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

作　　者：Maoqin Yuan Wenbin Chen Cheng Wang Steven M.Wise Zhengru Zhang 

机构地区：[1]School of Science&Arts,China University of Petroleum-Beijing at Karamay,Karamay,Xinjiang 834000,China [2]Shanghai Key Laboratory of Mathematics for Nonlinear Sciences,School of Mathematical Sciences,Fudan University,Shanghai 200433,China [3]Department of Mathematics,The University of Massachusetts,North Dartmouth,MA 02747,USA [4]Department of Mathematics,The University of Tennessee,Knoxville,TN 37996,USA [5]School of Mathematical Sciences,Beijing Normal University and Laboratory of Mathematics and Complex Systems,Ministry of Education,Beijing 100875,China

出　　处：《Advances in Applied Mathematics and Mechanics》2022年第6期1477-1508,共32页应用数学与力学进展（英文）

基　　金：NSFC(No.12071090);the National Key R&D Program of China(No.2019YFA0709502);Z.R.Zhang is partially supported by NSFC No.11871105 and Science Challenge Project No.TZ2018002;C.Wang is partially supported by the NSF DMS-2012269;S.M.Wise is partially supported by the NSF DMS-1719854,DMS-2012634.

摘　　要：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.

关 键 词：Cahn-Hilliard equations Flory Huggins energy potential mass lumped FEM convexconcave decomposition energy stability positivity preserving. 

分 类 号：O35[理学—流体力学]