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A New Projection-Based Stabilized Virtual Element Method for the Stokes Problem

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Abstract

We propose and analyze a stabilized virtual element method for the Stokes problem on polytopal meshes. We employ the \(C^0\) continuous arbitrary “equal-order” virtual element pairs to approximate both velocity and pressure, and develop a projection-based stabilization term to circumvent the discrete inf-sup condition, then we obtain the corresponding error estimates. The presented method involves neither the projection of the second derivative nor additional coupling terms, and it is parameter-free. In particularly, for the lowest-order case on triangular (tetrahedral) meshes the stabilized method introduced by Bochev et al. (SIAM J. Numer. Anal. 44: 82–101, 2006) is a special case of our method up to an approximation of the load term. Furthermore, numerical results are shown to confirm the theoretical predictions.

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References

  1. Adak, D., Natarajan, E., Kumar, S.: Convergence analysis of virtual element methods for semilinear parabolic problems on polygonal meshes. Numer. Methods Part. Differ. Equ. 35(1), 222–245 (2019)

    MathSciNet  MATH  Google Scholar 

  2. Ahmad, B., Alsaedi, A., Brezzi, F., Marini, L.D., Russo, A.: Equivalent projectors for virtual element methods. Comput. Math. Appl. 66(3), 376–391 (2013)

    MathSciNet  MATH  Google Scholar 

  3. Akbas, M., Linke, A., Rebholz, L.G., Schroeder, P.W.: The analogue of grad-div stabilization in DG methods for incompressible flows: limiting behavior and extension to tensor-product meshes. Comput. Methods Appl. Mech. Engrg. 341, 917–938 (2018)

    MathSciNet  MATH  Google Scholar 

  4. Antonietti, P.F., da Veiga, L.B., Mora, D., Verani, M.: A stream virtual element formulation of the Stokes problem on polygonal meshes. SIAM J. Numer. Anal. 52(1), 386–404 (2014)

    MathSciNet  MATH  Google Scholar 

  5. Antonietti, P.F., da Veiga, L.B., Scacchi, S., Verani, M.: A \(C^1\) virtual element method for the Cahn-Hilliard equation with polygonal meshes. SIAM J. Numer. Anal. 54(1), 34–56 (2016)

    MathSciNet  MATH  Google Scholar 

  6. Benedetto, M.F., Berrone, S., Borio, A., Pieraccini, S., Scialò, S.: Order preserving SUPG stabilization for the virtual element formulation of advection-diffusion problems. Comput. Methods Appl. Mech. Engrg. 311, 18–40 (2016)

    MathSciNet  MATH  Google Scholar 

  7. Berrone, S., Borio, A., Manzini, G.: SUPG stabilization for the nonconforming virtual element method for advection-diffusion-reaction equations. Comput. Methods Appl. Mech. Engrg. 340, 500–529 (2018)

    MathSciNet  MATH  Google Scholar 

  8. Bochev, P.B., Dohrmann, C.R., Gunzburger, M.D.: Stabilization of low-order mixed finite elements for the Stokes equations. SIAM J. Numer. Anal. 44(1), 82–101 (2006)

    MathSciNet  MATH  Google Scholar 

  9. Botti, L., Di Pietro, D.A., Droniou, J.: A Hybrid High-Order discretisation of the Brinkman problem robust in the Darcy and Stokes limits. Comput. Methods Appl. Mech. Engrg. 341, 278–310 (2018)

    MathSciNet  MATH  Google Scholar 

  10. Brenner, S.C., Guan, Q., Sung, L.-Y.: Some estimates for virtual element methods. Comput. Methods Appl. Math. 17(4), 553–574 (2017)

    MathSciNet  MATH  Google Scholar 

  11. Brezzi, F., Falk, R.S., Marini, L.D.: Basic principles of mixed virtual element methods. ESAIM Math. Model. Numer. Anal. 48(4), 1227–1240 (2014)

    MathSciNet  MATH  Google Scholar 

  12. Cangiani, A., Gyrya, V., Manzini, G.: The nonconforming virtual element method for the Stokes equations. SIAM J. Numer. Anal. 54(6), 3411–3435 (2016)

    MathSciNet  MATH  Google Scholar 

  13. Cangiani, A., Georgoulis, E.H., Pryer, T., Sutton, O.J.: A posteriori error estimates for the virtual element method. Numer. Math. 137(4), 857–893 (2017)

    MathSciNet  MATH  Google Scholar 

  14. Cao, S., Chen, L.: Anisotropic error estimates of the linear nonconforming virtual element methods. SIAM J. Numer. Anal. 57(3), 1058–1081 (2019)

    MathSciNet  MATH  Google Scholar 

  15. Chen, L., Huang, J.: Some error analysis on virtual element methods. Calcolo 55(1), 1–23 (2018)

    MathSciNet  MATH  Google Scholar 

  16. Chen, L., Wang, F.: A divergence free weak virtual element method for the Stokes problem on polytopal meshes. J. Sci. Comput. 78(2), 864–886 (2019)

    MathSciNet  MATH  Google Scholar 

  17. Chen, G., Feng, M., Xie, C.: A new projection-based stabilized method for steady convection-dominated convection-diffusion equations. Appl. Math. Comput. 239, 89–106 (2014)

    MathSciNet  MATH  Google Scholar 

  18. Clément, P.: Approximation by finite element functions using local regularization. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. 9(2), 77–84 (1975)

    MathSciNet  MATH  Google Scholar 

  19. da Veiga, L.B., Lipnikov, K.: A mimetic discretization of the Stokes problem with selected edge bubbles. SIAM J. Sci. Comput. 32(2), 875–893 (2010)

    MathSciNet  MATH  Google Scholar 

  20. da Veiga, L.B., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D., Russo, A.: Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23(1), 199–214 (2013)

    MathSciNet  MATH  Google Scholar 

  21. da Veiga, L.B., Brezzi, F., Marini, L.D.: Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51(2), 794–812 (2013)

    MathSciNet  MATH  Google Scholar 

  22. da Veiga, L.B., Brezzi, F., Marini, L.D., Russo, A.: The hitchhiker’s guide to the virtual element method. Math. Models Methods Appl. Sci. 24(8), 1541–1573 (2014)

    MathSciNet  MATH  Google Scholar 

  23. da Veiga, L.B., Brezzi, F., Marini, L.D., Russo, A.: Mixed virtual element methods for general second order elliptic problems on polygonal meshes. ESAIM Math. Model. Numer. Anal. 50(3), 727–747 (2016)

    MathSciNet  MATH  Google Scholar 

  24. da Veiga, L.B., Lovadina, C., Vacca, G.: Divergence free virtual elements for the Stokes problem on polygonal meshes. ESAIM Math. Model. Numer. Anal. 51(2), 509–535 (2017)

    MathSciNet  MATH  Google Scholar 

  25. da Veiga, L.B., Lovadina, C., Vacca, G.: Virtual elements for the Navier-Stokes problem on polygonal meshes. SIAM J. Numer. Anal. 56(3), 1210–1242 (2018)

    MathSciNet  MATH  Google Scholar 

  26. de Dios, B.A., Lipnikov, K., Manzini, G.: The nonconforming virtual element method. ESAIM Math. Model. Numer. Anal. 50(3), 879–904 (2016)

    MathSciNet  MATH  Google Scholar 

  27. Girault, V., Raviart, P.-A.: Finite Element Methods for Navier-Stokes Equations. Springer Series in Computational Mathematics, vol. 5. Springer, Berlin (1986)

    MATH  Google Scholar 

  28. Hughes, T.J.R., Franca, L.P., Balestra, M.: A new finite element formulation for computational fluid dynamics. V. Circumventing the Babuška-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Comput. Methods Appl. Mech. Engrg. 59(1), 85–99 (1986)

    MathSciNet  MATH  Google Scholar 

  29. Irisarri, D., Hauke, G.: Stabilized virtual element methods for the unsteady incompressible Navier-Stokes equations. Calcolo 56(4), 1–21 (2019)

    MathSciNet  MATH  Google Scholar 

  30. John, V., Linke, A., Merdon, C., Neilan, M., Rebholz, L.G.: On the divergence constraint in mixed finite element methods for incompressible flows. SIAM Rev. 59(3), 492–544 (2017)

    MathSciNet  MATH  Google Scholar 

  31. Kirk, K.L.A., Rhebergen, S.: Analysis of a pressure-robust hybridized discontinuous Galerkin method for the stationary Navier-Stokes equations. J. Sci. Comput. 81(2), 881–897 (2019)

    MathSciNet  MATH  Google Scholar 

  32. Li, Y., Feng, M.: A local projection stabilization virtual finite element method for convection-diffusion-reaction equation. preprint, (2020)

  33. Liu, X., Chen, Z.: The nonconforming virtual element method for the Navier-Stokes equations. Adv. Comput. Math. 45(1), 51–74 (2019)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  35. Mora, D., Rivera, G., Rodríguez, R.: A virtual element method for the Steklov eigenvalue problem. Math. Models Methods Appl. Sci. 25(8), 1421–1445 (2015)

    MathSciNet  MATH  Google Scholar 

  36. Mu, L., Wang, J., Ye, X., Zhang, S.: A discrete divergence free weak Galerkin finite element method for the Stokes equations. Appl. Numer. Math. 125, 172–182 (2018)

    MathSciNet  MATH  Google Scholar 

  37. Oikawa, I.: Analysis of a reduced-order HDG method for the Stokes equations. J. Sci. Comput. 67(2), 475–492 (2016)

    MathSciNet  MATH  Google Scholar 

  38. Talischi, C., Pereira, A., Paulino, G.H., Menezes, I.F.M., Carvalho, M.S.: Polygonal finite elements for incompressible fluid flow. Int. J. Numer. Methods Fluids 74(2), 134–151 (2014)

    MathSciNet  Google Scholar 

  39. Zhang, B., Feng, M.: Virtual element method for two-dimensional linear elasticity problem in mixed weakly symmetric formulation. Appl. Math. Comput. 328, 1–25 (2018)

    MathSciNet  MATH  Google Scholar 

  40. Zhang, Q., Gao, F.: A fully-discrete local discontinuous Galerkin method for convection-dominated Sobolev equation. J. Sci. Comput. 51(1), 107–134 (2012)

    MathSciNet  MATH  Google Scholar 

  41. Zhang, B., Yang, Y., Feng, M.: Mixed virtual element methods for elastodynamics with weak symmetry. J. Comput. Appl. Math. 353, 49–71 (2019)

    MathSciNet  MATH  Google Scholar 

  42. Zhao, J., Zhang, B., Mao, S., Chen, S.: The divergence-free nonconforming virtual element for the Stokes problem. SIAM J. Numer. Anal. 57(6), 2730–2759 (2019)

    MathSciNet  MATH  Google Scholar 

  43. Zheng, X., Chen, G., Xie, X.: A divergence-free weak Galerkin method for quasi-Newtonian Stokes flows. Sci. China Math. 60(8), 1515–1528 (2017)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. College of Mathematics, Sichuan University, Chengdu, 610064, China

    Jun Guo & Minfu Feng

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  1. Jun Guo
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  2. Minfu Feng
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Correspondence to Minfu Feng.

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Foundation item: Supported by the National Natural Science Foundation of China (No. 11971337 ) and the Key Fund Project of Sichuan Provincial Department of Education (No. 18ZA0276).

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Guo, J., Feng, M. A New Projection-Based Stabilized Virtual Element Method for the Stokes Problem. J Sci Comput 85, 16 (2020). https://doi.org/10.1007/s10915-020-01301-1

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