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Mathematics of Computation

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Stabilized finite element method based on the Crank-Nicolson extrapolation scheme for the time-dependent Navier-Stokes equations


Authors: Yinnian He and Weiwei Sun
Journal: Math. Comp. 76 (2007), 115-136
MSC (2000): Primary 35L70; Secondary 65N30, 76D06
Published electronically: September 15, 2006
MathSciNet review: 2261014
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Abstract: This paper provides an error analysis for the Crank-Nicolson extrapolation scheme of time discretization applied to the spatially discrete stabilized finite element approximation of the two-dimensional time-dependent Navier-Stokes problem, where the finite element space pair $ (X_h,M_h)$ for the approximation $ (u_h^n,p_h^n)$ of the velocity $ u$ and the pressure $ p$ is constructed by the low-order finite element: the $ Q_1-P_0$ quadrilateral element or the $ P_1-P_0$ triangle element with mesh size $ h$. Error estimates of the numerical solution $ (u_h^n,p_h^n)$ to the exact solution $ (u(t_n),p(t_n))$ with $ t_n\in (0,T]$ are derived.


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Additional Information

Yinnian He
Affiliation: Faculty of Science, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China
Email: heyn@mail.xjtu.edu.cn

Weiwei Sun
Affiliation: Department of Mathematics, City University of Hong Kong, Hong Kong, People’s Republic of China
Email: maweiw@math.cityu.edu.hk

DOI: https://doi.org/10.1090/S0025-5718-06-01886-2
Keywords: Navier--Stokes problem, stabilized finite element, Crank--Nicolson extrapolation scheme
Received by editor(s): November 22, 2004
Received by editor(s) in revised form: September 2, 2005
Published electronically: September 15, 2006
Additional Notes: The first author was supported in part by the NSF of the People’s Republic of China (10371095).
The second author was supported in part by the Research Grants Council of the Hong Kong Special Administrative Region, People’s Republic of China (Project No. City U 102103).
Article copyright: © Copyright 2006 American Mathematical Society