Search Advanced

AIMS Mathematics

2024, Volume 9, Issue 1: 453-480. doi: 10.3934/math.2024025
Previous Article Next Article
Research article

Convergence analysis of Euler and BDF2 grad-div stabilization methods for the time-dependent penetrative convection model

  • College of Mathematics and Physics, Wenzhou University, Wenzhou 325035, China
  • Received: 31 October 2023 Revised: 20 November 2023 Accepted: 20 November 2023 Published: 30 November 2023

  • MSC : 36Q30, 65M60, 76M10

  • Based on the grad-div stabilization method, the first-order backward Euler and second-order BDF2 finite element schemes were studied for the approximations of the time-dependent penetrative convection equations. The proposed schemes are both unconditionally stable. We proved the error bounds of the velocity and temperature in which the constants are independent of inverse powers of the viscosity and thermal conductivity coefficients when the Taylor-Hood element and $ P_2 $ element are used in finite element discretizations. Finally, numerical experiments with high Reynolds numbers were shown to confirm the theoretical results.

    Citation: Weiwen Wan, Rong An. Convergence analysis of Euler and BDF2 grad-div stabilization methods for the time-dependent penetrative convection model[J]. AIMS Mathematics, 2024, 9(1): 453-480. doi: 10.3934/math.2024025

    Related Papers:

  • Based on the grad-div stabilization method, the first-order backward Euler and second-order BDF2 finite element schemes were studied for the approximations of the time-dependent penetrative convection equations. The proposed schemes are both unconditionally stable. We proved the error bounds of the velocity and temperature in which the constants are independent of inverse powers of the viscosity and thermal conductivity coefficients when the Taylor-Hood element and $ P_2 $ element are used in finite element discretizations. Finally, numerical experiments with high Reynolds numbers were shown to confirm the theoretical results.


    加载中


    [1] R. A. Adams, J. F. Fournier, Sobolev spaces, New York: Academic Press, 2009.
    [2] A. Barletta, On the thermal instability induced by viscous dissipation, Int. J. Therm. Sci., 88 (2015), 238–247. https://doi.org/10.1016/j.ijthermalsci.2014.02.009 doi: 10.1016/j.ijthermalsci.2014.02.009
    [3] K. R. Blake, D. Poulikakos, A. Bejan, Natural convection near 4℃ in a horizontal water layer heated from below, Phys. Fluids, 27 (1984), 2608–2616. https://doi.org/10.1063/1.864561 doi: 10.1063/1.864561
    [4] S. C. Brenner, L. R. Scott, The mathematical theory of finite element methods, New York: Springer, 2008. https://doi.org/10.1007/978-0-387-75934-0
    [5] A. N. Brooks, T. J. R. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Method. Appl. M., 32 (1982), 199–259. https://doi.org/10.1016/0045-7825(82)90071-8 doi: 10.1016/0045-7825(82)90071-8
    [6] M. Cao, Y. Li, Optimal error analysis of linearized Crank-Nicolson finite element scheme for the time-Dependent penetrative convection problem, Commun. Appl. Math. Comput., 2023 (2023), 7. https://doi.org/10.1007/s42967-023-00269-7 doi: 10.1007/s42967-023-00269-7
    [7] F. Capone, M. Gentile, A. A. Hill, Penetrative convection in a fluid layer with throughflow, Ricerche Mat., 57 (2008), 251–260. https://doi.org/10.1007/s11587-008-0035-8 doi: 10.1007/s11587-008-0035-8
    [8] M. Carr, S. de Putter, Penetrative convection in a horizontallt isotropic porous layer, Continum Mech. Theromodyn., 15 (2003), 33–43. https://doi.org/10.1007/s00161-002-0102-4 doi: 10.1007/s00161-002-0102-4
    [9] S. Chandrasekhar, Hydrodynamic and hydromagnetic stability, Oxford: Oxford University Press, 1961.
    [10] A. A. Domnich, E. S. Baranovskii, M. A. Artemov, A nonlinear model of the non-isothermal slip flow between two parallel plates, J. Phys. Conf. Ser., 1479 (2020), 012005. https://doi.org/10.1088/1742-6596/1479/1/012005 doi: 10.1088/1742-6596/1479/1/012005
    [11] D. Erkmen, A. E. Labovsky, Note on the usage of grad-div stabilization for the penalty-projection algorithm in magnetohydrodynamics, Appl. Math. Comput., 349 (2019), 48–52. http://doi.org/10.1016/j.amc.2018.12.036 doi: 10.1016/j.amc.2018.12.036
    [12] L. P. Franca, T. J. R. Hughes, Two classes of mixed finite element methods, Comput. Method. Appl. M., 69 (1988), 89–129. https://doi.org/10.1016/0045-7825(88)90168-5 doi: 10.1016/0045-7825(88)90168-5
    [13] L. P. Franca, T. J. R. Hughes, Convergence analyses of Galerkin least-squares methods for symmetric advective-diffusive forms of the Stokes and incompressible Navier-Stokes equations, Comput. Method. Appl. M., 105 (1993), 285–298. https://doi.org/10.1016/0045-7825(93)90126-I doi: 10.1016/0045-7825(93)90126-I
    [14] L. P. Franca, A. Russo, Deriving upwinding, mass lumping and selective reduced integration by residual-free bubbles, Appl. Math. Lett., 9 (1996), 83–88. https://doi.org/10.1016/0893-9659(96)00078-X doi: 10.1016/0893-9659(96)00078-X
    [15] L. P. Franca, A. Nesliturk, On a two-level finite element method for the incompressible Navier-Stokes equations, Int. J. Numer. Meth. Eng., 52 (2001), 433–453. https://doi.org/10.1002/nme.220 doi: 10.1002/nme.220
    [16] F. Franchi, Stabilization estimates for penetrative motions in porous media, Math. Method. Appl. Sci., 17 (1994), 11–20. https://doi.org/10.1002/mma.1670170103 doi: 10.1002/mma.1670170103
    [17] J. de Frutos, B. García-Archilla, V. John, J. Novo, Grad-div stabilization for the evolutionary Oseen problem with inf-sup stable finite elements, J. Sci. Comput., 66 (2016), 991–1024. https://doi.org/10.1007/s10915-015-0052-1 doi: 10.1007/s10915-015-0052-1
    [18] J. de Frutos, B. García-Archilla, V. John, J. Novo, Analysis of the grad-div stabilization for the time-dependent Navier-Stokes equations with inf-sup stable finite elements, Adv. Comput. Math., 44 (2018), 195–225. https://doi.org/10.1007/s10444-017-9540-1 doi: 10.1007/s10444-017-9540-1
    [19] J. de Frutos, B. García-Archilla, J. Novo, Grad-div stabilization for the time-dependent Boussinesq equations with inf-sup stable finite elements, Appl. Math. Comput., 349 (2019), 281–291. https://doi.org/10.1016/j.amc.2018.12.062 doi: 10.1016/j.amc.2018.12.062
    [20] B. García-Archilla, V. John, J. Novo, On the convergence order of the finite element error in the kinetic energy for high Reynolds number incompressible flows, Comput. Method. Appl. M., 385 (2021), 114032. https://doi.org/10.1016/j.cma.2021.114032 doi: 10.1016/j.cma.2021.114032
    [21] B. García-Archilla, J. Novo, Robust error bounds for the Navier-Stokes equations using implicit-explicit second-order BDF method with variable steps, IMA J. Numer. Anal., 43 (2023), 2892–2933. https://doi.org/10.1093/imanum/drac058 doi: 10.1093/imanum/drac058
    [22] V. Girault, P. Raviart, Finite element methods for Navier-Stokes equations, Berlin Heidelberg: Springer-Verlag, 1986. https://doi.org/10.1007/978-3-642-61623-5
    [23] J. L. Guermond, Stabilization of Galerkin approximations of transport equations by subgrid modeling, ESAIM-Math. Model. Num., 33 (1999), 1293–1316. https://doi.org/10.1051/m2an:1999145 doi: 10.1051/m2an:1999145
    [24] J. G. Heywood, R. Rannacher, Finite-element approximation of the nonstationary Navier-Stokes problem Part Ⅳ: error analysis for second-order time discretization, SIAM J. Numer. Anal., 27 (1990), 353–384. https://doi.org/10.1137/0727022 doi: 10.1137/0727022
    [25] T. J. R. Hughes, L. P. Franca, G. M. Hulbert, A new finite element formulation for computational fluid dynamics: Ⅷ. The Galerkin/least-squares method for advective-diffusive equations, Comput. Method. Appl. M., 73 (1989), 173–189. https://doi.org/10.1016/0045-7825(89)90111-4 doi: 10.1016/0045-7825(89)90111-4
    [26] T. J. R. Hughes, L. Mazzei, K. E. Jansen, Large Eddy simulation and the variational multiscale method, Comput. Visual. Sci., 3 (2000), 47–59. https://doi.org/10.1007/s007910050051 doi: 10.1007/s007910050051
    [27] T. J. R. Hughes, L. Mazzei, A. A. Oberai, A. A. Wray, The multiscale formulation of large eddy simulation: Decay of homogeneous isotropic turbulence, Phys. Fluids, 13 (2001), 505–512. https://doi.org/10.1063/1.1332391 doi: 10.1063/1.1332391
    [28] E. W. Jenkins, V. John, A. Linke, L. G. Rebholz, On the parameter choice in grad-div stabilization for the Stokes equations, Adv. Comput. Math, 40 (2014), 491–516. https://doi.org/10.1007/s10444-013-9316-1 doi: 10.1007/s10444-013-9316-1
    [29] L. M. Jiji, Heat convection, Heidelberg: Springer Berlin, 2006. https://doi.org/10.1007/978-3-642-02971-4
    [30] V. John, Finite element methods for incompressible flow problems, Cham: Springer, 2016. https://doi.org/10.1007/978-3-319-45750-5
    [31] W. Layton, A connection between subgrid scale eddy viscosity and mixed methods, Appl. Math. Comput., 133 (2002), 147–157. https://doi.org/10.1016/S0096-3003(01)00228-4 doi: 10.1016/S0096-3003(01)00228-4
    [32] W. Layton, C. C. Manica, M. Neda, M. Olshanskii, L. G. Rebholz, On the accuracy of the rotation form in simulations of the Navier-Stokes equations, J. Comput. Phys., 228 (2009), 3433–3447. https://doi.org/10.1016/j.jcp.2009.01.027 doi: 10.1016/j.jcp.2009.01.027
    [33] Y. Li, D. D. Xue, Y. Rong, Y. Qin, A second order partitioned method with grad-div stabilization for the non-stationary dual-porosity-Stokes model, Comput. Math. Appl., 124 (2022), 111–128. https://doi.org/10.1016/j.camwa.2022.08.025 doi: 10.1016/j.camwa.2022.08.025
    [34] Y. Li, R. An, Two-level variational multiscale finite element methods for Navier-Stokes type variational inequality problem, J. Comput. Appl. Math., 290 (2015), 656–669. https://doi.org/10.1016/j.cam.2015.06.018 doi: 10.1016/j.cam.2015.06.018
    [35] J. Liu, Simple and efficient ALE methods with provable temporal accuracy up to fifth order for the Stokes equations on time varying domains, SIAM J. Numer. Anal., 51 (2013), 743–772. https://doi.org/10.1137/110825996 doi: 10.1137/110825996
    [36] A. J. Majda, A. L. Bertozzi, Vorticity and incompressible flow, Cambridge: Cambridge University Press, 2002. https://doi.org/10.1017/CBO9780511613203
    [37] G. Matthies, N. I. Ionkin, G. Lube, L. Röhe, Some remarks on residual-based stabilisation of inf-sup stable discretisations of the generalized Oseen problem, Comput. Meth. Appl. Math., 9 (2009), 368–390. https://doi.org/10.2478/cmam-2009-0024 doi: 10.2478/cmam-2009-0024
    [38] G. L. Mellor, T. Yamada, Development of a turbulence closure model for geophysical fluid problems, Rev. Geophys., 20 (1982), 851–875. https://doi.org/10.1029/RG020i004p00851 doi: 10.1029/RG020i004p00851
    [39] S. Mussman, Penetrative convection, J. Fluid Mech., 31 (1968), 343–360. https://doi.org/10.1017/S0022112068000194 doi: 10.1017/S0022112068000194
    [40] M. A. Olshanskii, A. Reusken, Grad-div stablilization for Stokes equations, Math. Comput., 73 (2004), 1699–1718. https://doi.org/10.1090/S0025-5718-03-01629-6 doi: 10.1090/S0025-5718-03-01629-6
    [41] L. E. Payne, J. C. Song, B. Straughan, Double diffusive porous penetrative convection-thawing subsea permafrost, Int. J. Eng. Sci., 26 (1998), 797–809. https://doi.org/10.1016/0020-7225(88)90031-6 doi: 10.1016/0020-7225(88)90031-6
    [42] J. Pedlosky, Geophysical fluid dynamics, New York: Springer, 1987. https://doi.org/10.1007/978-1-4612-4650-3
    [43] R. Rannacher, R. Scott, Some optimal error estimates for piecewise linear finite element approximations, Math. Comput., 38 (1982), 437–445. https://doi.org/10.1090/S0025-5718-1982-0645661-4 doi: 10.1090/S0025-5718-1982-0645661-4
    [44] S. S. Ravindran, Convergence of extrapolated BDF2 finite element schemes for unsteady penetrative convection model, Numer. Func. Anal. Opt., 33 (2012), 48–79. https://doi.org/10.1080/01630563.2011.618899 doi: 10.1080/01630563.2011.618899
    [45] P. Sagaut, Large Eddy simulation for incompressible flows, Heidelberg: Springer Berlin, 2003. https://doi.org/10.1007/b137536
    [46] B. Straughan, Continuous dependence on the heat source and non-linear stability in penetrative convection, Int. J. Nonlin. Mech., 26 (1991), 221–231. https://doi.org/10.1016/0020-7462(91)90053-V doi: 10.1016/0020-7462(91)90053-V
    [47] F. M. White, Fluid mechanics, New York: McGraw-Hill, 2002.
    [48] S. Woo, P. L. F. Liu, Finite-element model for modified boussinesq equations. Ⅰ: Model development, J. Waterway Port Coast., 130 (2004), 1. https://doi.org/10.1061/(ASCE)0733-950X(2004)130:1(1) doi: 10.1061/(ASCE)0733-950X(2004)130:1(1)
    [49] H. Zheng, Y. Hou, F. Shi, L. Song, A finite element variational multiscale method for incompressible flows based on two local Gauss integrations, J. Comput. Phys., 228 (2009), 5961–5977. https://doi.org/10.1016/j.jcp.2009.05.006 doi: 10.1016/j.jcp.2009.05.006
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
AIMS Mathematics
1.8 3.4

Metrics

Article views(564) PDF downloads(44) Cited by(0)

Preview PDF
Download XML
Export Citation
Article outline

Figures and Tables

Tables(4)

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog