A Second Order Accurate in Time, Energy Stable Finite Element Scheme for the Flory-Huggins-Cahn-Hilliard Equation

In this paper, we propose and analyze a second order accurate in time, masslumped mixed finite element scheme for the Cahn-Hilliard equation with a logarithmic Flory-Huggins energy potential. The standard backward differentiation formula (BDF)stencil is applied in the temporal discretization. In the chemical potential approximation,both the logarithmic singular terms and the surface diffusion term are treatedimplicitly, while the expansive term is explicitly updated via a second-order Adams-Bashforth extrapolation formula, following the idea of the convex-concave decompositionof the energy functional. In addition, an artificial Douglas-Dupont regularizationterm is added to ensure the energy dissipativity. In the spatial discretization, the masslumped finite element method is adopted. We provide a theoretical justification of theunique solvability of the mass lumped finite element scheme, using a piecewise linearelement. In particular, the positivity is always preserved for the logarithmic argumentsin the sense that the phase variable is always located between -1 and 1. In fact, thesingular nature of the implicit terms and the mass lumped approach play an essentialrole in the positivity preservation in the discrete setting. Subsequently, an unconditionalenergy stability is proven for the proposed numerical scheme. In addition, theconvergence analysis and error estimate of the numerical scheme are also presented.Two numerical experiments are carried out to verify the theoretical properties.