Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



On the convergence of entropy consistent schemes for lubrication type equations in multiple space dimensions

Author: Günther Grün
Journal: Math. Comp. 72 (2003), 1251-1279
MSC (2000): Primary 35K35, 35K55, 35K65, 35R35, 65M12, 65M60, 76D08
Published electronically: January 8, 2003
MathSciNet review: 1972735
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We present nonnegativity-preserving finite element schemes for a general class of thin film equations in multiple space dimensions. The equations are fourth order degenerate parabolic, and may contain singular terms of second order which are to model van der Waals interactions. A subtle discretization of the arising nonlinearities allows us to prove discrete counterparts of the essential estimates found in the continuous setting. By use of the entropy estimate, strong convergence results for discrete solutions are obtained. In particular, the limit of discrete fluxes $M_h(U_h)\nabla P_h$ will be identified with the flux $\mathcal M(u)\nabla(W'(u)-\Delta u)$ in the continuous setting. As a by-product, first results on existence and positivity almost everywhere of solutions to equations with singular lower order terms can be established in the continuous setting.

References [Enhancements On Off] (What's this?)

  • 1. J. Barrett, J. Blowey, and H. Garcke.
    Finite element approximation of a fourth order nonlinear degenerate parabolic equation.
    Numer. Math., 80:525-556, 1998. MR 99j:64144
  • 2. J. Barrett, J. Blowey, and H. Garcke.
    Finite element approximation of the Cahn-Hilliard equation with degenerate mobility.
    SIAM J. Num. Analysis, 37:286-318, 1999. MR 2001c:65118
  • 3. J. Barrett, J. Blowey, and H. Garcke.
    On fully practical finite element approximations of degenerate Cahn-Hilliard systems. Math. Model. Numer. Anal., 35:713-748, 2001.
  • 4. J. Becker, G. Grün, R. Seemann, H. Mantz, K. Jacobs, K.R. Mecke, and R. Blossey.
    Complex dewetting scenarios captured by thin film models.
    Nature Materials. In press.
  • 5. E. Beretta, M. Bertsch, and R. Dal Passo.
    Nonnegative solutions of a fourth order nonlinear degenerate parabolic equation.
    Arch. Ration. Mech. Anal., 129:175-200, 1995. MR 96b:35116
  • 6. F. Bernis.
    Viscous flows, fourth order nonlinear degenerate parabolic equations and singular elliptic problems.
    In J.I. Diaz, M.A. Herrero, A. Linan, and J.L. Vazquez, editors, Free boundary problems: theory and applications, Pitman Research Notes in Mathematics 323, pages 40-56. Longman, Harlow, 1995. MR 96b:00022
  • 7. MR 97e:35095 F. Bernis.
    Finite speed of propagation and continuity of the interface for thin viscous flows.
    Adv. Differential Equations, 1, no. 3:337-368, 1996. MR 97e:35095
  • 8. F. Bernis and A. Friedman.
    Higher order nonlinear degenerate parabolic equations.
    J. Differential Equations, 83:179-206, 1990. MR 91c:35078
  • 9. A.L. Bertozzi and M. Pugh.
    The lubrication approximation for thin viscous films: regularity and long time behaviour of weak solutions.
    Comm. Pure Appl. Math., 49(2):85-123, 1996. MR 97b:35114
  • 10. M. Bertsch, R. Dal Passo, H. Garcke, and G. Grün.
    The thin viscous flow equation in higher space dimensions.
    Adv. Differential Equations, 3:417-440, 1998. MR 2001a:35082
  • 11. Ph. G. Ciarlet.
    The finite element method for elliptic problems.
    North Holland, Amsterdam, 1978. MR 58:25001
  • 12. 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., 29:321-342, 1998. MR 99c:35118
  • 13. R. Dal Passo, L. Giacomelli, and G. Grün.
    A waiting time phenomenon for thin film equations.
    Ann. Scuola Norm. Sup. Pisa, XXX:437-463, 2001. CMP 2002:11
  • 14. C.M. Elliott and H. Garcke.
    On the Cahn-Hilliard equation with degenerate mobility.
    SIAM J. Math.Anal., 27 Nr. 2:404-423, 1996. MR 97c:35081
  • 15. C.M. Elliott and A.-M. Stuart.
    The global dynamics of discrete semilinear parabolic equations.
    Siam J. Numer. Anal., 30:1622-1663, 1993. MR 94j:65127
  • 16. G. Grün.
    Degenerate parabolic equations of fourth order and a plasticity model with nonlocal hardening.
    Z. Anal. Anwendungen, 14:541-573, 1995. MR 96m:35182
  • 17. G. Grün.
    On the numerical simulation of wetting phenomena.
    In W. Hackbusch and S. Sauter, editors, Proceedings of the 15th GAMM-Seminar Kiel, Numerical methods of composite materials. Vieweg-Verlag, Braunschweig. To appear.
  • 18. G. Grün and M. Rumpf.
    Nonnegativity preserving convergent schemes for the thin film equation.
    Num. Math., 87:113-152, 2000. MR 2002h:76108
  • 19. G. Grün and M. Rumpf.
    Simulation of singularities and instabilities in thin film flow.
    Euro. J. Appl. Math., 12:293-320, 2001.
  • 20. C. Neto, K. Jacobs, R. Seemann, R. Blossey, J. Becker, and G. Grün.
    Satellite hole formation during dewetting: experiment and simulation.
    Submitted for publication.
  • 21. A. Oron, S.H. Davis, and S.G. Bankoff.
    Long-scale evolution of thin liquid films.
    Reviews of Modern Physics, 69:932-977, 1997.
  • 22. K. Yosida.
    Functional analysis.
    Springer-Verlag, 1971. MR 50:2851
  • 23. L. Zhornitskaya and A.L. Bertozzi.
    Positivity preserving numerical schemes for lubrication-type equations.
    SIAM J. Num. Anal., 37:523-555, 2000. MR 2000m:65100

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 35K35, 35K55, 35K65, 35R35, 65M12, 65M60, 76D08

Retrieve articles in all journals with MSC (2000): 35K35, 35K55, 35K65, 35R35, 65M12, 65M60, 76D08

Additional Information

Günther Grün
Affiliation: Universität Bonn, Institut für Angewandte Mathematik, Beringstr. 6, 53115 Bonn, Germany

Keywords: {Lubrication approximation, fourth order degenerate parabolic equations, nonnegativity preserving, finite elements}
Received by editor(s): August 14, 2000
Received by editor(s) in revised form: September 21, 2001
Published electronically: January 8, 2003
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society