Navier-Stokes Approximations in High Order Norms

  • R. Rautmann
Part of the Notes on Numerical Fluid Mechanics (NNFM) book series (NONUFM, volume 48)


This note presents recent results on the convergence in high order norms of approximate solutions to the Navier-Stokes initial-boundary value problem: For a semi-discret splitting scheme using a 2-grid approach with Lagrangean transport stepping, H 2-convergence has been proved in [32]. To Rothe’s semi-discret approximation scheme, H 2-convergence has been proved in [29], and H 2,p-convergence in [34] with explicit convergence rates established in [35].

For parallelization methods to related splitting schemes, see Blazy, Borchers and Dralle’s contribution in this volume, for time splitting in kinetic gas equations, see Steiner’s contribution.


Finite Element Approximation Particle Path Stokes Operator Yosida Approximation Lagrangean Transport 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Alessandrini, G., Douglis, A., Fabes, E.: An approximate layering method for the Navier-Stokes equations in bounded cylinders, Ann. Math. Pura Appl. 135 (1983) 329–347.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    Ashyralyev, A., Sobolevskii, P.E.: Well-posedness of parabolic difference equations, Operator Theory Advances and Applications 69, Birkhäuser Basel, Boston, Berlin 1994.Google Scholar
  3. [3]
    Beale, J.T.: The approximation of the Navier-Stokes equations by fractional time steps, Lecture at the conference: The Navier-Stokes equations, theory and numerical methods, Oberwolfach 18.-24.8.91Google Scholar
  4. [4]
    Beale, J.T., Greengard, C.: Convergence of Euler-Stokes splitting of the Navier-Stokes equations, IBM Research Report RC 18072 6/11/92, Commun. Pure Appl. Math. XL VII(1994) 1083–1115.MathSciNetCrossRefGoogle Scholar
  5. [5]
    Borchers, W.: A splitting algorithm for incompressible Navier-Stokes equations, in: H. Niki, M. Kawahara (Eds.): Int. conf. on computational methods in flow analysis, Okayama, Japan (1988) 454–461.Google Scholar
  6. [6]
    Borchers, W.: On the characteristics method for the incompressible Navier-Stokes equations, in: E.H. Hirschel (Ed.): Finite Approximations in Fluid Mechanics II, Notes on Numerical Fluid Mechanics, Volume 25, Braunschweig 1989.Google Scholar
  7. [7]
    Borchers, W.: Zur Stabilität und Faktorisierungsmethode für die Navier-Stokes-Gleichungen inkompressibler viskoser Flüssigkeiten, Habilitationsschrift für das Fach Mathematik im Fachbereich Mathematik-Informatik der Universität-GH Paderborn, November 1992.Google Scholar
  8. [8]
    Chorin, A. J.: Numerical study of slightly viscous flow, J. Fluid Mechanics 57 (1973) 785–796.MathSciNetCrossRefGoogle Scholar
  9. [9]
    Chorin, A.J., Hughes, T.J.R., Mc Cracken, M.F., Marsden, J.E.: Product formulas and numerical algorithms, Comm. Pure Appl. Math. XXXI (1978) 205–256.MathSciNetCrossRefGoogle Scholar
  10. [10]
    Friedman, A.: Partial differential equations, Holt, Rinehart and Winston, New York 1964.zbMATHGoogle Scholar
  11. [11]
    Fujita, H.: On the semidiscrete finite element approximation for the evolution equation u t + A(t)u = 0 of parabolic type, in Miller, J.J. (Ed.): Topics in Numerical Analysis III, Academic Press New York (1977) 143–157.Google Scholar
  12. [12]
    Fujiwara, D., Morimoto, H.: An L r-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977) 685–700.MathSciNetzbMATHGoogle Scholar
  13. [13]
    Giga, Y.: The Stokes operator in L r spaces, Proc. Japan Acad. 57 Ser. A (1981) 85–89.MathSciNetzbMATHGoogle Scholar
  14. [14]
    Giga, Y.: Analyticity of the semigroup generated by the Stokes operator in L r spaces, Math. Z. 178 (1981) 297–329.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    Giga, Y.: Domains of fractional powers of the Stokes operator in L p spaces, Arch. Rat. Mech. Anal. 89 (1985) 251–265.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    Hebeker, F.K.: Analysis of a characteristics method for some incompressible and compressible Navier-Stokes problems, Preprint 1126 des FB Mathematik, TH Darmstadt (1988)Google Scholar
  17. [17]
    Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem I, Siam. J. Num. Anal. 19 (1982) 275–311.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    Lighthill, M.J.: Introduction, Boundary layer theory, in: Rosenhead, L. (Ed.), Laminar boundary layers, Oxford (1963) 46–113.Google Scholar
  19. [19]
    Masuda, K.: Remarks on compatibility conditions for solutions of Navier-Stokes equations, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987) 155–164.Google Scholar
  20. [20]
    Miyakawa, T.: On the initial value problem for the Navier-Stokes equations in L P spaces, Hiroshima Math. J. 11 (1981) 9–20.MathSciNetzbMATHGoogle Scholar
  21. [21]
    Miyakawa, T.: On nonstationary solutions of the Navier-Stokes equations in an exterior domain, Hiroshima Math. J. 12 (1982) 115–140.MathSciNetzbMATHGoogle Scholar
  22. [22]
    Pironneau, O.: On the transport diffusion algorithm and its applications to the Navier-Stokes equations, Num. Math. 38 (1982) 309–332.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    Rautmann, R.: On the convergence rate of nonstationary Navier-Stokes approximations, in: Proc. IUTAM Symp. Paderborn 1979, Springer Lecture Notes in Math. 771 (1980) 435–449.Google Scholar
  24. [24]
    Rautmann, R.: A semigroup approach to error estimates for nonstationary Navier-Stokes approximations, Proc. Conference Oberwolfach 1982, Methoden Verfahren Math. Physik 27 (1983) 63–77.MathSciNetGoogle Scholar
  25. [25]
    Rautmann, R.: On optimum regularity of Navier-Stokes solutions at time t = 0, Math. Z. 184 (1983) 141–149.MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    Rautmann, R.: Eine konvergente Produktformel für linearisierte Navier-Stokes-Probleme, Z. Angew. Math. Mech. 69 (1989) 181–183.MathSciNetGoogle Scholar
  27. [27]
    Rautmann, R.: A convergent product formula approach to three dimensional flow computations, Finite Approximations in Fluid Mechanics II (1989) 322–325.MathSciNetGoogle Scholar
  28. [28]
    Rautmann, R.: H 2-convergent linearizations to the Navier-Stokes initial value problem, in: Butazzo, G. Galdi, G.P. Zanghirati, L., (Eds.), Proc. Intern. Conf. on ”New developments in partial differential equations and applications to mathematical physics”, Ferrara 14.–18. October 1991, Plenum Press New York (1992) 135–156.Google Scholar
  29. [29]
    Rautmann, R.: H 2-convergence of Rothe’s scheme to the Navier-Stokes equations, Journal of Nonlinear Analyis 24 (1995) 1081–1102.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [30]
    Rautmann, R.: Optimum regularity of Navier-Stokes solutions at time t = 0 and applications, Acta Mech. (1993) supp. 4, 1–11.Google Scholar
  31. [31]
    Rautmann, R.: A remark on the convergence of Rothe’s scheme to the Navier-Stokes equations, to appear in: Stability and Applied Analysis of Continuous Media 3 (1993).Google Scholar
  32. [32]
    Rautmann, R., Masuda, K.: H 2-convergent approximation schemes to the Navier-Stokes equations, Comm. Math. Univ. Sancti Pauli 43 (1994) 55–108.MathSciNetzbMATHGoogle Scholar
  33. [33]
    Rautmann, R.: A direct construction of very smooth local Navier-Stokes solutions, Acta Appl. Math. 37 (1994) 153–168.MathSciNetzbMATHCrossRefGoogle Scholar
  34. [34]
    Rautmann, R.: A regularizing property of Rothe’s method to the Navier-Stokes equations, in: A. Sequeira (Ed.): Navier-Stokes Equations and Related Nonlinear Problems, Plenum Press New York (1995) 377–391.Google Scholar
  35. [35]
    Rautmann, R.: Convergence rates in H 2,q of Rothe’s method to the Navier-Stokes equations, to appear.Google Scholar
  36. [36]
    Rothe, E.: Zweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben, Math. Ann. 102 (1930) 650–670.MathSciNetzbMATHCrossRefGoogle Scholar
  37. [37]
    Rodenkirchen, J.: On optimum convergence rates of the Crank-Nicholson scheme to the Stokes initial value problem in higher order function spaces using realistic data. Thesis Paderborn 1995.Google Scholar
  38. [38]
    Solonnikov, V.A.: On differential properties of the solutions of the first boundary-value problem for nonstationary of Navier-Stokes equations, Trudy Mat. Inst. Steklov 73 (1964) 221–291, Transl.: British Library Lending Div., RTS 5211.MathSciNetzbMATHGoogle Scholar
  39. [39]
    Solonnikov, V.A.: Estimates for the solutions of nonstationary Navier-Stokes equations, Zap. Nauchn. Sem. Leningrad Lemingrad Mat. Steklova 38 (1973) 153–231, J. Sov. Math. 8 (1977) 467–529.MathSciNetzbMATHGoogle Scholar
  40. [40]
    Süli, E.: Lagrange-Galerkin mixed finite element approximation of the Navier-Stokes equations, in: K.W. Morton and M.J. Baines (eds.): Numerical Methods for Fluid Dynamics, Oxford University Press (1985) 439–448.Google Scholar
  41. [41]
    Süli, E.: Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations, Numer. Math. 53 (1988) 459–483.MathSciNetzbMATHCrossRefGoogle Scholar
  42. [42]
    Süli, E., Ware, A.: Analysis of spectral Lagrange-Galerkin method for the Navier-Stokes equations, in: Heywood, J.G., Masuda, K., Rautmann, R., Solonnikov, V.A., (Eds.), The Navier-Stokes Equations II, Springer Lecture Notes in Mathematics 1530 (1992) 184–195.Google Scholar
  43. [43]
    Tanabe, H.: Equations of evolution, Pitman London 1979.zbMATHGoogle Scholar
  44. [44]
    Temam, R.: Navier-Stokes equations, North-Holland, Amsterdam 1977, rev. ed. 1979, 3rd. rev. ed. 1984.zbMATHGoogle Scholar
  45. [45]
    Triebel, H.: Interpolation theory, function spaces, differential operators, North-Holland Amsterdam 1978.Google Scholar
  46. [46]
    Varnhorn, W.: Time stepping procedures for the nonstationary Stokes equations, preprint 1353, Technische Hochschule Darmstadt 1991.Google Scholar
  47. [47]
    Wahl, W. von: The equations of Navier-Stokes and abstract parabolic equations, Vieweg Braunschweig 1985.Google Scholar

Copyright information

© Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden 1996

Authors and Affiliations

  • R. Rautmann
    • 1
  1. 1.Fachbereich Mathematik-InformatikUniversität-GH PaderbornPaderbornGermany

Personalised recommendations