Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-23T08:09:59.657Z Has data issue: false hasContentIssue false
  • English
  • Français

Two-Level Defect-Correction Method for Steady Navier-Stokes Problem with Friction Boundary Conditions

Part of: Partial differential equations, boundary value problems Basic methods in fluid mechanics

Published online by Cambridge University Press:  19 September 2016

An Liu ,
Yuan Li  and
Rong An
An Liu
Affiliation:
College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, China
Yuan Li
Affiliation:
College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, China
Rong An*
Affiliation:
College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, China
*
*Corresponding author. Email:[email protected], [email protected] (R. An)
Get access

Abstract

In this paper, we present two-level defect-correction finite element method for steady Navier-Stokes equations at high Reynolds number with the friction boundary conditions, which results in a variational inequality problem of the second kind. Based on Taylor-Hood element, we solve a variational inequality problem of Navier-Stokes type on the coarse mesh and solve a variational inequality problem of Navier-Stokes type corresponding to Newton linearization on the fine mesh. The error estimates for the velocity in the H1 norm and the pressure in the L2 norm are derived. Finally, the numerical results are provided to confirm our theoretical analysis.

Keywords

Navier-Stokes equationsfriction boundary conditionsvariational inequality problemsdefect-correction methodtwo-level mesh method

MSC classification

Secondary: 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 76M10: Finite element methods
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 8 , Issue 6 , December 2016 , pp. 932 - 952
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] An, R., Li, Y. and Li, K., Solvability of Navier-Stokes equations with leak boundary conditions, Acta Math. Appl. Sinica English Series, 25(2) (2009), pp. 225–234.Google Scholar
[2] An, R. and Qiu, H., Two-Level Newton iteration methods for Navier-Stokes type variational inequality problem, Adv. Appl. Math. Mech., 5(1) (2013), pp. 36–54.Google Scholar
[3] Ayadi, M., Gdoura, M. and Sassi, T., Mixed formulation for Stokes problem with Tresca friction, C.R. Math. Acad. Sci. Paris, 348(19-20) (2010), pp. 1069–1072.CrossRefGoogle Scholar
[4] Dai, X., Tang, P. and Wu, M., Analysis of an iterative penalty method for Navier-Stokes equations with nonlinear slip boundary conditions, Int. J. Numer. Meth. Fluids, 72(4) (2013), pp. 403–413.Google Scholar
[5] Fujita, H., Flow Problems with Unilateral Boundary conditions, Lecons, Collège de France, 1993.Google Scholar
[6] Fujita, H., A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions, RIMS Kokyuroku, 888 (1994), pp. 199–216.Google Scholar
[7] Fujita, H., Non-stationary Stokes flows under leak boundary conditions of friction type, J. Comput. Math., 19(1) (2001), pp. 1–8.Google Scholar
[8] Fujita, H., A coherent analysis of Stokes folws under boundary conditions of friction type, J. Comput. Appl. Math., 149(1) (2002), pp. 57–69.CrossRefGoogle Scholar
[9] Girault, V. and Raviart, P., Finite Element Approximation of the Navier-Stokes Equations, Volume 749 of Lecture Notes Math., Springer, Heidelberg, 1979.Google Scholar
[10] Glowinski, R., Numerical Methods for Nonlinear Variational Problems, Springer Verlag, New York, 1984.CrossRefGoogle Scholar
[11] Guermond, J., Stabilization of Galerkin approximations of transport equations by subgrid modeling, Math. Model. Numer. Anal., 33(6) (1993), pp. 1293–1316.CrossRefGoogle Scholar
[12] Hecht, F., New development in FreeFem++, J. Numer. Math., 20(3-4) (2012), pp. 251–265.Google Scholar
[13] Hughes, T., Mazzei, L. and Jansen, K., Large eddy simulation and the variational multiscale method, Comput. Vis. Sci., 3(1-2) (2000), pp. 47–59.Google Scholar
[14] Hughes, T., Mazzei, L. and Oberai, A., The multiscale formulation of large eddy simulation: decay of homogeneous isotropic turbulence, Phys. Fluids, 13(2) (2001), pp. 505–511.Google Scholar
[15] Kashiwabara, T., On a strong solution of the non-stationary Navier-Stokes equations under slip or leak boundary conditions of friction type, J. Differential Equations, 254(2) (2013), pp. 756–778.Google Scholar
[16] Kashiwabara, T., On a finite element approximation of the Stokes equations under a slip boundary condition of the friction type, Japan J. Indust. Appl. Math., 30(1) (2013), pp. 227–261.Google Scholar
[17] Kashiwabara, T., Finite element method for Stokes equations under leak boundary condition of friction type, SIAM J. Numer. Anal., 51(4) (2013), pp. 2448–2469.Google Scholar
[18] Kaya, S., Layton, W. and Riviere, B., Subgrid stabilized defect correction methods for the Navier-Stokes equations, SIAM J. Numer. Anal., 44(4) (2006), pp. 1639–1654.Google Scholar
[19] Layton, W., Solution algorithm for incompressible viscous flows at high Reynolds number, Vestnik Moskov. Gos. Univ. Ser., 15 (1996), pp. 25–35.Google Scholar
[20] Layton, W., Lee, H. and Peterson, J., A defect-correction method for the incompressible Navier-Stokes equations, Appl. Math. Comput., 129(1) (2002), pp. 1–19.Google Scholar
[21] Layton, W., A connection between subgrid scale eddy viscosity and mixed methods, Appl. Math. Comput., 133(1) (2002), pp. 147–157.Google Scholar
[22] Li, Y. and An, R., Two-Level pressure projection finite element methods for Navier-Stokes equations with nonlinear slip boundary conditions, Appl. Numer. Math., 61(3) (2011), pp. 285–297.Google Scholar
[23] Li, Y. and An, R., Penalty finite element method for Navier-Stokes equations with nonlinear slip boundary conditions, Int. J. Numer. Meth. Fluids, 69(3) (2012), pp. 550–566.Google Scholar
[24] Li, Y. and An, R., Two-level iteration penalty methods for Navier-Stokes equations with friction boundary conditions, Abstract Appl. Anal., Article ID 125139, (2013), 17 pages.Google Scholar
[25] Li, Y. and Li, K., Pressure projection stabilized finite element method for Navier-Stokes equations with nonlinear slip boundary conditions, Computing, 87(3-4) (2010), pp. 113–133.CrossRefGoogle Scholar
[26] Li, Y. and Li, K., Existence of the solution to stationary Navier-Stokes equations with nonlinear slip boundary conditions, J. Math. Anal. Appl., 381(1) (2011), pp. 1–9.Google Scholar
[27] Li, Y. and Li, K., Uzawa iteration method for Stokes type variational inequality of the second kind, Acta Mathematicae Applicatae Sinica-English Series, 27(2) (2011), pp. 303–316.CrossRefGoogle Scholar
[28] Li, Y. and Li, K., Global strong solutions of two-dimensional Navier-Stokes equations with nonlinear slip boundary conditions, J. Math. Anal. Appl., 393(1) (2012), pp. 1–13.CrossRefGoogle Scholar
[29] Liu, Q. and Hou, Y., A two-level defect-correctionmethod for Navier-Stokes equations, Bull. Aust. Math. Soc., 81(3) (2010), pp. 442–454.Google Scholar
[30] Qiu, H. and Mei, L., Two-level defect-correction stabilized finite element method for Navier-Stokes equations with friction boundary conditions, J. Comput. Appl. Math., 280 (2015), pp. 80–93.CrossRefGoogle Scholar
[31] Le Roux, C., Steady Stokes flows with threshold slip boundary conditions, Math. Models Methods Appl. Sci., 15(8) (2005), pp. 1141–1168.CrossRefGoogle Scholar
[32] Le Roux, C. and Tani, A., Steady solutions of the Navier-Stokes equations with threshold slip boundary conditions, Math. Meth. Appl. Sci., 30(5) (2007), pp. 595–624.CrossRefGoogle Scholar
[33] Saito, N., On the Stokes equations with the leak and slip boundary conditions of friction type: regularity of solutions, Publ. RIMS, Kyoto Univ., 40(2) (2004), pp. 345–383.CrossRefGoogle Scholar
[34] Zheng, H., Hou, Y., Shi, F. and Song, L., A finite element variational multiscale method for incompressible flows based on two local Gauss integrations, J. Comput. Phys., 228(16) (2009), pp. 5961–5977.CrossRefGoogle Scholar
[35] Zheng, H., Hou, Y. and Shi, F., Adaptive variational multiscale methods for incompressible flow based on two local Gauss integrations, J. Comput. Phys., 229(19) (2010), pp. 7030–7041.CrossRefGoogle Scholar