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

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 Mh(Uh)∇Ph will be identified with the flux M(u)∇(W'(u) - Δu) 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.

[1]  Ralf Blossey,et al.  Complex dewetting scenarios captured by thin-film models , 2003, Nature materials.

[2]  S. Bankoff,et al.  Long-scale evolution of thin liquid films , 1997 .

[3]  G. Grün.  Degenerate Parabolic Differential Equations of Fourth Order and a Plasticity Model with Non-Local Hardening , 1995 .

[4]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[5]  Martin Rumpf,et al.  Simulation of singularities and instabilities arising in thin film flow , 2001, European Journal of Applied Mathematics.

[6]  Harald Garcke,et al.  On A Fourth-Order Degenerate Parabolic Equation: Global Entropy Estimates, Existence, And Qualitativ , 1998 .

[7]  M. Bertsch,et al.  Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation , 1995 .

[8]  Mary C. Pugh,et al.  The lubrication approximation for thin viscous films: Regularity and long-time behavior of weak solutions , 1996 .

[9]  Lorenzo Giacomelli,et al.  A waiting time phenomenon for thin film equations , 2001 .

[10]  Harald Garcke,et al.  On Fully Practical Finite Element Approximations of Degenerate Cahn-Hilliard Systems , 2001 .

[11]  C. M. Elliott,et al.  On the Cahn-Hilliard equation with degenerate mobility , 1996 .

[12]  John W. Barrett,et al.  Finite element approximation of a sixth order nonlinear degenerate parabolic equation , 2004, Numerische Mathematik.

[13]  Andrea L. Bertozzi,et al.  Positivity-Preserving Numerical Schemes for Lubrication-Type Equations , 1999, SIAM J. Numer. Anal..

[14]  Francisco Bernis,et al.  Finite speed of propagation and continuity of the interface for thin viscous flows , 1996, Advances in Differential Equations.

[15]  J. Becker,et al.  Satellite hole formation during dewetting: experiment and simulation , 2003 .

[16]  Martin Rumpf,et al.  Nonnegativity preserving convergent schemes for the thin film equation , 2000, Numerische Mathematik.

[17]  Charles M. Elliott,et al.  The global dynamics of discrete semilinear parabolic equations , 1993 .

[18]  A. Friedman,et al.  Higher order nonlinear degenerate parabolic equations , 1990 .

[19]  Harald Garcke,et al.  The thin viscous flow equation in higher space dimensions , 1998 .

[20]  Harald Garcke,et al.  Finite Element Approximation of the Cahn-Hilliard Equation with Degenerate Mobility , 1999, SIAM J. Numer. Anal..

[21]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[22]  Antonio Fasano,et al.  Free boundary problems : theory and applications , 1983 .