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

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

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

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topEmmrich, 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

top- 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.
- E. Emmrich, Analysis von Zeitdiskretisierungen des inkompressiblen Navier-Stokes-Problems. Cuvillier, Göttingen (2001).
- E. Emmrich, Error of the two-step BDF for the incompressible Navier-Stokes problem. Preprint 741, TU Berlin (2002).
- V. Girault and P.-A. Raviart, Finite Element Approximation of the Navier-Stokes Equations. Springer, Berlin (1979).
- 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.
- 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.
- S. Müller-Urbaniak, Eine Analyse des Zwischenschritt-θ-Verfahrens zur Lösung der instationären Navier-Stokes-Gleichungen. Preprint 94-01 (SFB 359), Univ. Heidelberg (1994).
- A. Prohl, Projection and Quasi-compressibility Methods for Solving the Incompressible Navier-Stokes Equations. Teubner, Stuttgart (1997).
- R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis. North-Holland Publ. Company, Amsterdam (1977).
- R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis. CBMS-NSF Reg. Confer. Ser. Appl. Math. SIAM41 (1985).

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