The Oseen Equations

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


Oseen equations, which are linear equations, show up as an auxiliary problem in many numerical approaches for solving the Navier–Stokes equations. Applying an implicit method for the temporal discretization of the Navier–Stokes equations requires the solution of a nonlinear problem in each discrete time. Likewise, the steady-state Navier–Stokes equations are nonlinear. Applying in either situation a so-called Picard method (a fixed point iteration) for solving the nonlinear problem, leads to an Oseen problem in each iteration, compare Sect.  6.3 The application of semi-implicit time discretizations to the Navier–Stokes equations leads directly to an Oseen problem in each discrete time, see Remark  7.61. Altogether, Oseen problems have to be solved in many methods for simulating the Navier–Stokes equations. In addition, some parts of the theory of the Oseen equations are used in the analysis of the Navier–Stokes equations, e.g., for the uniqueness of a weak solution of the steady-state Navier–Stokes equations in Theorem  6.20. For these reasons, the analysis and numerical analysis of Oseen problems is of fundamental interest.


Oseen Problem Steady-state Navier-Stokes Equations Semi-implicit Time Discretization Fixed Point Iteration Temporal Discretization 
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  1. Becker R, Braack M (2001) A finite element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo 38:173–199MathSciNetCrossRefzbMATHGoogle Scholar
  2. Braack M, Burman E (2006) Local projection stabilization for the Oseen problem and its interpretation as a variational multiscale method. SIAM J Numer Anal 43:2544–2566 (electronic)Google Scholar
  3. Braack M, Lube G (2009) Finite elements with local projection stabilization for incompressible flow problems. J Comput Math 27:116–147MathSciNetzbMATHGoogle Scholar
  4. Braack M, Burman E, John V, Lube G (2007) Stabilized finite element methods for the generalized Oseen problem. Comput Methods Appl Mech Eng 196:853–866MathSciNetCrossRefzbMATHGoogle Scholar
  5. Brooks AN, Hughes TJR (1982) Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput Methods Appl Mech Eng 32:199–259. FENOMECH ’81, part I (Stuttgart, 1981)Google Scholar
  6. Burman E, Hansbo P (2004) Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems. Comput Methods Appl Mech Eng 193:1437–1453MathSciNetCrossRefzbMATHGoogle Scholar
  7. Burman E, Hansbo P (2006) A stabilized non-conforming finite element method for incompressible flow. Comput Methods Appl Mech Eng 195:2881–2899MathSciNetCrossRefzbMATHGoogle Scholar
  8. Burman E, Fernández MA, Hansbo P (2006) Continuous interior penalty finite element method for Oseen’s equations. SIAM J Numer Anal 44:1248–1274MathSciNetCrossRefzbMATHGoogle Scholar
  9. Chacón Rebollo T, Gómez Mármol M, Girault V, Sánchez Muñoz I (2013) A high order term-by-term stabilization solver for incompressible flow problems. IMA J Numer Anal 33:974–1007MathSciNetCrossRefzbMATHGoogle Scholar
  10. Codina R, Blasco J (1997) A finite element formulation for the Stokes problem allowing equal velocity-pressure interpolation. Comput Methods Appl Mech Eng 143:373–391MathSciNetCrossRefzbMATHGoogle Scholar
  11. Douglas J Jr, Dupont T (1976) Interior penalty procedures for elliptic and parabolic Galerkin methods. In: Computing methods in applied sciences (second international symposium, Versailles, 1975). Lecture Notes in Physics, vol 58 Springer, Berlin, pp 207–216Google Scholar
  12. Franca LP, Frey SL (1992) Stabilized finite element methods. II. The incompressible Navier-Stokes equations. Comput Methods Appl Mech Eng 99:209–233MathSciNetCrossRefzbMATHGoogle Scholar
  13. Gelhard T, Lube G, Olshanskii MA, Starcke J-H (2005) Stabilized finite element schemes with LBB-stable elements for incompressible flows. J Comput Appl Math 177:243–267MathSciNetCrossRefzbMATHGoogle Scholar
  14. Hansbo P, Szepessy A (1990) A velocity-pressure streamline diffusion finite element method for the incompressible Navier-Stokes equations. Comput Methods Appl Mech Eng 84:175–192MathSciNetCrossRefzbMATHGoogle Scholar
  15. Hughes TJR, Brooks A (1979) A multidimensional upwind scheme with no crosswind diffusion. Finite element methods for convection dominated flows (Papers, winter annual meeting american society of mechanical engineers, New York, 1979). AMD, vol 34. American Society of Mechanical Engineers (ASME), New York, pp 19–35Google Scholar
  16. John V, Maubach JM, Tobiska L (1997) Nonconforming streamline-diffusion-finite-element-methods for convection-diffusion problems. Numer Math 78:165–188MathSciNetCrossRefzbMATHGoogle Scholar
  17. Lube G, Rapin G (2006) Residual-based stabilized higher-order FEM for a generalized Oseen problem. Math Models Methods Appl Sci 16:949–966MathSciNetCrossRefzbMATHGoogle Scholar
  18. Lube G, Tobiska L (1990) A nonconforming finite element method of streamline diffusion type for the incompressible Navier-Stokes equations. J Comput Math 8:147–158MathSciNetzbMATHGoogle Scholar
  19. Lube G, Rapin G, Löwe J (2008) Local projection stabilization for incompressible flows: equal-order vs. inf-sup stable interpolation. Electron Trans Numer Anal 32:106–122MathSciNetzbMATHGoogle Scholar
  20. Matthies G, Tobiska L (2015) Local projection type stabilization applied to inf-sup stable discretizations of the Oseen problem. IMA J Numer Anal 35:239–269MathSciNetCrossRefzbMATHGoogle Scholar
  21. Matthies G, Skrzypacz P, Tobiska L (2007) A unified convergence analysis for local projection stabilisations applied to the Oseen problem. M2AN Math. Model. Numer. Anal. 41:713–742MathSciNetCrossRefzbMATHGoogle Scholar
  22. Matthies G, Lube G, Röhe L (2009) Some remarks on residual-based stabilisation of inf-sup stable discretisations of the generalised Oseen problem. Comput Methods Appl Math 9:368–390MathSciNetCrossRefzbMATHGoogle Scholar
  23. Ohmori K, Ushijima T (1984) A technique of upstream type applied to a linear nonconforming finite element approximation of convective diffusion equations. RAIRO Anal Numér 18:309–332MathSciNetzbMATHGoogle Scholar
  24. Roos H-G, Stynes M, Tobiska L (2008) Robust numerical methods for singularly perturbed differential equations. Convection-diffusion-reaction and flow problems. Springer series in computational mathematics, vol 24, 2nd edn. Springer, Berlin, pp xiv+604Google Scholar
  25. Schieweck F, Tobiska L (1989) A nonconforming finite element method of upstream type applied to the stationary Navier-Stokes equation. RAIRO Modél Math Anal Numér 23:627–647MathSciNetzbMATHGoogle Scholar
  26. Schieweck F, Tobiska L (1996) An optimal order error estimate for an upwind discretization of the Navier-Stokes equations. Numer Methods Partial Differ Equ 12:407–421MathSciNetCrossRefzbMATHGoogle Scholar
  27. Scott LR, Zhang S (1990) Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math Comp 54:483–493MathSciNetCrossRefzbMATHGoogle Scholar
  28. Tobiska L, Lube G (1991) A modified streamline diffusion method for solving the stationary Navier-Stokes equation. Numer Math 59:13–29MathSciNetCrossRefzbMATHGoogle Scholar
  29. Tobiska L, Verfürth R (1996) Analysis of a streamline diffusion finite element method for the Stokes and Navier-Stokes equations. SIAM J Numer Anal 33:107–127MathSciNetCrossRefzbMATHGoogle Scholar
  30. Umla R (2009) Stabilisierte Finite–Element Verfahren für die Konvektions-Diffusions-Gleichungen und Oseen-Gleichungen. Diplomarbeit, Universität des Saarlandes, FR 6.1 – MathematikGoogle Scholar

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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Volker John
    • 1
    • 2
  1. 1.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany
  2. 2.Fachbereich Mathematik und InformatikFreie Universität BerlinBerlinGermany

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