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

Etienne Emmrich

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • 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 38.5 (2010): 757-764. <http://eudml.org/doc/194238>.

@article{Emmrich2010,
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},
keywords = {Incompressible Navier-Stokes equation; time discretisation; backward differentiation formula; error estimate; parabolic smoothing.; two-step backward differentiation formula; time-weigted error},
language = {eng},
month = {3},
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/194238},
volume = {38},
year = {2010},
}

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
DA - 2010/3//
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/194238
ER -

References

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  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.  
  2. E. Emmrich, Analysis von Zeitdiskretisierungen des inkompressiblen Navier-Stokes-Problems. Cuvillier, Göttingen (2001).  
  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).  
  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.  
  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.  
  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).  
  8. A. Prohl, Projection and Quasi-compressibility Methods for Solving the Incompressible Navier-Stokes Equations. Teubner, Stuttgart (1997).  
  9. R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis. North-Holland Publ. Company, Amsterdam (1977).  
  10. R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis. CBMS-NSF Reg. Confer. Ser. Appl. Math. SIAM41 (1985).  

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