American Institute of Mathematical Sciences

Advanced
January  2006, 6(1): 41-68. doi: 10.3934/dcdsb.2006.6.41

Stabilized finite element method for the non-stationary Navier-Stokes problem

Yinnian He 1, , Yanping Lin 2, and  Weiwei Sun 3,

1. 

Faculty of Science, Xi'an Jiaotong University, Xi'an 710049, China
2. 

Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada, T6G 2G1
3. 

Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, China

Received  April 2005 Revised  September 2005 Published  October 2005

In this article, a locally stabilized finite element formulation of the two-dimensional Navier-Stokes problem is used. A macroelement condition which provides the stability of the $Q_1-P_0$ quadrilateral element and the $P_1-P_0$ triangular element is introduced. Moreover, the $H^1$ and $L^2$-error estimates of optimal order for finite element solution $(u_h,p_h)$ are analyzed. Finally, a uniform $H^1$ and $L^2$-error estimates of optimal order for finite element solution $(u_h,p_h)$ is obtained if the uniqueness condition is satisfied.
Keywords: stabilized finite element, uniform error estimate., Navier-Stokes problem.
Mathematics Subject Classification: Primary: 35L70, 65N30, 76D0.
Citation: Yinnian He, Yanping Lin, Weiwei Sun. Stabilized finite element method for the non-stationary Navier-Stokes problem. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 41-68. doi: 10.3934/dcdsb.2006.6.41
[1]

Yueqiang Shang, Qihui Zhang. A subgrid stabilizing postprocessed mixed finite element method for the time-dependent Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3119-3142. doi: 10.3934/dcdsb.2020222
[2]

Konstantinos Chrysafinos. Error estimates for time-discretizations for the velocity tracking problem for Navier-Stokes flows by penalty methods. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1077-1096. doi: 10.3934/dcdsb.2006.6.1077
[3]

Yoshikazu Giga. A remark on a Liouville problem with boundary for the Stokes and the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1277-1289. doi: 10.3934/dcdss.2013.6.1277
[4]

Grzegorz Karch, Maria E. Schonbek, Tomas P. Schonbek. Singularities of certain finite energy solutions to the Navier-Stokes system. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 189-206. doi: 10.3934/dcds.2020008
[5]

Misha Perepelitsa. An ill-posed problem for the Navier-Stokes equations for compressible flows. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 609-623. doi: 10.3934/dcds.2010.26.609
[6]

Hantaek Bae. Solvability of the free boundary value problem of the Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 769-801. doi: 10.3934/dcds.2011.29.769
[7]

Yinnian He, Kaitai Li. Nonlinear Galerkin approximation of the two dimensional exterior Navier-Stokes problem. Discrete & Continuous Dynamical Systems, 1996, 2 (4) : 467-482. doi: 10.3934/dcds.1996.2.467
[8]

Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021014
[9]

Xufeng Xiao, Xinlong Feng, Jinyun Yuan. The stabilized semi-implicit finite element method for the surface Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2857-2877. doi: 10.3934/dcdsb.2017154
[10]

Juan Wen, Yaling He, Yinnian He, Kun Wang. Stabilized finite element methods based on multiscale enrichment for Allen-Cahn and Cahn-Hilliard equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021074
[11]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602
[12]

Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149
[13]

Viorel Barbu, Ionuţ Munteanu. Internal stabilization of Navier-Stokes equation with exact controllability on spaces with finite codimension. Evolution Equations & Control Theory, 2012, 1 (1) : 1-16. doi: 10.3934/eect.2012.1.1
[14]

Jean-Pierre Raymond, Laetitia Thevenet. Boundary feedback stabilization of the two dimensional Navier-Stokes equations with finite dimensional controllers. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 1159-1187. doi: 10.3934/dcds.2010.27.1159
[15]

Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641
[16]

Jie Shen, Xiaofeng Yang. Error estimates for finite element approximations of consistent splitting schemes for incompressible flows. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 663-676. doi: 10.3934/dcdsb.2007.8.663
[17]

Francesca Crispo, Paolo Maremonti. A remark on the partial regularity of a suitable weak solution to the Navier-Stokes Cauchy problem. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1283-1294. doi: 10.3934/dcds.2017053
[18]

Zilai Li, Zhenhua Guo. On free boundary problem for compressible navier-stokes equations with temperature-dependent heat conductivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3903-3919. doi: 10.3934/dcdsb.2017201
[19]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348
[20]

Xulong Qin, Zheng-An Yao. Global solutions of the free boundary problem for the compressible Navier-Stokes equations with density-dependent viscosity. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1041-1052. doi: 10.3934/cpaa.2010.9.1041

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (98)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences