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The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data

Author: Yinnian He
Journal: Math. Comp. 77 (2008), 2097-2124
MSC (2000): Primary 35L70, 65N30, 76D06
Published electronically: May 8, 2008
MathSciNet review: 2429876
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Abstract: This paper considers the stability and convergence results for the Euler implicit/explicit scheme applied to the spatially discretized two-dimensional (2D) time-dependent Navier-Stokes equations. A Galerkin finite element spatial discretization is assumed, and the temporal treatment is implicit/explict scheme, which is implicit for the linear terms and explicit for the nonlinear term. Here the stability condition depends on the smoothness of the initial data $ u_0\in H^\alpha$, i.e., the time step condition is $ \tau\leq C_0$ in the case of $ \alpha=2$, $ \tau\vert\log h\vert\leq C_0$ in the case of $ \alpha=1$ and $ \tau h^{-2}\leq C_0$ in the case of $ \alpha=0$ for mesh size $ h$ and some positive constant $ C_0$. We provide the $ H^2$-stability of the scheme under the stability condition with $ \alpha=0,1,2$ and obtain the optimal $ H^1-L^2$ error estimate of the numerical velocity and the optimal $ L^2$ error estimate of the numerical pressure under the stability condition with $ \alpha=1,2$.

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Additional Information

Yinnian He
Affiliation: Faculty of Science, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China

Keywords: Navier-Stokes equations, mixed finite element, Euler implicit/explicit scheme, Smooth or non-smooth initial data
Received by editor(s): February 26, 2007
Received by editor(s) in revised form: September 17, 2007
Published electronically: May 8, 2008
Additional Notes: This research was subsidized by the NSF of China 10671154 and the National Basic Research Program under the grant 2005CB321703.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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