem. SIAM J. Numer. Anal. 24, 77–88 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  31. Li, R., Zhu, P.: Generalized difference methods for second order elliptic partial differential equations (I)-triangle grids. Numer. Math. J. Chinese Universities 2, 140–152 (1982)

    Google Scholar 

  32. Li, R., Chen, Z., Wu, W.: Generalized Difference Methods for Differential Equations: Numerical Analysis of Finite Volume Methods. Marcel Dekker, New York (2000)

    MATH  Google Scholar 

  33. Lv, J., Li, Y.: L 2 error estimate of the finite volume element methods on quadrilateral meshes. Adv. Comput. Math. 33, 129–148 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Shen, J.: On error estimates of the penalty method for unsteady Navier–Stokes equations. SIAM J. Numer. Anal. 32, 386–403 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  35. Temam, R.: Navier–Stokes Equations. North-Holland, Amsterdam (1984)

  36. Thomée, V.: Galerkin finite element methods for parabolic problems. Lecture Notes in Math., vol. 1054, Springer-Verlag, Berlin (1984)

  37. Wang, J., Wang, Y., Ye, X.: A new finite volume method for the stokes problems. Int. J. Numer. Anal. Model. 7, 281–302 (2009)

    MathSciNet  Google Scholar 

  38. Xu, J., Zhu, Y., Zou, Q.: New adaptive finite volume methods and convergence analysis. (submitted)

  39. Xu, J., Zou, Q.: Analysis of linear and quadratic finite volume methods for elliptic equatiosns. Numer. Math. 111, 469–492 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  40. Ye, X.: On the relationship between finite volume and finite element methods applied to the Stokes equations. Numer. Methods Partial Differ. Equ. 5, 440–453 (2001)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Baoji University of Arts and Sciences, Baoji, 721013, People’s Republic of China

    Jian Li

  2. Department of Chemical & Petroleum Engineering, Schulich School of Engineering, University of Calgary, 2500 University Drive N.W., Calgary, AB, T2N 1N4, Canada

    Jian Li & Zhangxin Chen

  3. Faculty of Science, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China

    Zhangxin Chen

Authors
  1. Jian Li
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  2. Zhangxin Chen
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Correspondence to Jian Li.

Additional information

Communicated by Zhongying Chen.

This research was supported in part by the National Science Foundation of China (No. 11071193), Natural Science New Star of Science and Technologies Research Plan in Shaanxi Province of China (No. 2011kjxx12), Research Program of Education Department of Shaanxi Province (No. 11JK0490), the project-sponsored by SRF for ROCS, SEM, and NSERC/AERI/Foundation CMG Chair and iCORE Chair Funds in Reservoir Simulation.

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Li, J., Chen, Z. On the semi-discrete stabilized finite volume method for the transient Navier–Stokes equations. Adv Comput Math 38, 281–320 (2013). https://doi.org/10.1007/s10444-011-9237-9

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  • DOI: https://doi.org/10.1007/s10444-011-9237-9

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