Stabilized finite element method for the non-stationaryNavier-Stokes problem

Yinnian He; Yanping Lin; Weiwei Sun

doi:10.3934/dcdsb.2006.6.41

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2006, Volume 6, Issue 1: 41-68. Doi: 10.3934/dcdsb.2006.6.41

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

1.
Faculty of Science, Xi'an Jiaotong University, Xi'an 710049
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

Received: March 31, 2005

Revised: August 31, 2005

Published: December 2005

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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.

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Mathematics Subject Classification: Primary: 35L70, 65N30, 76D06.

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