# American Institue of Mathematical Sciences

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

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

 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.
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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