@Article{IJNAM-10-257,
author = {R. Ingram},
title = {Unconditional Convergence of High-Order Extrapolations of the Crank-Nicolson, Finite Element Method for the Navier-Stokes Equations},
journal = {International Journal of Numerical Analysis and Modeling},
year = {2013},
volume = {10},
number = {2},
pages = {257--297},
abstract = {<p style="text-align: justify;">Error estimates for the Crank-Nicolson in time, Finite Element in space (CNFE) discretization
of the Navier-Stokes equations require application of the discrete Gronwall inequality,
which leads to a time-step $(\Delta t)$ restriction. All known convergence analyses of the fully discrete
CNFE with linear extrapolation rely on a similar&nbsp; $\Delta t$-restriction. We show that CNFE with
arbitrary-order extrapolation (denoted CNLE) is convergences optimally in the energy norm without
any $\Delta t$-restriction. We prove that CNLE velocity and corresponding discrete time-derivative
converge optimally in $l^∞(H^1)$ and $l^2(L^2)$ respectively under the mild condition&nbsp; $\Delta t \leq Mh^{1/4}$ for
any arbitrary $M &gt; 0$ (e.g. independent of problem data, $h$, and &nbsp;$\Delta t$) where $h &gt; 0$ is the maximum
mesh element diameter. Convergence in these higher order norms is needed to prove convergence
estimates for pressure and the drag/lift force a fluid exerts on an obstacle. Our analysis exploits
the extrapolated convective velocity to avoid any &nbsp;$\Delta t$-restriction for convergence in the energy
norm. However, the coupling between the extrapolated convecting velocity of usual CNLE and
the a priori control of average velocities (characteristic of CN methods) rather than pointwise
velocities (e.g. backward-Euler methods) in $l^2(H^1)$ is precisely the source of &nbsp;$\Delta t$-restriction for
convergence in higher-order norms.</p>},
issn = {2617-8710},
doi = {https://doi.org/},
url = {http://global-sci.org/intro/article_detail/ijnam/568.html}
}