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## Error of the two-step BDF for the incompressible Navier-Stokes problem

 Source file is available as : Postscript Document

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