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Pressure-Correction Projection FEM for Time-Dependent Natural Convection Problem

  • Jilian Wu (a1), Xinlong Feng (a1) and Fei Liu (a2)


Pressure-correction projection finite element methods (FEMs) are proposed to solve nonstationary natural convection problems in this paper. The first-order and second-order backward difference formulas are applied for time derivative, the stability analysis and error estimates of the semi-discrete schemes are presented using energy method. Compared with characteristic variational multiscale FEM, pressure-correction projection FEMs are more efficient and unconditionally energy stable. Ample numerical results are presented to demonstrate the effectiveness of the pressure-correction projection FEMs for solving these problems.


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*Corresponding author. Email addresses: (J. Wu), (X. Feng), (F. Liu)


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Communicated by Jie Shen



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[1] Boland, J. and Layton, W.. An analysis of the finite element method for natural convection problems. Numer. Meth. Part. Differ. Equ., 6(2): 115126, 1990.
[2] Chorin, A.. Numerical solution of the Navier-Stokes equations. Math. Comp., 22: 745762, 1968.
[3] De Vahl Davis, G. Natural convection of air in a square cavity: a bench mark numerical solution. Int. J. Numer. Meth. Fluids, 3(3): 249264, 1983.
[4] W. E, and Liu, J.. Projection method. I. Convergence and numerical boundary layers. SIAM J. Numer. Anal., 32(4): 10171057, 1995.
[5] W. E, and Liu, J.. Gauge method for viscous incompressible flows. Commun. Math. Sci., 1(2): 317332, 2003.
[6] Goda, K.. A multistep technique with implicit difference schemes for calculating two-or three-dimensional cavity flows. J. Comput. Phys., 30(1): 7695, 1979.
[7] Guermond, J., Minev, P., and Shen, J.. An overview of projection methods for incompressible flows. Comput. Meth. Appl. Mech. Engrg., 195(44-47): 60116045, 2006.
[8] Guermond, J. and Shen, J.. A new class of truly consistent splitting schemes for incompressible flows. J. Comput. Phys., 192(1): 262276, 2003.
[9] Guermond, J. and Shen, J.. Velocity-correction projection methods for incompressible flows. SIAM J. Numer. Anal., 41(1): 112134, 2003.
[10] Guermond, J. and Shen, J.. On the error estimates for the rotational pressure-correction projection methods. Math. Comp., 73(248): 17191737, 2004.
[11] He, Y.. Stability and convergence of iterative methods related to viscosities for the 2D/3D steady Navier-Stokes equations. J. Math. Anal. Appl., 423(2): 11291149, 2015.
[12] He, Y. and Li, J.. A stabilized finite element method based on local polynomial pressure projection for the stationary Navier-Stokes equations. Appl. Numer.Math., 58(10): 15031514, 2008.
[13] Luo, Z., Zhu, J., Xie, Z., and Zhang, G.. Difference scheme and numerical simulation based on mixed finite elementmethod for natural convection problem. Appl. Math. Mech., 24(9): 11001110, 2003.
[14] Nochetto, R. and Pyo, J.. Error estimates for semi-discrete gauge methods for the Navier-Stokes equations. Math. Comp., 74(250): 521542, 2005.
[15] Pyo, J.. Error estimates for the second order semi-discrete stabilized gauge-Uzawa method for the Navier-Stokes equations. Int. J. Numer. Anal. Model., 10(1): 2441, 2013.
[16] Shen, J.. On error estimates of projection methods for Navier-Stokes equations: first-order schemes. SIAM J. Numer. Anal., 29(1): 5777, 1992.
[17] Shen, J.. Modeling and numerical approximation of two-phase incompressible flows by a phase-field approach. In Multiscale modeling and analysis for materials simulation, volume 22 of Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., pages 147195. World Sci. Publ., Hackensack, NJ, 2012.
[18] Si, Z., Song, X., and Huang, P.. Modified characteristics gauge-Uzawa finite element method for time dependent conduction-convection problems. J. Sci. Comput., 58(1): 124, 2014.
[19] Su, H., Qian, L., Gui, D., and Feng, X.. Second order fully discrete and divergence free conserving scheme for time-dependent conduction–convection equations. Int. Comm. Heat Mass Trans., 59: 120129, 2014.
[20] Temam, R.. Sur l’approximation de la solution des équations de navier-stokes par la méthode des pas fractionnaires II. Archive for Rational Mechanics and Analysis, 33(5): 377385, 1969.
[21] Timmermans, L., Minev, P., and Van De Vosse, F.. An approximate projection scheme for incompressible flow using spectral elements. Inter. J. Numer. Meth. Fluids, 22(7): 673688, 1996.
[22] Wan, D., Patnaik, B., and Wei, G.. A new benchmark quality solution for the buoyancy-driven cavity by discrete singular convolution. Numer. Heat Trans., Part B, 40(3): 199228, 2001.
[23] Wu, J., Gui, D., Liu, D., and Feng, X.. The characteristic variational multiscale method for time dependent conduction–convection problems. Int. Comm. Heat Mass Trans., 68: 5868, 2015.
[24] Zhang, T. and Tao, Z.. Decoupled scheme for time-dependent natural convection problem II: Time semidiscreteness. Math. Probl. Eng., 2014.
[25] Zhang, Y., Hou, Y., and Zheng, H.. A finite element variational multiscale method for steady-state natural convection problem based on two local gauss integrations. Numer. Meth. Part. Differ. Equ., 30(2): 361375, 2014.


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