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The Oseen Equations

  • Volker John
Chapter
Part of the Springer Series in Computational Mathematics book series (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.

Keywords

Oseen Problem Steady-state Navier-Stokes Equations Semi-implicit Time Discretization Fixed Point Iteration Temporal Discretization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© 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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