Author(s) :
Etienne Emmrich
Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 741-2002
MSC 2000
- 65M12 Stability and convergence of numerical methods
-
76D05 Navier-Stokes equations
-
35Q30 Stokes and Navier-Stokes equations
Abstract :
The incompressible Navier-Stokes problem is discretised in time by
the two-step backward differentiation formula with constant
step sizes. Error estimates are proved under
feasible assumptions on the regularity of the exact solution. The
question of compatibility of problem data is taken into account.
Whereas the time-weighted velocity error is of optimal second order
in the $l^{\infty}(L^2)$- and $l^2(H_0^1)$-norm, the time-weighted
error in the pressure is of first order in the
$l^{\infty}(L^2/\mathbbm{R})$-norm.
Furthermore, a linearisation that is based upon a
modification of the convective term
using a formally second-order extrapolation is considered. The
velocity error is then shown to be of order $3/2$, and
the pressure error is of order $1/2$.
The results presented cover both the two- and three-dimensional case.
Particular attention is directed to appearing constants and step size
restrictions.
Keywords :
Incompressible Navier-Stokes equation, time discretisation, backward differentiation formula, error estimate, parabolic smoothing