Skip to main content
Erschienen in: Calcolo 4/2021

01.12.2021

Lowest order virtual element approximations for transient Stokes problem on polygonal meshes

verfasst von: N. Verma, S. Kumar

Erschienen in: Calcolo | Ausgabe 4/2021

Einloggen, um Zugang zu erhalten

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

In this paper, we discuss and analyze virtual element approximations for the nonstationary Stokes problem on polygonal meshes. The proposed scheme is based on pressure-velocity formulations, and the virtual element spaces associated with velocity and pressure are constructed in a way that they obey the discrete inf-sup (LBB) condition. The spatial discretization for velocity is based on piecewise linear polynomials as well as non-linear functions, and the pressure approximation is relied on discontinuous piecewise constant polynomials, whereas a backward Euler method is employed for the time discretization. By introducing suitable energy and \(L^2\) projection operators, the optimal error estimates are established in \(H^1\) and \(L^2\) norms for both semi and fully discrete schemes under the minimal regularity assumptions on continuous solutions. Moreover, several numerical experiments are conducted to validate the obtained theoretical rate of convergence and exhibit the performance of the proposed scheme.
Literatur
1.
Zurück zum Zitat Ahmad, B., Alsaedi, A., Brezzi, F., Marini, L.D., Russo, A.: Equivalent projectors for virtual element methods. Comput. Math. Appl. 66, 376–391 (2013)MathSciNetMATH Ahmad, B., Alsaedi, A., Brezzi, F., Marini, L.D., Russo, A.: Equivalent projectors for virtual element methods. Comput. Math. Appl. 66, 376–391 (2013)MathSciNetMATH
2.
Zurück zum Zitat Ahmed, N., Linke, A., Merdon, C.: On really locking-free mixed finite element methods for the transient incompressible Stokes equations. SIAM J. Numer. Anal. 56(1), 185–209 (2018)MathSciNetMATH Ahmed, N., Linke, A., Merdon, C.: On really locking-free mixed finite element methods for the transient incompressible Stokes equations. SIAM J. Numer. Anal. 56(1), 185–209 (2018)MathSciNetMATH
3.
Zurück zum Zitat 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, 386–404 (2014)MathSciNetMATH 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, 386–404 (2014)MathSciNetMATH
4.
Zurück zum Zitat Bänsch, E., Karakatsani, F., Makridakis, C.G.: A posteriori error estimates for fully discrete schemes for the time dependent Stokes problem. Calcolo 55(19), 1–32 (2018)MathSciNetMATH Bänsch, E., Karakatsani, F., Makridakis, C.G.: A posteriori error estimates for fully discrete schemes for the time dependent Stokes problem. Calcolo 55(19), 1–32 (2018)MathSciNetMATH
5.
Zurück zum Zitat da Veiga, L.B., Lipnikov, K., Manzini, G.: Error analysis for a mimetic discretization of the steady Stokes problem on polyhedral meshes. SIAM J. Numer. Anal. 48(4), 1419–1443 (2010)MathSciNetMATH da Veiga, L.B., Lipnikov, K., Manzini, G.: Error analysis for a mimetic discretization of the steady Stokes problem on polyhedral meshes. SIAM J. Numer. Anal. 48(4), 1419–1443 (2010)MathSciNetMATH
6.
Zurück zum Zitat 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)MathSciNetMATH 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)MathSciNetMATH
7.
Zurück zum Zitat da Veiga, L.B., Brezzi, F., Marini, L.D.: Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51(2), 794–812 (2013)MathSciNetMATH da Veiga, L.B., Brezzi, F., Marini, L.D.: Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51(2), 794–812 (2013)MathSciNetMATH
8.
Zurück zum Zitat da Veiga, L.B., Brezzi, F., Marini, L.D., Russo, A.: Virtual Element Method for general second-order elliptic problems on polygonal meshes. Math. Models Methods Appl. Sci. 26(4), 729–750 (2016)MathSciNetMATH da Veiga, L.B., Brezzi, F., Marini, L.D., Russo, A.: Virtual Element Method for general second-order elliptic problems on polygonal meshes. Math. Models Methods Appl. Sci. 26(4), 729–750 (2016)MathSciNetMATH
9.
Zurück zum Zitat 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, 509–535 (2017)MathSciNetMATH 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, 509–535 (2017)MathSciNetMATH
10.
Zurück zum Zitat 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)MathSciNetMATH 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)MathSciNetMATH
11.
Zurück zum Zitat da Veiga, L.B., Mora, D., Vacca, G.: The Stokes Complex for Virtual Elements with Application to Navier-Stokes Flows. J. Sci. Comput. 81, 990–1018 (2019)MathSciNetMATH da Veiga, L.B., Mora, D., Vacca, G.: The Stokes Complex for Virtual Elements with Application to Navier-Stokes Flows. J. Sci. Comput. 81, 990–1018 (2019)MathSciNetMATH
12.
Zurück zum Zitat Bernardi, C., Raugel, G.: A conforming finite element method for the time-dependent Navier-Stokes equations. SIAM J. Numer. Anal. 22(3), 455–473 (1985)MathSciNetMATH Bernardi, C., Raugel, G.: A conforming finite element method for the time-dependent Navier-Stokes equations. SIAM J. Numer. Anal. 22(3), 455–473 (1985)MathSciNetMATH
13.
Zurück zum Zitat Bochev, P.B., Gunzburger, M.D., Shadid, J.N.: On inf-sup stabilized finite element methods for transient problems. Comput. Methods Appl. Mech. Engrg. 193, 1471–1489 (2004)MathSciNetMATH Bochev, P.B., Gunzburger, M.D., Shadid, J.N.: On inf-sup stabilized finite element methods for transient problems. Comput. Methods Appl. Mech. Engrg. 193, 1471–1489 (2004)MathSciNetMATH
14.
Zurück zum Zitat 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, 82–101 (2006)MathSciNetMATH 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, 82–101 (2006)MathSciNetMATH
15.
Zurück zum Zitat Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Verlag, New York (1991)MATH Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Verlag, New York (1991)MATH
16.
Zurück zum Zitat Brenner, S., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer Verlag, New York (2008)MATH Brenner, S., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer Verlag, New York (2008)MATH
18.
Zurück zum Zitat Burman, E., Fernández, M.A.: Continuous interior penalty finite element method for the time-dependent Navier-Stokes equations: space discretization and convergence. Numer. Math. 107, 39–77 (2007)MathSciNetMATH Burman, E., Fernández, M.A.: Continuous interior penalty finite element method for the time-dependent Navier-Stokes equations: space discretization and convergence. Numer. Math. 107, 39–77 (2007)MathSciNetMATH
20.
Zurück zum Zitat Gatica, G.N., Munar, M., Sequeira, F.A.: A mixed virtual element method for the Navier-Stokes equations. Math. Models Methods Appl. Sci. 28(14), 2719–2762 (2018)MathSciNetMATH Gatica, G.N., Munar, M., Sequeira, F.A.: A mixed virtual element method for the Navier-Stokes equations. Math. Models Methods Appl. Sci. 28(14), 2719–2762 (2018)MathSciNetMATH
21.
Zurück zum Zitat Girault, V., Raviart, P.A.: Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms. Springer Series in Computational Mathematics, Springer-Verlag, Berlin Heidelberg (1986)MATH Girault, V., Raviart, P.A.: Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms. Springer Series in Computational Mathematics, Springer-Verlag, Berlin Heidelberg (1986)MATH
22.
Zurück zum Zitat He, Y.: A fully discrete stabilized finite-element method for the time-dependent Navier-Stokes problem. IMA J. Numer. Anal. 23, 665–691 (2003)MathSciNetMATH He, Y.: A fully discrete stabilized finite-element method for the time-dependent Navier-Stokes problem. IMA J. Numer. Anal. 23, 665–691 (2003)MathSciNetMATH
23.
Zurück zum Zitat He, Y., Sun, W.: Stabilized finite element method based on the Crank-Nicolson Extrapolation scheme for the time-dependent Navier-Stokes equations. Math. Comput. 76, 115–136 (2007)MathSciNetMATH He, Y., Sun, W.: Stabilized finite element method based on the Crank-Nicolson Extrapolation scheme for the time-dependent Navier-Stokes equations. Math. Comput. 76, 115–136 (2007)MathSciNetMATH
24.
Zurück zum Zitat He, G., He, Y., Chen, Z.: A penalty finite volume method for the transient Navier-Stokes equations. Appl. Numer. Math. 58(11), 1583–1613 (2008)MathSciNetMATH He, G., He, Y., Chen, Z.: A penalty finite volume method for the transient Navier-Stokes equations. Appl. Numer. Math. 58(11), 1583–1613 (2008)MathSciNetMATH
25.
Zurück zum Zitat 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(2), 275–311 (1982)MathSciNetMATH 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(2), 275–311 (1982)MathSciNetMATH
26.
Zurück zum Zitat Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem, part ii: stability of solutions and error estimates uniform in time. SIAM J. Numer. Anal. 23(4), 750–777 (1986)MathSciNetMATH Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem, part ii: stability of solutions and error estimates uniform in time. SIAM J. Numer. Anal. 23(4), 750–777 (1986)MathSciNetMATH
27.
Zurück zum Zitat Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes Problem, Part III. Smoothing property and higher order error estimates for spatial discretization. SIAM J. Numer. Anal. 25(3), 489–512 (1988)MathSciNetMATH Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes Problem, Part III. Smoothing property and higher order error estimates for spatial discretization. SIAM J. Numer. Anal. 25(3), 489–512 (1988)MathSciNetMATH
28.
Zurück zum Zitat Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier Stokes problem. IV. Error analysis for second-order time discretization. SIAM J. Numer. Anal. 27, 353–384 (1990)MathSciNetMATH Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier Stokes problem. IV. Error analysis for second-order time discretization. SIAM J. Numer. Anal. 27, 353–384 (1990)MathSciNetMATH
29.
Zurück zum Zitat Hill, A.T., Süli, E.: Approximation of the global attractor for the incompressible Navier-Stokes equations. IMA J. Numer. Anal. 20(4), 633–667 (2000)MathSciNetMATH Hill, A.T., Süli, E.: Approximation of the global attractor for the incompressible Navier-Stokes equations. IMA J. Numer. Anal. 20(4), 633–667 (2000)MathSciNetMATH
30.
Zurück zum Zitat Irisarri, D., Hauke, G.: Stabilized virtual element methods for the unsteady incompressible Navier-Stokes equations. Calcolo 56(38) (2019) Irisarri, D., Hauke, G.: Stabilized virtual element methods for the unsteady incompressible Navier-Stokes equations. Calcolo 56(38) (2019)
31.
Zurück zum Zitat Jiang, Y., Mei, L., Wei, H.: A stabilized finite element method for transient Navier-Stokes equations based on two local Gauss integrations. Int. J. Numer. Meth. Fluids 70, 713–723 (2011)MathSciNetMATH Jiang, Y., Mei, L., Wei, H.: A stabilized finite element method for transient Navier-Stokes equations based on two local Gauss integrations. Int. J. Numer. Meth. Fluids 70, 713–723 (2011)MathSciNetMATH
32.
Zurück zum Zitat Kumar, S., Ruiz-Baier, R.: Equal order discontinuous finite volume element methods for the Stokes problem. J. Sci. Comput. 65, 956–978 (2015)MathSciNetMATH Kumar, S., Ruiz-Baier, R.: Equal order discontinuous finite volume element methods for the Stokes problem. J. Sci. Comput. 65, 956–978 (2015)MathSciNetMATH
33.
Zurück zum Zitat Adak, D., Natarajan, E., Kumar, S.: Convergence analysis of virtual element methods for semilinear parabolic problems on polygonal meshes. Numer. Methods Partial Differ. Equ. 35, 222–245 (2018)MathSciNetMATH Adak, D., Natarajan, E., Kumar, S.: Convergence analysis of virtual element methods for semilinear parabolic problems on polygonal meshes. Numer. Methods Partial Differ. Equ. 35, 222–245 (2018)MathSciNetMATH
34.
Zurück zum Zitat Adak, D., Natarajan, E., Kumar, S.: Virtual element method for semilinear hyperbolic problems on polygonal meshes. Int. J. Comput. Math. 96(5), 971–991 (2019)MathSciNetMATH Adak, D., Natarajan, E., Kumar, S.: Virtual element method for semilinear hyperbolic problems on polygonal meshes. Int. J. Comput. Math. 96(5), 971–991 (2019)MathSciNetMATH
35.
Zurück zum Zitat Li, J., He, Y., Chen, Z.: A new stabilized finite element method for the transient Navier-Stokes equations. Comput. Methods Appl. Mech. Engg. 197, 22–35 (2007)MathSciNetMATH Li, J., He, Y., Chen, Z.: A new stabilized finite element method for the transient Navier-Stokes equations. Comput. Methods Appl. Mech. Engg. 197, 22–35 (2007)MathSciNetMATH
36.
Zurück zum Zitat Liu, X., Chen, Z.: The nonconforming virtual element method for the Navier-Stokes equations. Adv. Comput. Math. 45, 51–74 (2019)MathSciNetMATH Liu, X., Chen, Z.: The nonconforming virtual element method for the Navier-Stokes equations. Adv. Comput. Math. 45, 51–74 (2019)MathSciNetMATH
37.
Zurück zum Zitat Lu, X., Lin, P.: Error estimate of the P1 nonconforming finite element method for the penalized unsteady Navier-Stokes equations. Numer. Math. 115, 261–287 (2010)MathSciNetMATH Lu, X., Lin, P.: Error estimate of the P1 nonconforming finite element method for the penalized unsteady Navier-Stokes equations. Numer. Math. 115, 261–287 (2010)MathSciNetMATH
38.
Zurück zum Zitat Mu, L., Ye, X.: A finite volume method for solving Navier-Stokes problems. Nonlinear Anal. 74(17), 6686–6695 (2011)MathSciNetMATH Mu, L., Ye, X.: A finite volume method for solving Navier-Stokes problems. Nonlinear Anal. 74(17), 6686–6695 (2011)MathSciNetMATH
39.
Zurück zum Zitat Qiu, H., Xue, C., Xue, L.: Low-order stabilized finite element methods for the unsteady Stokes/Navier-Stokes equations with friction boundary conditions. Math Method. Appl. Sci. 41, 2119–2139 (2018)MathSciNetMATH Qiu, H., Xue, C., Xue, L.: Low-order stabilized finite element methods for the unsteady Stokes/Navier-Stokes equations with friction boundary conditions. Math Method. Appl. Sci. 41, 2119–2139 (2018)MathSciNetMATH
40.
Zurück zum Zitat Shang, Y.: New stabilized finite element method for time-dependent incompressible flow problems. Int. J. Numer. Meth. Fluids 62, 166–187 (2010)MathSciNetMATH Shang, Y.: New stabilized finite element method for time-dependent incompressible flow problems. Int. J. Numer. Meth. Fluids 62, 166–187 (2010)MathSciNetMATH
41.
Zurück zum Zitat Shang, Yueqiang: Error analysis of a fully discrete finite element variational multiscale method for time-dependent incompressible Navier-Stokes equations. Numer. Methods Partial Differ. Equ. 29(6), 2025–2046 (2013)MathSciNetMATH Shang, Yueqiang: Error analysis of a fully discrete finite element variational multiscale method for time-dependent incompressible Navier-Stokes equations. Numer. Methods Partial Differ. Equ. 29(6), 2025–2046 (2013)MathSciNetMATH
42.
Zurück zum Zitat Talischi, C., Paulino, G.H., Pereira, A., Menezes, I.F.: Polymesher: a general-purpose mesh generator for polygonal elements written in matlab. Struct. Multidiscip. Optim. 45(3), 309–328 (2012)MathSciNetMATH Talischi, C., Paulino, G.H., Pereira, A., Menezes, I.F.: Polymesher: a general-purpose mesh generator for polygonal elements written in matlab. Struct. Multidiscip. Optim. 45(3), 309–328 (2012)MathSciNetMATH
43.
Zurück zum Zitat TEMAM, R.: Navier-Stokes: Equations, Theory and Numerical Analysis. North-Holland, Amsterdam (1977) TEMAM, R.: Navier-Stokes: Equations, Theory and Numerical Analysis. North-Holland, Amsterdam (1977)
44.
Zurück zum Zitat Thomée, Vidar: Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics, Springer-Verlag, Berlin (2006)MATH Thomée, Vidar: Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics, Springer-Verlag, Berlin (2006)MATH
45.
Zurück zum Zitat Vacca, G., da Veiga, L.B.: Virtual element methods for parabolic problems on polygonal meshes. Numer. Methods Partial Differential Equations 31, 2110–2134 (2015)MathSciNetMATH Vacca, G., da Veiga, L.B.: Virtual element methods for parabolic problems on polygonal meshes. Numer. Methods Partial Differential Equations 31, 2110–2134 (2015)MathSciNetMATH
46.
Zurück zum Zitat Vacca, G.: An \(H^1\)-conforming virtual element for Darcy and Brinkman equations. Math. Models Methods Appl. Sci. 28, 159–194 (2018)MathSciNetMATH Vacca, G.: An \(H^1\)-conforming virtual element for Darcy and Brinkman equations. Math. Models Methods Appl. Sci. 28, 159–194 (2018)MathSciNetMATH
48.
Zurück zum Zitat Xie, C., Feng, M.: New nonconforming finite element method for solving transient Naiver-Stokes equations. Appl. Math. Mech. Engl. Ed. 35(2), 237–258 (2014)MathSciNetMATH Xie, C., Feng, M.: New nonconforming finite element method for solving transient Naiver-Stokes equations. Appl. Math. Mech. Engl. Ed. 35(2), 237–258 (2014)MathSciNetMATH
49.
Zurück zum Zitat Xu, C., Shi, D., Liao, X.: Low order nonconforming mixed finite element method for nonstationary incompressible Navier-Stokes equations. Appl. Math. Mech. Engl. Ed. 37(8), 1095–1112 (2016)MathSciNetMATH Xu, C., Shi, D., Liao, X.: Low order nonconforming mixed finite element method for nonstationary incompressible Navier-Stokes equations. Appl. Math. Mech. Engl. Ed. 37(8), 1095–1112 (2016)MathSciNetMATH
Metadaten
Titel
Lowest order virtual element approximations for transient Stokes problem on polygonal meshes
verfasst von
N. Verma
S. Kumar
Publikationsdatum
01.12.2021
Verlag
Springer International Publishing
Erschienen in
Calcolo / Ausgabe 4/2021
Print ISSN: 0008-0624
Elektronische ISSN: 1126-5434
DOI
https://doi.org/10.1007/s10092-021-00440-7

Weitere Artikel der Ausgabe 4/2021

Calcolo 4/2021 Zur Ausgabe