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A Second Order Accurate in Time, energy Stable Finite Element Scheme for the flory-huggins-Cahn-Hilliard Equation

Advances in Applied Mathematics and Mechanics 2022年第6期14卷 1477-1508页

作者： Maoqin Yuan Wenbin Chen Cheng Wang Steven M.Wise Zhengru Zhang School of Science&Arts China University of Petroleum-Beijing at KaramayKaramayXinjiang 834000China Shanghai Key Laboratory of Mathematics for Nonlinear Sciences School of Mathematical SciencesFudan UniversityShanghai 200433China Department of Mathematics The University of MassachusettsNorth DartmouthMA 02747USA Department of Mathematics The University of TennesseeKnoxvilleTN 37996USA School of Mathematical Sciences Beijing Normal University and Laboratory of Mathematics and Complex SystemsMinistry of EducationBeijing 100875China

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 *** 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.

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A Modified Crank-Nicolson Numerical Scheme for the flory-huggins Cahn-Hilliard Model

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Communications in Computational Physics 2022年第1期31卷 60-93页

作者： Wenbin Chen Jianyu Jing Cheng Wang Xiaoming Wang Steven M.Wise Shanghai Key Laboratory for Contemporary Applied Mathematics Fudan UniversityShanghai 200433P.R.China School of Mathematical Sciences Fudan UniversityShanghai 200433P.R.China Mathematics Department University of MassachusettsNorth DartmouthMA 02747USA International Center for Mathematics and Department of Mathematics and Guangdong Provincial Key Laboratory of Computational Science and Material Designand National Center for Applied Mathematics ShenzhenSouthern University of Science and TechnologyShenzhen 518055P.R.China Mathematics Department University of TennesseeKnoxvilleTN 37996USA

In this paper we propose and analyze a second order accurate numericalscheme for the Cahn-Hilliard equation with logarithmic flory huggins energy potential. A modified Crank-Nicolson approximation is applied to the logarithmic nonlinear term, while the expansive term is updated by an explicit second order AdamsBashforth extrapolation, and an alternate temporal stencil is used for the surface diffusion term. A nonlinear artificial regularization term is added in the numerical scheme,which ensures the positivity-preserving property, i.e., the numerical value of the phasevariable is always between -1 and 1 at a point-wise level. Furthermore, an unconditional energy stability of the numerical scheme is derived, leveraging the special formof the logarithmic approximation term. In addition, an optimal rate convergence estimate is provided for the proposed numerical scheme, with the help of linearizedstability analysis. A few numerical results, including both the constant-mobility andsolution-dependent mobility flows, are presented to validate the robustness of the proposed numerical scheme.

关键词： Cahn-Hilliard equation flory huggins energy potential positivity preserving energy stability second order accuracy optimal rate convergence estimate

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