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

Abstract

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.

Numerical Analysis of Stabilized Second Order Semi-Implicit Finite Element Methods for the Phase-Field Equations

“…Note that the energy decay property of the phase field equations is considered to be fundamental to their derivation, their behavior, and their discretization, various numerical schemes which satisfy the corresponding discrete energy law have been designed (cf. [9,23,24,32,34,37,39,40,48] and the references therein). A commonly used technique to obtain an energy stable discretization includes convex splitting method [3,10,11] and secant-line type method [8,12,19,28].…”

Explicit P1 Finite Element Solution of the Maxwell-Wave Equation Coupling Problem with Absorbing b. c.

“…It should be noted, however, that mass lumping finite element schemes also have nice properties, such as the preservation of the positivity of certain physical variables (see, e.g., ref. [27]).…”

Error Analysis of a Pressure Penalty Scheme for the Reformulated Ericksen-Leslie System with Variable Density

“…In addition, a scalar auxiliary variable was used to transform the system into an equivalent form, which allowed the double well potential to be treated semi-explicitly. About Cahn-Hilliard equation, we refer the reader to [30][31][32][33][34][35] and references therein. To the best of our knowledge, there is little research on the numerical method of the Ericksen-Leslie system with variable density.…”