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

He, Yinnian; Sun, Weiwei

doi:10.1090/S0025-5718-06-01886-2

Stabilized finite element method based on the Crank–Nicolson extrapolation scheme for the time-dependent Navier–Stokes equations
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by Yinnian He and Weiwei Sun;

Math. Comp. 76 (2007), 115-136

DOI: https://doi.org/10.1090/S0025-5718-06-01886-2

Published electronically: September 15, 2006

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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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