Applications of Mathematics

, Volume 47, Issue 6, pp 517–543 | Cite as

Numerical methods for fourth order nonlinear degenerate diffusion problems

  • Jurgen Becker
  • Gunther Grun
  • Martin Lenz
  • Martin Rumpf


Numerical schemes are presented for a class of fourth order diffusion problems. These problems arise in lubrication theory for thin films of viscous fluids on surfaces. The equations being in general fourth order degenerate parabolic, additional singular terms of second order may occur to model effects of gravity, molecular interactions or thermocapillarity. Furthermore, we incorporate nonlinear surface tension terms. Finally, in the case of a thin film flow driven by a surface active agent (surfactant), the coupling of the thin film equation with an evolution equation for the surfactant density has to be considered. Discretizing the arising nonlinearities in a subtle way enables us to establish discrete counterparts of the essential integral estimates found in the continuous setting. As a consequence, the resulting algorithms are efficient, and results on convergence and nonnegativity or even strict positivity of discrete solutions follow in a natural way. The paper presents a finite element and a finite volume scheme and compares both approaches. Furthermore, an overview over qualitative properties of solutions is given, and various applications show the potential of the proposed approach.

thin film fourth order degenerate parabolic equation nonnegativity preserving scheme surfactant driven flow finite element method finite volume method 


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  1. [1]
    J.W. Barrett, J. F. Blowey and H. Garcke: Finite element approximation of a fourth order nonlinear degenerate parabolic equation. Numer. Math. 80 (1998), 525–556.Google Scholar
  2. [2]
    J.W. Barrett, J. F. Blowey and H. Garcke: On fully practical.nite element approximations of degenerate Cahn-Hilliard systems. Math. Model. Num. Anal. 35 (2001),713–748.Google Scholar
  3. [3]
    J. Becker: Finite-Elemente-Verfahren zur Dünne-Filme-Gleichung mit nichtlinearem Ober.ächenspannungsterm. Diplomarbeit. Universität Bonn, 2000.Google Scholar
  4. [4]
    J. Becker, G. Grün: Numerical schemes for the thin.lm equation with nonlinear surface tension term. In preparation.Google Scholar
  5. [5]
    E. Beretta, M. Bertsch and R. Dal Passo: Nonnegative solutions of a fourth order nonlinear degenerate parabolic equation. Arch. Rational Mech. Anal. 129 (1995), 175–20.Google Scholar
  6. [6]
    F. Bernis: Viscous.ows, fourth order nonlinear degenerate parabolic equations and singular elliptic problems. Free Boundary Problems: Theory and Applications. Pitman ResearchNotes in Mathematics 323 (J. I. Diaz, M.A. Herrero, A. Linan and J. L. Vazquez, eds.). Longman, Harlow, 1995, pp. 40–56.Google Scholar
  7. [7]
    F. Bernis: Finite speed of propagation and continuity of the interface for thin viscous. flows. Adv. Differential Equations 1 (1996), 337–368.Google Scholar
  8. [8]
    F. Bernis: Finite speed of propagation for thin viscous flows when 2 n < 3. C. R. Acad. Sci. Paris Sér. I Math. 322 (1996),1169–1174.Google Scholar
  9. [9]
    F. Bernis, A. Friedman: Higher order nonlinear degenerate parabolic equations. J. Differential Equations 83 (1990), 179–206.Google Scholar
  10. [10]
    A. L. Bertozzi, M. Pugh: The lubrication approximation for thin viscous films: Regularity and long time behaviour of weak solutions. Nonlin. Anal. 18 (1992), 217–234.Google Scholar
  11. [11]
    M. Bertsch, R. Dal Passo, H. Garcke and G. Grün: The thin viscous flow equation in higher space dimensions. Adv. Differential Equ. 3 (1998), 417–440.Google Scholar
  12. [12]
    Ph.G. Ciarlet: The Finite Element Method for Elliptic Problems. North Holland, Amsterdam, 1978.Google Scholar
  13. [13]
    R. Dal Passo, H. Garcke and G. Grün: On a fourth order degenerate parabolic equation: global entropy estimates and qualitative behaviour of solutions. SIAM J. Math. Anal. 29 (1998), 321–342.Google Scholar
  14. [14]
    C.M. Elliott, H. Garcke: On the Cahn-Hilliard equation with degenerate mobility. SIAM J. Math. Anal. 27 (1996), 404–423.Google Scholar
  15. [15]
    B. Engquist, S. Osher: One-sided difference approximations for nonlinear conservation laws. Math. Comp. 31 (1981), 321–351.Google Scholar
  16. [16]
    R. Eymard, M. Gutnic and D. Hilhorst: The finite volume method for the Richards equation. Computational Geosciences 3 (1999).Google Scholar
  17. [17]
    L. Giacomelli, F. Otto: Droplet spreading: intermediate scaling law by pde methods. Comm. Pure Appl. Math. 55 (2002), 217–254.Google Scholar
  18. 18]
    G. Grün: On the convergence of entropy consistent schemes for lubrication type equations in multiple space dimensions. Math. Comp. To appear.Google Scholar
  19. [19]
    G. Grün: Degenerate parabolic equations of fourth order and a plasticity model with nonlocal hardening. Z. Anal. Anwendungen 14 (1995), 541–573.Google Scholar
  20. [20]
    G. Grün: On free boundary problems arising in thin fillm flow. Habilitationsschrift. Universität Bonn, 2001.Google Scholar
  21. [21]
    G. Grün: Droplet spreading under weak slippage: the optimal asymptotic propagation rate in the multi-dimensional case. Interfaces and Free Boundaries 4 (2002), 309–323.Google Scholar
  22. [22]
    G. Grün: Droplet spreading under weak slippage: a basic result on finite speed of propagation. Submitted.Google Scholar
  23. [23]
    G. Grün: On the numerical simulation of wetting phenomena. In: Proceedings of the 15th GAMM-Seminar Kiel, Numerical Methods of Composite Materials (W. Hackbusch, S. Sauter, eds.). Vieweg-Verlag, Braunschweig. To appear.Google Scholar
  24. [24]
    G. Grün, M. Lenz, M. Rumpf: A.nite volume scheme for surfactant driven thin flow. Finite volumes for complex applications III (R. Herbin, D. Kröner, eds.). Hermes Penton Sciences, 2002, pp. 567–574.Google Scholar
  25. [25]
    G. Grün, M. Rumpf: Nonnegativity preserving convergent schemes for the thin film equation. Numer. Math. 87 (2000), 113–152.Google Scholar
  26. [26]
    G. Grün, M. Rumpf: Simulation of singularities and instabilities arising in flow. European J. Appl. Math. 12 (2001), 293–320.Google Scholar
  27. [27]
    O. E. Jensen, J. B. Grotberg: Insoluble surfactant spreading on a thin viscous film: shock evolution and film rupture. J. Fluid Mech. 240 (1992), 259–288.Google Scholar
  28. [28]
    D. Kröner: Numerical Schemes for Conservation Laws. Willey-Teubner Series Advances in Numerical Mathematics. Willey-Teubner, Chichester-Stuttgart, 1997.Google Scholar
  29. [29]
    M. Lenz: Finite Volumen Methoden für degenerierte parabolische Systeme—Ausbreitung eines Surfactant auf einem dünnen Flüssigkeitsfilm. Diplomarbeit. Universität Bonn, 2002.Google Scholar
  30. [30]
    K. Mikula, N. Ramarosy: Semi-implicit.nite volume scheme for solving nonlinear diffusion equations in image processing. Numer. Math. 89 (2001), 561–590.Google Scholar
  31. [31]
    A. Oron, S.H. Davis and S.G. Banko.: Long-scale evolution of thin liquid.films. Rev.of Mod. Phys. 69 (1997), 931–980.Google Scholar
  32. [32]
    L. Zhornitskaya, A. L. Bertozzi: Positivity preserving numerical schemes for lubrication-type equations. SIAM J. Numer. Anal. 37 (2000), 523–555.Google Scholar

Copyright information

© Mathematical Institute, Academy of Sciences of Czech Republic 2002

Authors and Affiliations

  • Jurgen Becker
    • 1
  • Gunther Grun
    • 1
  • Martin Lenz
    • 1
  • Martin Rumpf
    • 2
  1. 1.Institut fur Angewandte Mathematik, Universitat BonnBonn
  2. 2.Institut fur Mathematik, Gerhard-Mercator-Universitat DuisburgDuisburg

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