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Volume 22, Issue 5
An Energy-Dissipation Finite Element Pressure-Correction Scheme for the Hydrodynamics of Smectic-A Liquid Crystals

Panpan Guo, Guang-An Zou & Min Zhang

DOI: 10.4208/ijnam2025-1028

Int. J. Numer. Anal. Mod., 22 (2025), pp. 637-670.

Published online: 2025-05

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  • Abstract

In this paper, we develop a new linear, fully-decoupled, unconditional energy-stable BDF2-SAV-FEM scheme for solving the smectic-A liquid crystals, based on the finite element method (FEM) for spatial discretization and two-step backward differentiation formula (BDF2) for temporal discretization. To decouple the computations of the layer function and velocity field, we introduce an additional stabilization term into the constitutive equation. The nonlinear energy potential and the Navier-Stokes equations are treated by the scalar auxiliary variable (SAV) method and the rotational pressure-correction method, respectively. The unique solvability, unconditional energy stability, and error estimations of the proposed numerical scheme have been demonstrated. Several numerical experiments are carried out to validate our theoretical analysis.

  • Keywords

Liquid crystals flows, unconditional energy stability, second-order, finite element method.

  • AMS Subject Headings

65M12, 65N15, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-22-637, author = {Guo , PanpanZou , Guang-An and Zhang , Min}, title = {An Energy-Dissipation Finite Element Pressure-Correction Scheme for the Hydrodynamics of Smectic-A Liquid Crystals}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2025}, volume = {22}, number = {5}, pages = {637--670}, abstract = {

In this paper, we develop a new linear, fully-decoupled, unconditional energy-stable BDF2-SAV-FEM scheme for solving the smectic-A liquid crystals, based on the finite element method (FEM) for spatial discretization and two-step backward differentiation formula (BDF2) for temporal discretization. To decouple the computations of the layer function and velocity field, we introduce an additional stabilization term into the constitutive equation. The nonlinear energy potential and the Navier-Stokes equations are treated by the scalar auxiliary variable (SAV) method and the rotational pressure-correction method, respectively. The unique solvability, unconditional energy stability, and error estimations of the proposed numerical scheme have been demonstrated. Several numerical experiments are carried out to validate our theoretical analysis.

}, issn = {2617-8710}, doi = {https://doi.org/10.4208/ijnam2025-1028}, url = {http://global-sci.org/intro/article_detail/ijnam/24080.html} }
TY - JOUR T1 - An Energy-Dissipation Finite Element Pressure-Correction Scheme for the Hydrodynamics of Smectic-A Liquid Crystals AU - Guo , Panpan AU - Zou , Guang-An AU - Zhang , Min JO - International Journal of Numerical Analysis and Modeling VL - 5 SP - 637 EP - 670 PY - 2025 DA - 2025/05 SN - 22 DO - http://doi.org/10.4208/ijnam2025-1028 UR - https://global-sci.org/intro/article_detail/ijnam/24080.html KW - Liquid crystals flows, unconditional energy stability, second-order, finite element method. AB -

In this paper, we develop a new linear, fully-decoupled, unconditional energy-stable BDF2-SAV-FEM scheme for solving the smectic-A liquid crystals, based on the finite element method (FEM) for spatial discretization and two-step backward differentiation formula (BDF2) for temporal discretization. To decouple the computations of the layer function and velocity field, we introduce an additional stabilization term into the constitutive equation. The nonlinear energy potential and the Navier-Stokes equations are treated by the scalar auxiliary variable (SAV) method and the rotational pressure-correction method, respectively. The unique solvability, unconditional energy stability, and error estimations of the proposed numerical scheme have been demonstrated. Several numerical experiments are carried out to validate our theoretical analysis.

Guo , PanpanZou , Guang-An and Zhang , Min. (2025). An Energy-Dissipation Finite Element Pressure-Correction Scheme for the Hydrodynamics of Smectic-A Liquid Crystals. International Journal of Numerical Analysis and Modeling. 22 (5). 637-670. doi:10.4208/ijnam2025-1028
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