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An Energy Stable Finite Element Scheme for the Three-Component Cahn–Hilliard-Type Model for Macromolecular Microsphere Composite Hydrogels

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Abstract

In this article, we present and analyze a finite element numerical scheme for a three-component macromolecular microsphere composite (MMC) hydrogel model, which takes the form of a ternary Cahn–Hilliard-type equation with Flory–Huggins–deGennes energy potential. The numerical approach is based on a convex–concave decomposition of the energy functional in multi-phase space, in which the logarithmic and the nonlinear surface diffusion terms are treated implicitly, while the concave expansive linear terms are explicitly updated. A mass lumped finite element spatial approximation is applied, to ensure the positivity of the phase variables. In turn, a positivity-preserving property can be theoretically justified for the proposed fully discrete numerical scheme. In addition, unconditional energy stability is established as well, which comes from the convexity analysis. Several numerical simulations are carried out to verify the accuracy and positivity-preserving property of the proposed scheme.

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Funding

W.B. Chen is partially supported by the National Natural Science Foundation of China (NSFC) 12071090, Shanghai Science and technology research program 19JC1420101 and a 111 project B08018. 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-2012669, S.M. Wise is partially supported by the NSF NSF-DMS 1719854, DMS-2012634.

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Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Maoqin Yuan

  2. Shanghai Key Laboratory of Mathematics for Nonlinear Sciences, School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Wenbin Chen

  3. Department of Mathematics, The University of Massachusetts, North Dartmouth, MA, 02747, USA

    Cheng Wang

  4. Department of Mathematics, The University of Tennessee, Knoxville, TN, 37996, USA

    Steven M. Wise

  5. School of Mathematical Sciences, Beijing Normal University and Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, People’s Republic of China

    Zhengru Zhang

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  1. Maoqin Yuan
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  2. Wenbin Chen
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  5. Zhengru Zhang
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Correspondence to Wenbin Chen.

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Yuan, M., Chen, W., Wang, C. et al. An Energy Stable Finite Element Scheme for the Three-Component Cahn–Hilliard-Type Model for Macromolecular Microsphere Composite Hydrogels. J Sci Comput 87, 78 (2021). https://doi.org/10.1007/s10915-021-01508-w

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