RESEARCH ARTICLE

Convergence and stability of two-level penalty mixed finite element method for stationary Navier-Stokes equations

  • Pengzhan HUANG 1 ,
  • Yinnian HE , 1,2 ,
  • Xinlong FENG 1
Expand
  • 1. College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China
  • 2. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China

Received date: 15 May 2012

Accepted date: 03 Nov 2012

Published date: 01 Aug 2013

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg

Abstract

The two-level penalty mixed finite element method for the stationary Navier-Stokes equations based on Taylor-Hood element is considered in this paper. Two algorithms are proposed and analyzed. Moreover, the optimal stability analysis and error estimate for these two algorithms are provided. Finally, the numerical tests confirm the theoretical results of the presented algorithms.

Cite this article

Pengzhan HUANG , Yinnian HE , Xinlong FENG . Convergence and stability of two-level penalty mixed finite element method for stationary Navier-Stokes equations[J]. Frontiers of Mathematics in China, 0 , 8(4) : 837 -854 . DOI: 10.1007/s11464-013-0257-2

1
Ervin V, Layton W, Maubach J. A posteriori error estimators for a two-level finite element method for the Navier-Stokes equations. Numer Methods Partial Differential Equations, 1996, 12: 333-346

DOI

2
Ghia U, Ghia K N, Shin C T. High-resolutions for incompressible flow using the Navier- Stokes equations and a multigrid method. J Comput Phys, 1982, 48: 387-411

DOI

3
He Y N. Optimal error estimate of the penalty finite element method for the timedependent Navier-Stokes equations. Math Comp, 2005, 74: 1201-1216

DOI

4
He Y N, Li J. A penalty finite element method based on the Euler implicit explicit scheme for the time-dependent Navier-Stokes equations. J Comput Appl Math, 2010, 235: 708-725

DOI

5
He Y N, Li J, Yang X Z. Two-level penalized finite element methods for the stationary Navier-Stoke equations. Int J Inf Syst Sci, 2006, 2: 131-143

6
He Y N, Li K T. Two-level stabilized finite element methods for the steady Navier- Stokes problem. Computing, 2005, 74: 337-351

DOI

7
He Y N, Wang A W. A simplified two-level method for the steady Navier-Stokes equations. Comput Methods Appl Mech Engrg, 2008, 197: 1568-1576

DOI

8
Hecht F, Pironneau O, Hyaric A L, Ohtsuka K. FREEFEM++, version 2.3-3. 2008, http://www.freefem.org

9
Heywood J G, Rannacher R. Finite-element approximations of the nonstationary Navier-Stokes problem. Part I: Regularity of solutions and second-order spatial discretization. SIAM J Numer Anal, 1982, 19: 275-311

DOI

10
Huang P Z, Feng X L. Error estimates for two-level penalty finite volume method for the stationary Navier-Stokes equations. Math Methods Appl Sci, 2013, DOI: 10.1002/mma.2736

DOI

11
Huang P Z, Feng X L, Liu D M. Two-level stabilized method based on three corrections for the stationary Navier-Stokes equations. Appl Numer Math, 2012, 62: 988-1001

DOI

12
Huang P Z, Feng X L, Su H Y. Two-level defect-correction locally stabilized finite element method for the steady Navier-Stokes equations. Nonlinear Anal Real World Appl, 2013, 14: 1171-1181

DOI

13
Huang P Z, He Y N, Feng X L. Two-level stabilized finite element method for Stokes eigenvalue problem. Appl Math Mech (English Ed), 2012, 33: 621-630

DOI

14
Hughes T J R, Liu W T, Brooks A J. Finite element analysis of incompressible viscous flows by the penalty function formulation. J Comput Phys, 1979, 30: 1-60

DOI

15
Layton W. A two level discretization method for the Navier-Stokes equations. Comput Math Appl, 1993, 26: 33-38

DOI

16
Layton W, Lenferink W. Two-level Picard and modified Picard methods for the Navier- Stokes equations. Appl Math Comput, 1995, 69: 263-274

DOI

17
Layton W, Tobiska L. A two-level method with backtracking for the Navier-Stokes equations. SIAM J Numer Anal, 1998, 35: 2035-2054

DOI

18
Li J. Investigations on two kinds of two-level stabilized finite element methods for the stationary Navier-Stokes equations. Appl Math Comput, 2006, 182: 1470-1481

DOI

19
Lu X, Lin P. Error estimate of the P1 nonconforming finite element method for the penalized unsteady Navier-Stokes equations. Numer Math, 2010, 115: 261-287

DOI

20
Oden J T, Kikuchi N. Penalty method for constrained problems in elasticity. Int J Numer Methods Engrg, 1982, 18: 701-725

DOI

21
Shang Y Q. A parallel two-level linearization method for incompressible flow problems. Appl Math Lett, 2011, 24: 364-369

DOI

22
Shen J. On error estimates of some higher order projection and penalty-projection methods for Navier-Stokes equations. Numer Math, 1992, 62: 49-73

DOI

23
Shen J. On error estimates of the penalty method for unsteady Navier-Stokes equations. SIAM J Numer Anal, 1995, 32: 386-403

DOI

24
Taylor C, Hood P, A numerical solution of the Navier-Stokes equations using the finite element technique. Comput Fluids, 1973, 1: 73-100

DOI

25
Xu J. A novel two-grid method for semilinear elliptic equations. SIAM J Sci Comput, 1994, 15: 231-237

DOI

26
Xu J. Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J Numer Anal, 1996, 33: 1759-1778

DOI

27
Zhang Y, He Y N. A two-level finite element method for the stationary Navier-Stokes equations based on a stabilized local projection. Numer Methods Partial Different Equations, 2011, 27: 460-477

DOI

Options
Outlines

/