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

Etienne Emmrich

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2004)

  • Volume: 38, Issue: 5, page 757-764
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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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 38.5 (2004): 757-764. <http://eudml.org/doc/244827>.

@article{Emmrich2004,
abstract = {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.},
author = {Emmrich, Etienne},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {incompressible Navier-Stokes equation; time discretisation; backward differentiation formula; error estimate; parabolic smoothing; two-step backward differentiation formula; time-weigted error},
language = {eng},
number = {5},
pages = {757-764},
publisher = {EDP-Sciences},
title = {Error of the two-step BDF for the incompressible Navier-Stokes problem},
url = {http://eudml.org/doc/244827},
volume = {38},
year = {2004},
}

TY - JOUR
AU - Emmrich, Etienne
TI - Error of the two-step BDF for the incompressible Navier-Stokes problem
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2004
PB - EDP-Sciences
VL - 38
IS - 5
SP - 757
EP - 764
AB - 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.
LA - eng
KW - incompressible Navier-Stokes equation; time discretisation; backward differentiation formula; error estimate; parabolic smoothing; two-step backward differentiation formula; time-weigted error
UR - http://eudml.org/doc/244827
ER -

References

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  1. [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. Zbl0503.76038
  2. [2] E. Emmrich, Analysis von Zeitdiskretisierungen des inkompressiblen Navier-Stokes-Problems. Cuvillier, Göttingen (2001). Zbl0982.76003
  3. [3] E. Emmrich, Error of the two-step BDF for the incompressible Navier-Stokes problem. Preprint 741, TU Berlin (2002). Zbl1076.76054
  4. [4] V. Girault and P.-A. Raviart, Finite Element Approximation of the Navier-Stokes Equations. Springer, Berlin (1979). Zbl0413.65081MR548867
  5. [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. Zbl0694.76014
  6. [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. Zbl0982.76022
  7. [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). Zbl0796.76072
  8. [8] A. Prohl, Projection and Quasi-compressibility Methods for Solving the Incompressible Navier-Stokes Equations. Teubner, Stuttgart (1997). Zbl0874.76002MR1472237
  9. [9] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis. North-Holland Publ. Company, Amsterdam (1977). Zbl0383.35057MR609732
  10. [10] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis. CBMS-NSF Reg. Confer. Ser. Appl. Math. SIAM 41 (1985). Zbl0833.35110MR764933

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