Error of the two-step BDF for the incompressible Navier-Stokes problem
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 5, p. 757-764

The incompressible Navier-Stokes problem is discretized in time by the two-step backward differentiation formula. Error estimates are proved under feasible assumptions on the regularity of the exact solution avoiding hardly fulfillable compatibility conditions. Whereas the time-weighted velocity error is of optimal second order, the time-weighted error in the pressure is of first order. Suboptimal estimates are shown for a linearisation. The results cover both the two- and three-dimensional case.

DOI : https://doi.org/10.1051/m2an:2004037
Classification:  35Q30,  65M12,  76D05
Keywords: incompressible Navier-Stokes equation, time discretisation, backward differentiation formula, error estimate, parabolic smoothing
@article{M2AN_2004__38_5_757_0,
author = {Emmrich, Etienne},
title = {Error of the two-step BDF for the incompressible Navier-Stokes problem},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {38},
number = {5},
year = {2004},
pages = {757-764},
doi = {10.1051/m2an:2004037},
zbl = {1076.76054},
mrnumber = {2104427},
language = {en},
url = {http://www.numdam.org/item/M2AN_2004__38_5_757_0}
}

Emmrich, Etienne. Error of the two-step BDF for the incompressible Navier-Stokes problem. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 5, pp. 757-764. doi : 10.1051/m2an:2004037. http://www.numdam.org/item/M2AN_2004__38_5_757_0/

[1] G.A. Baker, V.A. Dougalis and O.A. Karakashian, On a higher order accurate fully discrete Galerkin approximation to the Navier-Stokes equations. Math. Comp. 39 (1982) 339-375. | Zbl 0503.76038

[2] E. Emmrich, Analysis von Zeitdiskretisierungen des inkompressiblen Navier-Stokes-Problems. Cuvillier, Göttingen (2001). | Zbl 0982.76003

[3] E. Emmrich, Error of the two-step BDF for the incompressible Navier-Stokes problem. Preprint 741, TU Berlin (2002).

[4] V. Girault and P.-A. Raviart, Finite Element Approximation of the Navier-Stokes Equations. Springer, Berlin (1979). | MR 548867 | Zbl 0413.65081

[5] J.G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem, Part IV: Error analysis for second-order time discretization. SIAM J. Numer. Anal. 27 (1990) 353-384. | Zbl 0850.76350

[6] A.T. Hill and E. Süli, Approximation of the global attractor for the incompressible Navier-Stokes equations. IMA J. Numer. Anal. 20 (2000) 633-667. | Zbl 0982.76022

[7] S. Müller-Urbaniak, Eine Analyse des Zwischenschritt-$\theta$-Verfahrens zur Lösung der instationären Navier-Stokes-Gleichungen. Preprint 94-01 (SFB 359), Univ. Heidelberg (1994). | Zbl 0796.76072

[8] A. Prohl, Projection and Quasi-compressibility Methods for Solving the Incompressible Navier-Stokes Equations. Teubner, Stuttgart (1997). | MR 1472237 | Zbl 0874.76002

[9] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis. North-Holland Publ. Company, Amsterdam (1977). | MR 609732 | Zbl 0383.35057

[10] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis. CBMS-NSF Reg. Confer. Ser. Appl. Math. SIAM 41 (1985). | MR 764933 | Zbl 0833.35110