Finite element approximation of surfactant spreading on a thin film

We consider a fully practical finite element approximation of the following system of nonlinear degenerate parabolic equations: ∂u/∂t + 1/2 ∇. (u2 ∇ [σ (v)]) - 1/3 ∇. (u3 ∇ w) = 0, w = -c Δ u + a u-3 - δ u-v, ∂v/∂t + ∇. (u v ∇[σ (v)]) - ρ Δ v - 1/2 ∇. (u2 v ∇w) = 0. The above models a surfactant-driven thin film flow in the presence of both attractive, a > 0, and repulsive, δ > 0 with v > 3, van der Waals forces, where u is the height of the film, v is the concentration of the insoluble surfactant monolayer, and σ(v) := 1 - v is the typical surface tension. Here ρ > 0 and c > 0 are the inverses of the surface Peclet number and the modified capillary number. In addition to showing stability bounds for our approximation, we prove convergence in one space dimension when ρ > 0 and either a = δ = 0 or δ > 0. Furthermore, iterative schemes for solving the resulting nonlinear discrete system are discussed. Finally, some numerical experiments are presented.