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On the numerical analysis of the nonstationary Navier-Stokes equations

The role of regularity and stability theory

  • Numerical Methods
  • Conference paper
  • First Online:
The Navier-Stokes Equations Theory and Numerical Methods

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1431))

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References

  1. Chen, C.M., Thomee, V.: The lumped mass finite element method for a parabolic problem, J.Austral.Math.Soc.Ser. B 26, 329–354 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  2. Girault,V., Raviart,P.A.: Finite Element Methods for Navier-Stokes Equations, Springer 1986.

    Google Scholar 

  3. Glowinski R., Periaux, J.: Numerical methods for nonlinear problems in fluid dynamics, Proc.Intern.Seminar on Scientific Supercomputers, Paris, North-Holland 1987.

    Google Scholar 

  4. Hahn,W.: Stability of Motion, Springer 1967.

    Google Scholar 

  5. Heywood, J.G.: The Navier-Stokes equations: On the existence regularity and decay of solutions, Indiana Univ.Math.J. 29, 636–682(1980).

    Article  MathSciNet  Google Scholar 

  6. Heywood,J.G.: Classical solutions of the Navier-Stokes equations, Proc. IUTAM Symp., Paderborn 1979,Springer, LNM 771, 1980, pp. 235–248.

    Google Scholar 

  7. Heywood, J.G.: Stability, regularity and numerical analysis of the nonstationary Navier-Stokes problem, Nonlinear PDE in Applied Science, U.S.-Japan Seminar, Tokyo, July 1982, Lecture Notes in Num.Appl.Anal. 5, 377–397(1982).

    Google Scholar 

  8. Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem, I. Regularity of solutions and second order error estimates for spatial discretization, SIAM J.Number.Anal. 19, 275–311 (1982), II. Stability of solutions and error estimates uniform in time, SIAM J.Numer.Anal. 23, 750–777 (1986), III. Smoothing property and higher order error estimates for spatial discretization, SIAM J.Numer.Anal. 25, 489–512 (1988), IV. Error analysis for second order time discretization, SIAM J.Numer. Anal., to appear.

    Article  MathSciNet  MATH  Google Scholar 

  9. Heywood, J.G., Rannacher, R.: An analysis of stability concepts for the Navier-Stokes equations, J.Reine Angew.Math. 372, 1–33 (1986).

    MathSciNet  MATH  Google Scholar 

  10. Heywood,J.G., Rannacher,R.: On the global numerical approximability of dynamical systems, Research Report, University of Saarbrücken, August 1986.

    Google Scholar 

  11. Johnson, C., Larsson, S., Thomee, V., Wahlbin, L.B.: Error estimates for spatially discrete approximations of semilinear parabolic equations with nonsmooth initial data, Math. Comp. 49, 331–357(1987).

    Article  MathSciNet  MATH  Google Scholar 

  12. Johnson, C., Saranen, J.:Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations, Math. Comp.47, 1–18 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  13. Ladyshenskaya,O.A.: The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach 1969

    Google Scholar 

  14. Lambert,J.D.: Computational Methods in Ordinary Differential Equations, John Wiley 1973.

    Google Scholar 

  15. Luskin, M., Rannacher, R.: On the smoothing property of the Crank-Nicolson scheme, Applicable Anal. 14, 117–135(1982).

    Article  MathSciNet  MATH  Google Scholar 

  16. Pironneau, O.: On the transport-diffusion algorithm and its application to the Navier-Stokes equations, Numer.Math. 38, 309–332 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  17. Rannacher, R.: Finite element solution of diffusion problems with irregular data, Numer. Math. 43, 309–327(1984).

    Article  MathSciNet  MATH  Google Scholar 

  18. Rannacher,R.: On the stabilization of the Crank-Nicolson scheme for long time calculations, Technical Report, University of Saarbrücken, September 1986.

    Google Scholar 

  19. Rannacher,R.: Numerical analysis of nonstationary fluid flow (a survey), Proc. III German-Italian Symp. "The Applications of Mathematics in Industry and Technology", Siena, June 18–22, 1988, Teubner-Kluver, to appear.

    Google Scholar 

  20. Rautmann, R.: On optimum regularity of Navier-Stokes solutions at time t = 0, Proc. Conf. Oberwolfach, November 21–27, 1982, Math.Z. 184, 141–149 (1983)

    Article  MathSciNet  Google Scholar 

  21. Solonnikov, V.A.: Estimates of solutions of a nonstationary linearized system of Navier-Stokes equations, A.M.S. Transl. 75, 1–116(1968).

    MATH  Google Scholar 

  22. Stetter,H.J.: Analysis of Discretization Methods for Ordinary Differential Equations, Springer 1973.

    Google Scholar 

  23. Temam,R.: Theory and Numerical Analysis of the Navier-Stokes Equations, North-Holland 1977.

    Google Scholar 

  24. Temam, R.: Behaviour at time t=0 of the solutions of semi-linear evolution equations, J.Differential Equations 43, 73–92 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  25. Thomee, V.: Galerkin-Finite Element Methods for Parabolic Problems, Springer, LNM 1054, 1984.

    Google Scholar 

Contributions to this Oberwolfach-Conference

  1. Hebeker,F.K.: On Lagrangean methods and Volterra integral equations of the first kind for incompressible Navier-Stokes problems, these Proceedings.

    Google Scholar 

  2. Masuda, K.: Remarks on compatibility conditions for solutions of Navier-Stokes equations, J.Fac.Sci.Univ. Tokyo, Sect. IA, Math. 34, 159–164 (1987).

    MathSciNet  MATH  Google Scholar 

  3. Polezhaev,V.I.: The problems of mathematical modelling on the basis of the unsteady Navier-Stokes equations.

    Google Scholar 

  4. Rautmann, R.: Eine konvergente Produktformel für linearisierte Navier-Stokes-Probleme, Proceedings GAMM-Conf. 1988, ZAMM 69, T181–T183 (1989).

    MathSciNet  MATH  Google Scholar 

  5. Rivkind, V.J.: Numerical methods for the Navier-Stokes equations with an unknown boundary between two viscous incompressible fluids, these Proceedings.

    Google Scholar 

  6. Roesner,K.G., Bar-Joseph, P., Solan, A.: Numerical simulation and experimental verification of cavity flows, these Proceedings.

    Google Scholar 

  7. Rozhdestvensky,B.L.: Numerical simulation of turbulent incompressible fluid flows in channels and pipes by nonstationary solutions of Navier-Stokes equations.

    Google Scholar 

  8. Tarunin,E.L.: Investigation of implicit schemes in terms of the variables ψ, ω with the help of the Babenko-Gelfand principle.

    Google Scholar 

  9. Titi,E.S.: Numerical criteria for detecting stable stationary and time periodic solutions to the Navier-Stokes equations.

    Google Scholar 

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Authors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld 294, D-6900, Heidelberg

    Rolf Rannacher

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  1. Rolf Rannacher
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John G. Heywood Kyûya Masuda Reimund Rautmann Vsevolod A. Solonnikov

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© 1990 Springer-Verlag

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Rannacher, R. (1990). On the numerical analysis of the nonstationary Navier-Stokes equations. In: Heywood, J.G., Masuda, K., Rautmann, R., Solonnikov, V.A. (eds) The Navier-Stokes Equations Theory and Numerical Methods. Lecture Notes in Mathematics, vol 1431. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0086069

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  • DOI: https://doi.org/10.1007/BFb0086069

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-52770-1

  • Online ISBN: 978-3-540-47141-7

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