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The Oseen Equations

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Finite Element Methods for Incompressible Flow Problems

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

Abstract

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.

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References

  • Becker R, Braack M (2001) A finite element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo 38:173–199

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • Braack M, Lube G (2009) Finite elements with local projection stabilization for incompressible flow problems. J Comput Math 27:116–147

    MathSciNet  MATH  Google Scholar 

  • Braack M, Burman E, John V, Lube G (2007) Stabilized finite element methods for the generalized Oseen problem. Comput Methods Appl Mech Eng 196:853–866

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • Burman E, Hansbo P (2004) Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems. Comput Methods Appl Mech Eng 193:1437–1453

    Article  MathSciNet  MATH  Google Scholar 

  • Burman E, Hansbo P (2006) A stabilized non-conforming finite element method for incompressible flow. Comput Methods Appl Mech Eng 195:2881–2899

    Article  MathSciNet  MATH  Google Scholar 

  • Burman E, Fernández MA, Hansbo P (2006) Continuous interior penalty finite element method for Oseen’s equations. SIAM J Numer Anal 44:1248–1274

    Article  MathSciNet  MATH  Google Scholar 

  • 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 33:974–1007

    Article  MathSciNet  MATH  Google Scholar 

  • Codina R, Blasco J (1997) A finite element formulation for the Stokes problem allowing equal velocity-pressure interpolation. Comput Methods Appl Mech Eng 143:373–391

    Article  MathSciNet  MATH  Google Scholar 

  • 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 207–216

    Google Scholar 

  • Franca LP, Frey SL (1992) Stabilized finite element methods. II. The incompressible Navier-Stokes equations. Comput Methods Appl Mech Eng 99:209–233

    Article  MathSciNet  MATH  Google Scholar 

  • 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 177:243–267

    Article  MathSciNet  MATH  Google Scholar 

  • Hansbo P, Szepessy A (1990) A velocity-pressure streamline diffusion finite element method for the incompressible Navier-Stokes equations. Comput Methods Appl Mech Eng 84:175–192

    Article  MathSciNet  MATH  Google Scholar 

  • 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 19–35

    Google Scholar 

  • John V, Maubach JM, Tobiska L (1997) Nonconforming streamline-diffusion-finite-element-methods for convection-diffusion problems. Numer Math 78:165–188

    Article  MathSciNet  MATH  Google Scholar 

  • Lube G, Rapin G (2006) Residual-based stabilized higher-order FEM for a generalized Oseen problem. Math Models Methods Appl Sci 16:949–966

    Article  MathSciNet  MATH  Google Scholar 

  • Lube G, Tobiska L (1990) A nonconforming finite element method of streamline diffusion type for the incompressible Navier-Stokes equations. J Comput Math 8:147–158

    MathSciNet  MATH  Google Scholar 

  • Lube G, Rapin G, Löwe J (2008) Local projection stabilization for incompressible flows: equal-order vs. inf-sup stable interpolation. Electron Trans Numer Anal 32:106–122

    MathSciNet  MATH  Google Scholar 

  • Matthies G, Tobiska L (2015) Local projection type stabilization applied to inf-sup stable discretizations of the Oseen problem. IMA J Numer Anal 35:239–269

    Article  MathSciNet  MATH  Google Scholar 

  • 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. 41:713–742

    Article  MathSciNet  MATH  Google Scholar 

  • 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 9:368–390

    Article  MathSciNet  MATH  Google Scholar 

  • 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 18:309–332

    MathSciNet  MATH  Google Scholar 

  • 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+604

    Google Scholar 

  • 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 23:627–647

    MathSciNet  MATH  Google Scholar 

  • Schieweck F, Tobiska L (1996) An optimal order error estimate for an upwind discretization of the Navier-Stokes equations. Numer Methods Partial Differ Equ 12:407–421

    Article  MathSciNet  MATH  Google Scholar 

  • Scott LR, Zhang S (1990) Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math Comp 54:483–493

    Article  MathSciNet  MATH  Google Scholar 

  • Tobiska L, Lube G (1991) A modified streamline diffusion method for solving the stationary Navier-Stokes equation. Numer Math 59:13–29

    Article  MathSciNet  MATH  Google Scholar 

  • Tobiska L, Verfürth R (1996) Analysis of a streamline diffusion finite element method for the Stokes and Navier-Stokes equations. SIAM J Numer Anal 33:107–127

    Article  MathSciNet  MATH  Google Scholar 

  • Umla R (2009) Stabilisierte Finite–Element Verfahren für die Konvektions-Diffusions-Gleichungen und Oseen-Gleichungen. Diplomarbeit, Universität des Saarlandes, FR 6.1 – Mathematik

    Google Scholar 

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Authors and Affiliations

  1. Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany

    Volker John

  2. Fachbereich Mathematik und Informatik, Freie Universität Berlin, Berlin, Germany

    Volker John

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  1. Volker John
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John, V. (2016). The Oseen 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_5

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