Article
MSC:
35K35,
35K55,
35K65,
65M12,
65M50,
65M60,
76D08 | MR 1948194 | Zbl 1090.35086 | DOI: 10.1023/B:APOM.0000034537.55985.44
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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; nonnegativity preserving scheme; surfactant driven flow; finite element method; finite volume method
Summary:
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.
References:
[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. DOI 10.1007/s002110050377 | MR 1650035
[2] J. W. Barrett, J. F. Blowey and H. Garcke: On fully practical finite element approximations of degenerate Cahn-Hilliard systems. Math. Model. Num. Anal. 35 (2001), 713–748. DOI 10.1051/m2an:2001133 | MR 1863277
[3] J. Becker: Finite-Elemente-Verfahren zur Dünne-Filme-Gleichung mit nichtlinearem Oberflächenspannungsterm. Diplomarbeit, Universität Bonn, 2000.
[4] J. Becker, G. Grün: Numerical schemes for the thin film equation with nonlinear surface tension term. In preparation.
[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. DOI 10.1007/BF00379920 | MR 1328475
[6] F. Bernis: 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. MR 1342325 | Zbl 0839.35102
[7] F. Bernis: Finite speed of propagation and continuity of the interface for thin viscous flows. Adv. Differential Equations 1 (1996), 337–368. MR 1401398 | Zbl 0846.35058
[8] F. Bernis: Finite speed of propagation for thin viscous flows when $2\le n<3$. C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), 1169–1174. MR 1396660
[9] F. Bernis, A. Friedman: Higher order nonlinear degenerate parabolic equations. J. Differential Equations 83 (1990), 179–206. DOI 10.1016/0022-0396(90)90074-Y | MR 1031383
[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.
[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. MR 1751951
[12] Ph. G. Ciarlet: The Finite Element Method for Elliptic Problems. North Holland, Amsterdam, 1978. MR 0520174 | Zbl 0383.65058
[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. DOI 10.1137/S0036141096306170 | MR 1616558
[14] C. M. Elliott, H. Garcke: On the Cahn-Hilliard equation with degenerate mobility. SIAM J. Math. Anal. 27 (1996), 404–423. DOI 10.1137/S0036141094267662 | MR 1377481
[15] B. Engquist, S. Osher: One-sided difference approximations for nonlinear conservation laws. Math. Comp. 31 (1981), 321–351. MR 0606500
[16] R. Eymard, M. Gutnic and D. Hilhorst: The finite volume method for the Richards equation. Computational Geosciences 3 (1999), . MR 1750075
[17] L. Giacomelli, F. Otto: Droplet spreading: intermediate scaling law by pde methods. Comm. Pure Appl. Math. 55 (2002), 217–254. DOI 10.1002/cpa.10017 | MR 1865415
[18] G. Grün: On the convergence of entropy consistent schemes for lubrication type equations in multiple space dimensions. Math. Comp (to appear). MR 1972735
[19] G. Grün: Degenerate parabolic equations of fourth order and a plasticity model with nonlocal hardening. Z. Anal. Anwendungen 14 (1995), 541–573. DOI 10.4171/ZAA/639 | MR 1362530
[20] G. Grün: On free boundary problems arising in thin film flow. Habilitationsschrift, Universität Bonn, 2001.
[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. MR 1914626 | Zbl 1056.35072
[22] G. Grün: Droplet spreading under weak slippage: a basic result on finite speed of propagation. Submitted, (2002). MR 1969611
[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. MR 0993361
[24] G. Grün, M. Lenz, M. Rumpf: 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. MR 2008981
[25] G. Grün, M. Rumpf: Nonnegativity preserving convergent schemes for the thin film equation. Numer. Math. 87 (2000), 113–152. DOI 10.1007/s002110000197 | MR 1800156
[26] G. Grün, M. Rumpf: Simulation of singularities and instabilities arising in thin film flow. European J. Appl. Math. 12 (2001), 293–320. DOI 10.1017/S0956792501004429 | MR 1936040
[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. DOI 10.1017/S0022112092000090 | MR 1175090
[28] D. Kröner: Numerical Schemes for Conservation Laws. Willey-Teubner Series Advances in Numerical Mathematics. Willey-Teubner, Chichester-Stuttgart, 1997. MR 1437144
[29] M. Lenz: Finite Volumen Methoden für degenerierte parabolische Systeme—Ausbreitung eines Surfactant auf einem dünnen Flüssigkeitsfilm. Diplomarbeit, Universität Bonn, 2002.
[30] K. Mikula, N. Ramarosy: Semi-implicit finite volume scheme for solving nonlinear diffusion equations in image processing. Numer. Math. 89 (2001), 561–590. DOI 10.1007/PL00005479 | MR 1864431
[31] A. Oron, S. H. Davis and S. G. Bankoff: Long-scale evolution of thin liquid films. Rev. of Mod. Phys. 69 (1997), 931–980. DOI 10.1103/RevModPhys.69.931
[32] L. Zhornitskaya, A. L. Bertozzi: Positivity preserving numerical schemes for lubrication-type equations. SIAM J. Numer. Anal. 37 (2000), 523–555. MR 1740768
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