Numerical methods for fourth order nonlinear degenerate diffusion problems

Jürgen Becker; Günther Grün; Martin Lenz; Martin Rumpf

Applications of Mathematics (2002)

  • Volume: 47, Issue: 6, page 517-543
  • ISSN: 0862-7940

Abstract

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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.

How to cite

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Becker, Jürgen, et al. "Numerical methods for fourth order nonlinear degenerate diffusion problems." Applications of Mathematics 47.6 (2002): 517-543. <http://eudml.org/doc/33129>.

@article{Becker2002,
abstract = {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.},
author = {Becker, Jürgen, Grün, Günther, Lenz, Martin, Rumpf, Martin},
journal = {Applications of Mathematics},
keywords = {thin film; fourth order degenerate parabolic equation; nonnegativity preserving scheme; surfactant driven flow; finite element method; finite volume method; thin film; fourth order degenerate parabolic equation; surfactant driven flow; finite element method; finite volume method},
language = {eng},
number = {6},
pages = {517-543},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Numerical methods for fourth order nonlinear degenerate diffusion problems},
url = {http://eudml.org/doc/33129},
volume = {47},
year = {2002},
}

TY - JOUR
AU - Becker, Jürgen
AU - Grün, Günther
AU - Lenz, Martin
AU - Rumpf, Martin
TI - Numerical methods for fourth order nonlinear degenerate diffusion problems
JO - Applications of Mathematics
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 47
IS - 6
SP - 517
EP - 543
AB - 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.
LA - eng
KW - thin film; fourth order degenerate parabolic equation; nonnegativity preserving scheme; surfactant driven flow; finite element method; finite volume method; thin film; fourth order degenerate parabolic equation; surfactant driven flow; finite element method; finite volume method
UR - http://eudml.org/doc/33129
ER -

References

top
  1. 10.1007/s002110050377, Numer. Math. 80 (1998), 525–556. (1998) MR1650035DOI10.1007/s002110050377
  2. 10.1051/m2an:2001133, Math. Model. Num. Anal. 35 (2001), 713–748. (2001) MR1863277DOI10.1051/m2an:2001133
  3. Finite-Elemente-Verfahren zur Dünne-Filme-Gleichung mit nichtlinearem Oberflächenspannungsterm, Diplomarbeit, Universität Bonn, 2000. (2000) 
  4. Numerical schemes for the thin film equation with nonlinear surface tension term, In preparation. 
  5. 10.1007/BF00379920, Arch. Rational Mech. Anal. 129 (1995), 175–20. (1995) MR1328475DOI10.1007/BF00379920
  6. Viscous flows, fourth order nonlinear degenerate parabolic equations and singular elliptic problems, Free Boundary Problems: Theory and Applications. Pitman Research Notes in Mathematics 323, J. I. Diaz, M. A. Herrero, A. Linan and J. L. Vazquez (eds.), Longman, Harlow, 1995, pp. 40–56. (1995) Zbl0839.35102MR1342325
  7. Finite speed of propagation and continuity of the interface for thin viscous flows, Adv. Differential Equations 1 (1996), 337–368. (1996) Zbl0846.35058MR1401398
  8. Finite speed of propagation for thin viscous flows when 2 ≤ n < 3 , C. R.  Acad. Sci. Paris Sér. I Math. 322 (1996), 1169–1174. (1996) MR1396660
  9. 10.1016/0022-0396(90)90074-Y, J.  Differential Equations 83 (1990), 179–206. (1990) MR1031383DOI10.1016/0022-0396(90)90074-Y
  10. The lubrication approximation for thin viscous films: Regularity and long time behaviour of weak solutions, Nonlin. Anal. 18 (1992), 217–234. (1992) 
  11. The thin viscous flow equation in higher space dimensions, Adv. Differential Equ. 3 (1998), 417–440. (1998) MR1751951
  12. The Finite Element Method for Elliptic Problems, North Holland, Amsterdam, 1978. (1978) Zbl0383.65058MR0520174
  13. 10.1137/S0036141096306170, SIAM J.  Math. Anal. 29 (1998), 321–342. (1998) MR1616558DOI10.1137/S0036141096306170
  14. 10.1137/S0036141094267662, SIAM J. Math. Anal. 27 (1996), 404–423. (1996) MR1377481DOI10.1137/S0036141094267662
  15. One-sided difference approximations for nonlinear conservation laws, Math. Comp. 31 (1981), 321–351. (1981) MR0606500
  16. The finite volume method for the Richards equation, Computational Geosciences 3 (1999), . (1999) MR1750075
  17. 10.1002/cpa.10017, Comm. Pure Appl. Math. 55 (2002), 217–254. (2002) MR1865415DOI10.1002/cpa.10017
  18. On the convergence of entropy consistent schemes for lubrication type equations in multiple space dimensions, Math. Comp (to appear). (to appear) MR1972735
  19. 10.4171/ZAA/639, Z.  Anal. Anwendungen 14 (1995), 541–573. (1995) MR1362530DOI10.4171/ZAA/639
  20. On free boundary problems arising in thin film flow, Habilitationsschrift, Universität Bonn, 2001. (2001) 
  21. Droplet spreading under weak slippage: the optimal asymptotic propagation rate in the multi-dimensional case, Interfaces and Free Boundaries 4 (2002), 309–323. (2002) Zbl1056.35072MR1914626
  22. Droplet spreading under weak slippage: a basic result on finite speed of propagation, Submitted,  (2002). (2002) MR1969611
  23. 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. MR0993361
  24. A finite volume scheme for surfactant driven thin film flow, Finite volumes for complex applications III, R. Herbin, D. Kröner (eds.), Hermes Penton Sciences, 2002, pp. 567–574. (2002) MR2008981
  25. 10.1007/s002110000197, Numer. Math. 87 (2000), 113–152. (2000) MR1800156DOI10.1007/s002110000197
  26. 10.1017/S0956792501004429, European J.  Appl. Math. 12 (2001), 293–320. (2001) MR1936040DOI10.1017/S0956792501004429
  27. 10.1017/S0022112092000090, J.  Fluid Mech. 240 (1992), 259–288. (1992) MR1175090DOI10.1017/S0022112092000090
  28. Numerical Schemes for Conservation Laws. Willey-Teubner Series Advances in Numerical Mathematics, Willey-Teubner, Chichester-Stuttgart, 1997. (1997) MR1437144
  29. Finite Volumen Methoden für degenerierte parabolische Systeme—Ausbreitung eines Surfactant auf einem dünnen Flüssigkeitsfilm, Diplomarbeit, Universität Bonn, 2002. (2002) 
  30. 10.1007/PL00005479, Numer. Math. 89 (2001), 561–590. (2001) MR1864431DOI10.1007/PL00005479
  31. 10.1103/RevModPhys.69.931, Rev. of Mod. Phys. 69 (1997), 931–980. (1997) DOI10.1103/RevModPhys.69.931
  32. Positivity preserving numerical schemes for lubrication-type equations, SIAM J.  Numer. Anal. 37 (2000), 523–555. (2000) MR1740768

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