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A Quadratic Discontinuous Finite Volume Element Scheme for Stokes Problems

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Abstract

In this paper, we construct a quadratic discontinuous finite volume scheme for Stokes equations. Specifically, the quadratic discontinuous finite volume method (DFVM) is composed of the discontinuous Galerkin method with incomplete interior penalty term and the finite volume method. By means of special mapping, we simplify the bilinear form of DFVM and establish a relationship with that of the discontinuous Galerkin scheme, and proving the LBB condition. Then, we further rigorously prove that both the broken \(H^1\) norm error of the velocity and the standard \(L^2\) norm error of the pressure of this scheme converge to the optimal order. Numerical experiments are presented to confirm the theoretical results.

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The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

References

  1. Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Tech. Rep., Los Alamos Scientific Lab., N. Mex. (USA) (1973)

  2. Wheeler, M.F.: An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15(1), 152–161 (1978)

    Article  MathSciNet  Google Scholar 

  3. Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19(4), 742–760 (1982)

    Article  MathSciNet  Google Scholar 

  4. Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2002)

    Article  MathSciNet  Google Scholar 

  5. Babuška, I.: The finite element method with penalty. Math. Comput. 27(122), 221–228 (1973)

    Article  MathSciNet  Google Scholar 

  6. Babuška, I., Zlámal, M.: Nonconforming elements in the finite element method with penalty. SIAM J. Numer. Anal. 10(5), 863–875 (1973)

    Article  MathSciNet  Google Scholar 

  7. Baumann, C.E., Oden, J.T.: A discontinuous HP finite element method for convection-diffusion problems. Comput. Methods Appl. Mech. Eng. 175(3–4), 311–341 (1999)

    Article  MathSciNet  Google Scholar 

  8. Cockburn, B., Shu, C.W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35(6), 2440–2463 (1998)

    Article  MathSciNet  Google Scholar 

  9. Chou, S.: Analysis and convergence of a covolume method for the generalized Stokes problem. Math. Comput. 66(217), 85–104 (1997)

    Article  MathSciNet  Google Scholar 

  10. Ye, X.: On the relationship between finite volume and finite element methods applied to the Stokes equations. Numer. Methods Partial Differ. Equ. Int. J. 17(5), 440–453 (2001)

    Article  MathSciNet  Google Scholar 

  11. Rui, H.: Symmetric modified finite volume element methods for self-adjoint elliptic and parabolic problems. J. Comput. Appl. Math. 146(2), 373–386 (2002)

    Article  MathSciNet  Google Scholar 

  12. Rui, H.: Analysis on a finite volume element method for Stokes problems. Acta Math. Appl. Sin. 21(3), 359–372 (2005)

    Article  MathSciNet  Google Scholar 

  13. Eymard, R., Herbin, R., Latché, J.C.: On a stabilized colocated finite volume scheme for the Stokes problem. ESAIM: Math. Model. Numer. Anal. 40(3), 501–527 (2006)

    Article  MathSciNet  Google Scholar 

  14. Zhang, T., Tang, L.: A stabilized finite volume method for Stokes equations using the lowest order P1–P0 element pair. Adv. Comput. Math. 41(4), 781–798 (2015)

    Article  MathSciNet  Google Scholar 

  15. Yang, H., Li, Y.: The mixed finite volume method for stokes problem based on mini element pair. Int. J. Numer. Anal. Model. 20(1), 134 (2022)

    Article  MathSciNet  Google Scholar 

  16. Chen, L.: A new class of high order finite volume methods for second order elliptic equations. SIAM J. Numer. Anal. 47(6), 4021–4043 (2010)

    Article  MathSciNet  Google Scholar 

  17. Chen, Z., Wu, J., Xu, Y.: Higher-order finite volume methods for elliptic boundary value problems. Adv. Comput. Math. 37(2), 191–253 (2012)

    Article  MathSciNet  Google Scholar 

  18. Chen, Z., Xu, Y., Zhang, Y.: A construction of higher-order finite volume methods. Math. Comput. 84(292), 599–628 (2015)

    Article  MathSciNet  Google Scholar 

  19. Chen, Z., Xu, Y., Zhang, Y.: Higher-order finite volume methods II: Inf-sup condition and uniform local ellipticity. J. Comput. Appl. Math. 265, 96–109 (2014)

    Article  MathSciNet  Google Scholar 

  20. Xu, J., Zou, Q.: Analysis of linear and quadratic simplicial finite volume methods for elliptic equations. Numer. Math. 111(3), 469–492 (2009)

    Article  MathSciNet  Google Scholar 

  21. Zou, Q.: An unconditionally stable quadratic finite volume scheme over triangular meshes for elliptic equations. J. Sci. Comput. 70(1), 112–124 (2017)

    Article  MathSciNet  Google Scholar 

  22. Wang, X., Li, Y.: L\(^{\hat{}}2\) Error estimates for high order finite volume methods on triangular meshes. SIAM J. Numer. Anal. 54(5), 2729–2749 (2016)

    Article  MathSciNet  Google Scholar 

  23. Ye, X.: A new discontinuous finite volume method for elliptic problems. SIAM J. Numer. Anal. 42(3), 1062–1072 (2004)

    Article  MathSciNet  Google Scholar 

  24. Wang, G., He, Y., Li, R.: Discontinuous finite volume methods for the stationary Stokes–Darcy problem. Int. J. Numer. Methods Eng. 107(5), 395–418 (2016)

    Article  MathSciNet  Google Scholar 

  25. Ye, X.: A discontinuous finite volume method for the Stokes problems. SIAM J. Numer. Anal. 44(1), 183–198 (2006)

    Article  MathSciNet  Google Scholar 

  26. Li, R., Gao, Y., Chen, J., Zhang, L., He, X., Chen, Z.: Discontinuous finite volume element method for a coupled Navier–Stokes–Cahn–Hilliard phase field model. Adv. Comput. Math. 46, 1–35 (2020)

    Article  MathSciNet  Google Scholar 

  27. Li, R., Gao, Y., Li, J., Chen, Z.: Discontinuous finite volume element method for a coupled non-stationary Stokes–Darcy problem. J. Sci. Comput. 74, 693–727 (2018)

    Article  MathSciNet  Google Scholar 

  28. Wang, J., Wang, Y., Ye, X.: A new finite volume method for the Stokes problems. Int. J. Numer. Anal. Model 7(2), 281–302 (2010)

    MathSciNet  Google Scholar 

  29. Cui, M., Ye, X.: Unified analysis of finite volume methods for the Stokes equations. SIAM J. Numer. Anal. 48(3), 824–839 (2010)

    Article  MathSciNet  Google Scholar 

  30. Kumar, S., Ruiz-Baier, R.: Equal order discontinuous finite volume element methods for the Stokes problem. J. Sci. Comput. 65(3), 956–978 (2015)

    Article  MathSciNet  Google Scholar 

  31. Wang, J., Wang, Y., Ye, X.: A unified a posteriori error estimator for finite volume methods for the Stokes equations. Math. Methods Appl. Sci. 41(3), 866–880 (2018)

    Article  MathSciNet  Google Scholar 

  32. Chen, Z., Xu, Y., Zhang, J.: A second-order hybrid finite volume method for solving the Stokes equation. Appl. Numer. Math. 119, 213–224 (2017)

    Article  MathSciNet  Google Scholar 

  33. Zhang, J.: A family of quadratic finite volume method for solving the Stokes equation. Comput. Math. Appl. 117, 155–186 (2022)

    Article  MathSciNet  Google Scholar 

  34. Rivière, B.: Discontinuous Galerkin methods for solving elliptic and parabolic equations: theory and implementation. SIAM (2008)

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Acknowledgements

The authors wish to thank the reviewers for their constructive comments and suggestions to improve the paper.

Funding

Projects supported by the National Natural Science Foundation of China Grant No. 12131014.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Ningxia University, Yinchuan, 750021, China

    Yuzhi Lou

  2. School of Mathematics, Shandong University, Jinan, 250100, China

    Hongxing Rui

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  1. Yuzhi Lou
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  2. Hongxing Rui
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Correspondence to Hongxing Rui.

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Lou, Y., Rui, H. A Quadratic Discontinuous Finite Volume Element Scheme for Stokes Problems. J Sci Comput 99, 44 (2024). https://doi.org/10.1007/s10915-024-02506-4

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  • DOI: https://doi.org/10.1007/s10915-024-02506-4

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