We consider a system of degenerate parabolic equations modelling a thin film, consisting of two layers of immiscible Newtonian liquids, ona solid horizontal substrate. In addition, the model includes the presence of insoluble surfactants onboth the free liquid-liquid and liquid-air interfaces,and the presence of both attractive and repulsive van der Waals forcesin terms of the heights of the two layers. We show that this system formally satisfies a Lyapunov structure,and a second energy inequality controlling the Laplacianof the liquid heights.We introduce a fully practical finite element approximation of this nonlinear degenerate parabolic system, that satisfies discrete analoguesof these energy inequalities. Finally, we prove convergence of this approximation,and hence existence of a solutionto this nonlinear degenerate parabolic system.


In recent years several papers have been devoted to stabilityand smoothing properties in maximum-norm offinite element discretizations of parabolic problems.Using the theory of analytic semigroups it has been possibleto rephrase such properties as bounds for the resolventof the associated discrete elliptic operator. In all thesecases the triangulations of the spatial domain has beenassumed to be quasiuniform. In the present paper weshow a resolvent estimate, in one and two space dimensions,under weaker conditions on the triangulations than quasiuniformity.In the two-dimensional case, the bound for the resolvent containsa logarithmic factor.