We consider a system
of degenerate parabolic equations modelling a
thin film, consisting of two layers of immiscible Newtonian liquids, on
a solid horizontal substrate.
In addition, the model includes the presence of insoluble surfactants on
both the free liquid-liquid and liquid-air interfaces,
and the presence of both attractive and repulsive van der Waals forces
in 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 Laplacian
of the liquid heights.
We introduce a fully practical finite element approximation
of this nonlinear degenerate parabolic system, that satisfies discrete analogues
of these energy inequalities. Finally, we prove convergence of this approximation,
and hence existence of a solution
to this nonlinear degenerate parabolic system.