no. 5
Error of the two-step BDF for the incompressible Navier-Stokes problem
ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 5, pp. 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 : 10.1051/m2an:2004037
Classification : 35Q30, 65M12, 76D05
Mots clés : 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: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {757--764},
     publisher = {EDP-Sciences},
     volume = {38},
     number = {5},
     year = {2004},
     doi = {10.1051/m2an:2004037},
     mrnumber = {2104427},
     zbl = {1076.76054},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2004037/}
}
TY  - JOUR
AU  - Emmrich, Etienne
TI  - Error of the two-step BDF for the incompressible Navier-Stokes problem
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2004
SP  - 757
EP  - 764
VL  - 38
IS  - 5
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2004037/
DO  - 10.1051/m2an:2004037
LA  - en
ID  - M2AN_2004__38_5_757_0
ER  - 
%0 Journal Article
%A Emmrich, Etienne
%T Error of the two-step BDF for the incompressible Navier-Stokes problem
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2004
%P 757-764
%V 38
%N 5
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an:2004037/
%R 10.1051/m2an:2004037
%G en
%F M2AN_2004__38_5_757_0
Emmrich, Etienne. Error of the two-step BDF for the incompressible Navier-Stokes problem. ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 5, pp. 757-764. doi : 10.1051/m2an:2004037. http://www.numdam.org/articles/10.1051/m2an:2004037/

[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

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

[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 | Zbl

[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

[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

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

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

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

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

Cité par Sources :