(untitled) (Advances in Applied Mathematics and Mechanics, 2022)
In this paper, we propose and analyze a second order accurate in time, mass lumped 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 treated implicitly, while the expansive term is explicitly updated via a second-order Adams-Bashforth extrapolation formula, following the idea of the convex-concave decomposition of the energy functional. In addition, an artificial Douglas-Dupont regularization term is added to ensure the energy dissipativity. In the spatial discretization, the mass lumped finite element method is adopted. We provide a theoretical justification of the unique solvability of the mass lumped finite element scheme, using a piecewise linear element. In particular, the positivity is always preserved for the logarithmic arguments in the sense that the phase variable is always located between −1 and 1. In fact, the singular nature of the implicit terms and the mass lumped approach play an essential role in the positivity preservation in the discrete setting. Subsequently, an unconditional energy stability is proven for the proposed numerical scheme. In addition, the convergence analysis and error estimate of the numerical scheme are also presented. Two numerical experiments are carried out to verify the theoretical properties.