Skip to main content

The Steady-State Navier–Stokes Equations

  • Chapter
  • First Online:
Finite Element Methods for Incompressible Flow Problems

Part of the book series: Springer Series in Computational Mathematics ((SSCM,volume 51))

  • 3644 Accesses

Abstract

The steady-state or stationary Navier–Stokes equations describe steady-state flows. Such flow fields can be expected in practice if:

  • all data of the Navier–Stokes equations (2.25) do not depend on the time,

  • the viscosity ν is sufficiently large, or equivalently, the Reynolds number Re is sufficiently small,

see Remark 2.22.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 249.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 249.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Arndt D, Dallmann H, Lube G (2015) Local projection FEM stabilization for the time-dependent incompressible Navier-Stokes problem. Numer Methods Partial Differ Equ 31:1224–1250

    Article  MathSciNet  MATH  Google Scholar 

  • Aubin J-P (1967) Behavior of the error of the approximate solutions of boundary value problems for linear elliptic operators by Gelerkin’s and finite difference methods. Ann Scuola Norm Sup Pisa (3) 21:599–637

    Google Scholar 

  • Bangerth W, Rannacher R (2003) Adaptive finite element methods for differential equations. Lectures in mathematics ETH Zürich. Birkhäuser Verlag, Basel, pp viii+207

    Google Scholar 

  • Becker R (2000) An optimal-control approach to a posteriori error estimation for finite element discretizations of the Navier-Stokes equations. East-West J Numer Math 8:257–274

    MathSciNet  MATH  Google Scholar 

  • Becker R, Rannacher R (1996) A feed-back approach to error control in finite element methods: basic analysis and examples. East-West J Numer Math 4:237–264

    MathSciNet  MATH  Google Scholar 

  • Becker R, Rannacher R (1998) Weighted a posteriori error control in FE methods. In: Bock HG, Kanschat G, Rannacher R, Brezzi F, Glowinski R, Kuznetsov YA, Périaux J (eds) ENUMATH 97. Proceedings of the 2nd European conference on numerical mathematics and advanced applications held at the University of Heidelberg, Heidelberg, September 28–October 3, 1997. World Scientific Publishing Co., Inc., River Edge, NJ, pp 621–637

    Google Scholar 

  • Becker R, Rannacher R (2001) An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer 10:1–102

    Article  MathSciNet  MATH  Google Scholar 

  • Besier M, Rannacher R (2012) Goal-oriented space-time adaptivity in the finite element Galerkin method for the computation of nonstationary incompressible flow. Int J Numer Methods Fluids 70:1139–1166

    Article  MathSciNet  Google Scholar 

  • Braack M, Mucha PB (2014) Directional do-nothing condition for the Navier-Stokes equations. J Comput Math 32:507–521

    Article  MathSciNet  MATH  Google Scholar 

  • Galdi GP (2011) An introduction to the mathematical theory of the Navier-Stokes equations. Steady-state problems. Springer monographs in mathematics, 2nd edn. Springer, New York, pp xiv+1018

    Google Scholar 

  • Girault V, Raviart P-A (1986) Finite element methods for Navier-Stokes equations. Theory and algorithms. Springer series in computational mathematics, vol 5. Springer, Berlin, pp x+374

    Google Scholar 

  • John V (2000) A numerical study of a posteriori error estimators for convection-diffusion equations. Comput Methods Appl Mech Eng 190:757–781

    Article  MathSciNet  MATH  Google Scholar 

  • John V, Matthies G (2001) Higher-order finite element discretizations in a benchmark problem for incompressible flows. Int J Numer Methods Fluids 37:885–903

    Article  MATH  Google Scholar 

  • John V, Tabata M, Tobiska L (1998) Error estimates for nonconforming finite element approximations of drag and lift in channel flows. Preprint 3/98. Fakultät für Mathematik, Otto-von-Guericke-Universität Magdeburg

    Google Scholar 

  • Layton W, Tobiska L (1998) A two-level method with backtracking for the Navier-Stokes equations. SIAM J Numer Anal 35:2035–2054 (electronic)

    Google Scholar 

  • Nitsche J (1968) Ein Kriterium für die Quasi-Optimalität des Ritzschen Verfahrens. Numer Math 11:346–348

    Article  MathSciNet  MATH  Google Scholar 

  • Olshanskii MA (2002) A low order Galerkin finite element method for the Navier-Stokes equations of steady incompressible flow: a stabilization issue and iterative methods. Comput Methods Appl Mech Eng 191:5515–5536

    Article  MathSciNet  MATH  Google Scholar 

  • Rockel S (2013) Über Formen des konvektiven Terms in Finite–Elemente–Diskretisierungen der inkompressiblen Navier–Stokes–Gleichungen. Diplomarbeit, Freie Universität Berlin

    Google Scholar 

  • Temam R (1984) Navier-Stokes equations. Theory and numerical analysis. Studies in mathematics and its applications, vol 2, 3rd edn. North-Holland Publishing Co., Amsterdam, pp xii+526. With an appendix by F. Thomasset

    Google Scholar 

  • Verfürth R (2013) A posteriori error estimation techniques for finite element methods. Numerical mathematics and scientific computation. Oxford University Press, Oxford, pp xx+393

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

John, V. (2016). The Steady-State Navier–Stokes Equations. In: Finite Element Methods for Incompressible Flow Problems. Springer Series in Computational Mathematics, vol 51. Springer, Cham. https://doi.org/10.1007/978-3-319-45750-5_6

Download citation

Publish with us

Policies and ethics