A Second Order Accurate in Time, Energy Stable Finite Element Scheme for a Liquid Thin Film Coarsening Model

In this paper, we propose and analyze a second order accurate (in time) mass lumped mixed finite element numerical scheme for the liquid thin film coarsening model with a singular Leonard-Jones energy potential. The backward differentiation formula (BDF) stencil is applied in the temporal discretization, and a convex-concave decomposition is derived, so that the concave part corresponds to a quadratic energy. In turn, the Leonard-Jones potential term is treated implicitly and the concave part is approximated by a second order Adams-Bashforth explicit extrapolation. An artificial Douglas-Dupont regularization term is added to ensure the energy stability. Furthermore, we provide a theoretical justification that this numerical algorithm has a unique solution, such that the positivity property is always preserved for the phase variable at a point-wise level, so that a singularity is avoided in the scheme. In fact, the singular nature of the Leonard-Jones potential term around the value of 0 and the mass lumped FEM approach play an essential role in the positivity-preserving property in the discrete level. In addition, an optimal rate convergence estimate in the $ℓ^∞(0, T ; H^{−1}_h )∩ ℓ^2 (0, T ; H^1_h )$ norm is presented. Finally, two numerical experiments are carried out to verify the theoretical properties.