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Volume 14, Issue 6
A Second Order Accurate in Time, Energy Stable Finite Element Scheme for the Flory-Huggins-Cahn-Hilliard Equation

Maoqin Yuan, Wenbin Chen, Cheng Wang, Steven M. Wise & Zhengru Zhang

Adv. Appl. Math. Mech., 14 (2022), pp. 1477-1508.

Published online: 2022-08

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  • 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.

  • Keywords

Cahn-Hilliard equations, Flory Huggins energy potential, mass lumped FEM, convex-concave decomposition, energy stability, positivity preserving.

  • AMS Subject Headings

35K25, 35K55, 60F10, 65M60

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-14-1477, author = {Maoqin and Yuan and and 24250 and and Maoqin Yuan and Wenbin and Chen and and 24251 and and Wenbin Chen and Cheng and Wang and and 24252 and and Cheng Wang and Steven M. and Wise and and 24253 and and Steven M. Wise and Zhengru and Zhang and and 24254 and and Zhengru Zhang}, title = {A Second Order Accurate in Time, Energy Stable Finite Element Scheme for the Flory-Huggins-Cahn-Hilliard Equation}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2022}, volume = {14}, number = {6}, pages = {1477--1508}, 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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2021-0331}, url = {http://global-sci.org/intro/article_detail/aamm/20856.html} }
TY - JOUR T1 - A Second Order Accurate in Time, Energy Stable Finite Element Scheme for the Flory-Huggins-Cahn-Hilliard Equation AU - Yuan , Maoqin AU - Chen , Wenbin AU - Wang , Cheng AU - Wise , Steven M. AU - Zhang , Zhengru JO - Advances in Applied Mathematics and Mechanics VL - 6 SP - 1477 EP - 1508 PY - 2022 DA - 2022/08 SN - 14 DO - http://doi.org/10.4208/aamm.OA-2021-0331 UR - https://global-sci.org/intro/article_detail/aamm/20856.html KW - Cahn-Hilliard equations, Flory Huggins energy potential, mass lumped FEM, convex-concave decomposition, energy stability, positivity preserving. AB -

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.

Maoqin Yuan, Wenbin Chen, Cheng Wang, Steven M. Wise & Zhengru Zhang. (2022). A Second Order Accurate in Time, Energy Stable Finite Element Scheme for the Flory-Huggins-Cahn-Hilliard Equation. Advances in Applied Mathematics and Mechanics. 14 (6). 1477-1508. doi:10.4208/aamm.OA-2021-0331
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