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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
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