A subgrid stabilized method for Navier-Stokes equations with nonlinear slip boundary conditions
Abstract
In this paper, we consider a subgrid stabilized Oseen iterative method for the Navier-Stokes equations with nonlinear slip boundary conditions and high Reynolds number. We provide one-level and two-level schemes based on this stability algorithm. The two-level schemes involve solving a subgrid stabilized nonlinear coarse mesh inequality system by applying m Oseen iterations, and a standard one-step Newton linearization problems without stabilization on the fine mesh. We analyze the stability of the proposed algorithm and provide error estimates and parameter scalings. Numerical examples are given to confirm our theoretical findings.
Keyword : Navier-Stokes equations, nonlinear slip boundary conditions, subgrid stabilization, two-level method, error estimate
How to Cite
Dai, X., & Zhang, C. (2021). A subgrid stabilized method for Navier-Stokes equations with nonlinear slip boundary conditions. Mathematical Modelling and Analysis, 26(4), 528-547. https://doi.org/10.3846/mma.2021.12299
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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J. Shen. On error estimates of some higher order projection and penaltyprojection methods for Navier-Stokes equations. Numer. Math., 62:49–73, 1992. https://doi.org/10.1007/BF01396220
R. Temam. Navier-Stokes Equations. Theory and Numerical Analysis. NorthHolland, Amsterdam, 1984.
K. Wang. A new defect correction method for the Navier-Stokes equations at high reynolds numbers. Appl. Math. Comput., 216(11):3252–3264, 2010. https://doi.org/10.1016/j.amc.2010.04.050
Y. Zhang and Y.N. He. Assessment of subgrid-scale models for incompressible Navier-Stokes equations. J. Comput. Appl. Math., 234(2):593–604, 2010. https://doi.org/10.1016/j.cam.2009.12.051
H.B. Zheng, Y.R. Hou, F. Shi and L.N. Song. A finite element variational multiscale method for incompressible flows based on two local gauss integrations. J. Comput. Phys., 228(16):5961–5977, 2009. https://doi.org/10.1016/j.jcp.2009.05.006
K.R. Zhou and Y.Q. Shang. Local and parallel finite element algorithms for the Stokes equations with nonlinear slip boundary conditions. Int. J. Comp. Meth., 17(08):1950050, 2020. https://doi.org/10.1142/S0219876219500506
M. Benzi, G.H. Golub and J. Liesen. Numerical solution of saddle point problems. Acta Numer., 14:1–137, 2005. https://doi.org/10.1017/S0962492904000212
G.F. Carey and R. Krishnan. Penalty approximation of Stokes flow. Comput. Meth. Appl. Mech. Eng., 35(2):169–206, 1982. https://doi.org/10.1016/00457825(82)90133-5
X.X. Dai, P.P. Tang and M.H. Wu. Analysis of an iterative penalty method for Navier-Stokes equations with nonlinear slip boundary conditions. Int. J. Numer. Methods Fluids, 72:403–413, 2013. https://doi.org/10.1002/fld.3742
J. Djoko and J. Koko. Numerical methods for the Stokes and Navier-Stokes equations driven by threshold slip bundary conditons. Comput. Meth. Appl. Mech. Eng., 305:936–958, 2016. https://doi.org/10.1016/j.cma.2016.03.026
H.C. Elman, D.J. Silvester and A.J. Wathen. Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations. Numer. Math., 90:641–664, 2002. https://doi.org/10.1007/s002110100300
E. Erturk, T. Corke and C. Gokcol. Numerical solutions of 2-d steady incompressible driven cavity flow at high Reynolds numbers. Int.J.Numer. Methods Fluids, 48:747–774, 2005. https://doi.org/10.1002/fld.953
H. Fujita. Flow Problems with Unilateral Boundary Conditions. Lecons, Collge de France, 1993.
H. Fujita. A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions. RIMS Kōkyūroku 888, pp. 199–216, 1994.
H. Fujita. Non-stationary Stokes flows under leak boundary conditions of friction type. J. Comput. Math., 19:1–8, 2001.
J.L. Guermod, P. Minev and J. Shen. An overview of projection methods for incompressible flows. Comput. Meth. Appl. Mech. Eng., 195(44–47):6011–6045, 2006. https://doi.org/10.1016/j.cma.2005.10.010
F.F. Jing, W.M. Han, W.J. Yan and F. Wang. Discontinuous Galerkin methods for a stationary Navier-Stokes problem with a nonlinear slip boundary condition of friction type. J. Sci. Comput., 76:888–912, 2018. https://doi.org/10.1007/s10915-018-0644-7
F.F. Jing, J. Li, X. Chen and Z.H. Zhang. Numerical analysis of a characteristic stabilized finite element method for the time-dependent Navier-Stokes equations with nonlinear slip boundary conditions. J. Comput. Appl. Math., 320:43–60, 2017. https://doi.org/10.1016/j.cam.2017.01.012
N. Kechkar and D. Silvester. Analysis of locally stabilized mixed finite element methods for the Stokes problem. Math. Comput., 58:1–10, 1992. https://doi.org/10.1090/S0025-5718-1992-1106973-X
W. Layton. A connection between subgrid-scale eddy viscosity and mixed methods. Appl. Math. Comput., 133(1):147–157, 2002. https://doi.org/10.1016/S0096-3003(01)00228-4
W. Layton and L. Tobiska. A two-level method with backtracking for the Navier-Stokes equations. SIAM J. Numer. Anal., 35(5):2035–2054, 1998. https://doi.org/10.1137/S003614299630230X
Y. Li and R. An. Penalty finite element method for Navier-Stokes equations with nonlinear slip boundary conditions. Int. J. Numer. Methods Fluids, 69(3):550– 566, 2012. https://doi.org/10.1002/fld.2574
Y. Li and K.T. Li. Uzawa iteration method for Stokes type variational inequality of the second kind. Acta Math. Appl. Sin.-Engl. Ser., 17:303–16, 2011. https://doi.org/10.1007/s10255-011-0063-0
H.L. Qiu, R. An, L.Q. Mei and C.F. Xue. Two-step algorithms for the stationary incompressible Navier-Stokes equations with friction boundary conditions. Appl. Numer. Math., 120:97–114, 2017. https://doi.org/10.1016/j.apnum.2017.05.003
H.L. Qiu and L.Q. Mei. Two-level defect-correction stabilized finite element method for Navier-Stokes equations with friction boundary conditions. J. Comput. Appl. Math., 280:80–93, 2015. https://doi.org/10.1016/j.cam.2014.11.045
H.L. Qiu, L.Q. Mei, H. Liu and S. Cartwright. A defect-correction stabilized finite element method for Navier-Stokes equations with friction boundary conditions. Appl. Numer. Math., 90:9–21, 2015. https://doi.org/10.1016/j.apnum.2014.11.009
Y.Q. Shang. A two-level subgrid stabilized Oseen iterative method for the steady Navier-Stokes equations. J. Comp. Phys., 233:210–226, 2013. https://doi.org/10.1016/j.jcp.2012.08.024
J. Shen. On error estimates of some higher order projection and penaltyprojection methods for Navier-Stokes equations. Numer. Math., 62:49–73, 1992. https://doi.org/10.1007/BF01396220
R. Temam. Navier-Stokes Equations. Theory and Numerical Analysis. NorthHolland, Amsterdam, 1984.
K. Wang. A new defect correction method for the Navier-Stokes equations at high reynolds numbers. Appl. Math. Comput., 216(11):3252–3264, 2010. https://doi.org/10.1016/j.amc.2010.04.050
Y. Zhang and Y.N. He. Assessment of subgrid-scale models for incompressible Navier-Stokes equations. J. Comput. Appl. Math., 234(2):593–604, 2010. https://doi.org/10.1016/j.cam.2009.12.051
H.B. Zheng, Y.R. Hou, F. Shi and L.N. Song. A finite element variational multiscale method for incompressible flows based on two local gauss integrations. J. Comput. Phys., 228(16):5961–5977, 2009. https://doi.org/10.1016/j.jcp.2009.05.006
K.R. Zhou and Y.Q. Shang. Local and parallel finite element algorithms for the Stokes equations with nonlinear slip boundary conditions. Int. J. Comp. Meth., 17(08):1950050, 2020. https://doi.org/10.1142/S0219876219500506