Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-05-22T10:08:01.228Z Has data issue: false hasContentIssue false
  • English
  • Français

Two-Level Newton Iteration Methods for Navier-Stokes Type Variational Inequality Problem

Published online by Cambridge University Press:  03 June 2015

Rong An  and
Hailong Qiu
Rong An*
Affiliation:
College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, Zhejiang, China
Hailong Qiu
Affiliation:
College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, Zhejiang, China
*
*Corresponding author.Email: anrong702@yahoo.com.cn
Get access

Abstract

This paper deals with the two-level Newton iteration method based on the pressure projection stabilized finite element approximation to solve the numerical solution of the Navier-Stokes type variational inequality problem. We solve a small Navier-Stokes problem on the coarse mesh with mesh size H and solve a large linearized Navier-Stokes problem on the fine mesh with mesh size h. The error estimates derived show that if we choose h = O (|logh|1/2H3), then the two-level method we provide has the same H1 and L2 convergence orders of the velocity and the pressure as the one-level stabilized method. However, the L2 convergence order of the velocity is not consistent with that of one-level stabilized method. Finally, we give the numerical results to support the theoretical analysis.

Keywords

65N30Navier-Stokes equationsnonlinear slip boundary conditionsvariational inequality problemstabilized finite elementtwo-level methods
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 5 , Issue 1 , February 2013 , pp. 36 - 54
Copyright
Copyright © Global-Science Press 2013

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]Fujita, H., Flow Problems with Unilateral Boundary conditions, Lecons, Collége de France, October, 1993.Google Scholar
[2]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
[3]Fujita, H., Non-stationary Stokes flows under leak boundary conditions of friction type, J. Comput. Math., 19 (2001), pp. 1–8.Google Scholar
[4]Fujita, H., A coherent analysis of Stokes flows under boundary conditions of friction type, J. Comput. Appl. Math., 149(1) (2002), pp. 57–69.Google Scholar
[5]Saito, N., On the Stokes equations with the leak and slip boundary conditions of friction type: regularity of solutions, Pub. RIMS, 40 (2004), pp. 345–383.Google Scholar
[6]Li, Y. and Li, K. T., 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
[7]Suito, H. and Ueda, T., Numerical simulation of blood flow in thoracic aora, Med. Imag. Tech., 28 (2010), pp. 175–180.Google Scholar
[8]Suito, H., Ueda, T. and Rubin, G. D., Simulation of blood flow in thoracic aorta for prediction of long-term adverse events, Proceedings of the 1st International Conference on Mathematical and Computational Biomedical Engineering, 2009.Google Scholar
[9]Suito, H. and Kawarada, H., Numerical simulation of spilled oil by fictious domain method, Japan J. Indust. Appl. Math., 21 (2004), pp. 219–236.Google Scholar
[10]Ayadi, M., Gdoura, M. K. and Sassi, T., Mixed formulation for Stokes problem with Tresca friction, C.R. Math. Acad. Sci. Paris, 348 (2010), pp. 1069–1072.CrossRefGoogle Scholar
[11]Kashiwabara, T., On a finite element approximation of the Stokes problem under leak or slip boundary conditions of friction type, arXiv:1012.4982v1, 2010.Google Scholar
[12]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.CrossRefGoogle Scholar
[13]Li, Y. and Li, K. T., Pressure projection stabilized finite element method for Navier-Stokes equations with nonlinear slip boundary conditions, Computing, 87 (2010), pp. 113–133.CrossRefGoogle Scholar
[14]Djoko, J. K., On the time approximation for Stokes equations with slip boundary condition, submitted to Nonlinear Analysis: Real Wrold, 2011.Google Scholar
[15]Li, Y. and An, R., Semi-discrete stabilized finite element methods for Navier-Stokes equations with nonlinear slip boundary conditions based on regularization procedure, Numer. Math., 117(1) (2011), pp. 1–36.Google Scholar
[16]Li, Y. and An, R., Penalty finite element method for Navier-Stokes equations with nonlinear slip boundary conditions, Int. J. Numer. Methods Fluids, 69(3) (2012), pp. 550–566.Google Scholar
[17]He, Y. N., Zhang, Y., Shang, Y. Q. and Xu, H., Two-level Newton iterative method for the 2D/3D steady Navier-Stokes equations, Numer. Methods Partial Differential Equations, (2011), DOI:10.1002/num.20695.Google Scholar
[18]He, Y. N., Mei, L. Q., Shang, Y. Q. and Cui, J., Newton iterative parallel finite element algorithm for the steady Navier-Stokes equations, J. Sci. Computing, 44(1) (2010), pp. 92–106.CrossRefGoogle Scholar
[19]He, Y. N. and Li, J., Convergence of three iterative methods based on the finite element discretization for the stationary Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 198(15-16) (2009), pp. 1351–1359.CrossRefGoogle Scholar
[20]Bochev, P., Dohrmann, C. and Gunzburger, M., Stabilization of low-order mixed finite element, SIAM J. Numer. Anal., 44(1) (2006), pp. 82–101.Google Scholar
[21]Li, J. and He, Y. N., A stabilized finite element method based on two local Gauss integrations for the Stokes equations, J. Comput. Appl. Math., 214(1) (2008), pp. 58–65.Google Scholar
[22]Zheng, H., Hou, Y. and Shi, F., Adaptive variational multiscale methods for incompressible flow based on two local Gauss integrations, J. Comput. Phys., 229 (2010), pp. 7030–7041.Google Scholar
[23]Li, J., Investigations on two kinds of two-level stabilized finite element methods for the stationary Navier-Stokes equations, Appl. Math. Comput., 182 (2006), pp. 1470–1481.Google Scholar
[24]He, Y. N. and Li, K. T., Two-level stabilized finite element methods for the steady Navier-Stokes problem, Computing, 74 (2005), pp. 337–351.Google Scholar
[25]He, Y. N., A fully discrete stabilized finite-element method for the time-dependent Navier-Stokes problem, IMA J. Numer. Anal., 23 (2003), pp. 665–691.Google Scholar
[26]Layton, W. and Tobiska, L., A two-level method with backtracking for the Navier-Stokes equations, SIAM J. Numer. Anal., 35(5) (1998), pp. 2035–2054.Google Scholar
[27]Layton, W. and Lenferink, W., A multilevel mesh independence principle for the Navier-Stokes equations, SIAM J. Numer. Anal., 33 (1996), pp. 17–30.Google Scholar
[28]Li, Y. and Li, K. T., Uzawa iteration method for Stokes type variational inequality of the second kind, Acta Math. Appl. Sinica, 27(2) (2011), pp. 302–315.Google Scholar