Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

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

Pages: 41 - 68, Volume 6, Issue 1, January 2006      doi:10.3934/dcdsb.2006.6.41

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Yinnian He - Faculty of Science, Xi'an Jiaotong University, Xi'an 710049, China (email)
Yanping Lin - Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada, T6G 2G1, Canada (email)
Weiwei Sun - Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, China (email)

Abstract: 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:  Navier-Stokes problem, stabilized finite element, uniform error estimate.
Mathematics Subject Classification:  Primary: 35L70, 65N30, 76D06.

Received: April 2005;      Revised: September 2005;      Available Online: October 2005.