Access the full text.
Sign up today, get DeepDyve free for 14 days.
F. Leslie (1992)
Continuum theory for nematic liquid crystalsContinuum Mechanics and Thermodynamics, 4
L. Veiga, F. Brezzi, L. Marini (2013)
Virtual Elements for Linear Elasticity ProblemsSIAM J. Numer. Anal., 51
Jia Zhao, Qi Wang (2016)
Semi-Discrete Energy-Stable Schemes for a Tensor-Based Hydrodynamic Model of Nematic Liquid Crystal FlowsJournal of Scientific Computing, 68
(2013)
Beirão da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D., Russo, AMath. Models Methods Appl. Sci., 23
L. Veiga, C. Lovadina, G. Vacca (2015)
Divergence free Virtual Elements for the Stokes problem on polygonal meshesarXiv: Numerical Analysis
L. Veiga, F. Brezzi, L. Marini, A. Russo (2014)
The Hitchhiker's Guide to the Virtual Element MethodMathematical Models and Methods in Applied Sciences, 24
L. Veiga, F. Brezzi, L. Marini, A. Russo (2014)
Mixed Virtual Element Methods for general second order elliptic problems on polygonal meshesarXiv: Numerical Analysis
Chun Liu, N. Walkington (2002)
Mixed Methods for the Approximation of Liquid Crystal FlowsMathematical Modelling and Numerical Analysis, 36
JL Erickson (1987)
Continuum theory of nematic liquid crystalsRes. Mechanica, 21
F. Guillén-González, Juan Gutiérrez-Santacreu (2013)
A linear mixed finite element scheme for a nematic Ericksen–Leslie liquid crystal modelMathematical Modelling and Numerical Analysis, 47
L. Beir, F. Brezzi, S. Arabia (2013)
Basic principles of Virtual Element MethodsMathematical Models and Methods in Applied Sciences, 23
Guang‐an Zou, Bo Wang, X. Yang (2022)
A FULLY-DECOUPLED DISCONTINUOUS GALERKIN APPROXIMATION OF THE CAHN–HILLIARD–BRINKMAN–OHTA–KAWASAKI TUMOR GROWTH MODEL
A. Cangiani, G. Manzini, Oliver Sutton (2015)
Conforming and nonconforming virtual element methods for elliptic problemsarXiv: Numerical Analysis
Cameron Talischi, G. Paulino, A. Pereira, I. Menezes (2012)
PolyMesher: a general-purpose mesh generator for polygonal elements written in MatlabStructural and Multidisciplinary Optimization, 45
Xin Liu, Zhangxin Chen (2018)
The nonconforming virtual element method for the Navier-Stokes equationsAdvances in Computational Mathematics, 45
Chun Liu, N. Walkington (2000)
Approximation of Liquid Crystal FlowsSIAM J. Numer. Anal., 37
X Yang (2011)
10.1016/j.jnnfm.2011.02.004J. Non-Newtonian Fluid Mech., 166
L. Veiga, C. Lovadina, A. Russo (2016)
Stability Analysis for the Virtual Element MethodarXiv: Numerical Analysis
L. Veiga, A. Russo, G. Vacca (2017)
The Virtual Element Method with curved edgesESAIM: Mathematical Modelling and Numerical Analysis
D. Adak, E. Natarajan, Sarvesh Kumar (2018)
Convergence analysis of virtual element methods for semilinear parabolic problems on polygonal meshesNumerical Methods for Partial Differential Equations, 35
F. Leslie (1979)
Theory of Flow Phenomena in Liquid Crystals, 4
F. Leslie (1966)
SOME CONSTITUTIVE EQUATIONS FOR ANISOTROPIC FLUIDSQuarterly Journal of Mechanics and Applied Mathematics, 19
G. Gatica, Mauricio Munar, Filánder Sequeira (2018)
A mixed virtual element method for the Navier–Stokes equationsMathematical Models and Methods in Applied Sciences
F. Lin, Chun Liu (2000)
Existence of Solutions for the Ericksen-Leslie SystemArchive for Rational Mechanics and Analysis, 154
Arun Gain, Cameron Talischi, G. Paulino (2013)
On the Virtual Element Method for three-dimensional linear elasticity problems on arbitrary polyhedral meshesComputer Methods in Applied Mechanics and Engineering, 282
L. Veiga, C. Lovadina, G. Vacca (2017)
Virtual Elements for the Navier-Stokes Problem on Polygonal MeshesSIAM J. Numer. Anal., 56
F. Lin (1989)
Nonlinear theory of defects in nematic liquid crystals; Phase transition and flow phenomenaCommunications on Pure and Applied Mathematics, 42
Xi Zhang, M. Feng (2021)
A projection-based stabilized virtual element method for the unsteady incompressible Brinkman equationsAppl. Math. Comput., 408
B. Rivière (2008)
Discontinuous Galerkin methods for solving elliptic and parabolic equations - theory and implementation, 35
Xin Liu, Y. Nie (2021)
A modified nonconforming virtual element with BDM-like reconstruction for the Navier-Stokes equationsApplied Numerical Mathematics, 167
Meng Li, Jikun Zhao, Nan Wang, Shaochun Chen (2021)
Conforming and nonconforming conservative virtual element methods for nonlinear Schrödinger equation: A unified frameworkComputer Methods in Applied Mechanics and Engineering
L. Veiga, C. Lovadina, D. Mora (2015)
A Virtual Element Method for elastic and inelastic problems on polytope meshesarXiv: Numerical Analysis
Xiaofeng Yang, M. Forest, Chun Liu, Jie Shen (2011)
Journal of Non-newtonian Fluid Mechanics Shear Cell Rupture of Nematic Liquid Crystal Droplets in Viscous Fluids
Yangyang Tang, Guang‐an Zou, Jian Li (2023)
Unconditionally Energy-Stable Finite Element Scheme for the Chemotaxis-Fluid SystemJournal of Scientific Computing, 95
Guang‐an Zou, Zhaohua Li, X. Yang (2023)
Fully Discrete Discontinuous Galerkin Numerical Scheme with Second-Order Temporal Accuracy for the Hydrodynamically Coupled Lipid Vesicle ModelJournal of Scientific Computing, 95
Xin Liu, Zhengkang He, Zhangxin Chen (2020)
A fully discrete virtual element scheme for the Cahn-Hilliard equation in mixed formComput. Phys. Commun., 246
Cheng Wang, Jilu Wang, Zeyu Xia, Liwei Xu (2022)
Optimal error estimates of a Crank-Nicolson finite element projection method for magnetohydrodynamic equationsArXiv, abs/2203.07680
A. Cangiani, V. Gyrya, G. Manzini (2016)
The NonConforming Virtual Element Method for the Stokes EquationsSIAM J. Numer. Anal., 54
R. Becker, Xiaobing Feng, A. Prohl (2008)
Finite Element Approximations of the Ericksen-Leslie Model for Nematic Liquid Crystal FlowSIAM J. Numer. Anal., 46
G. Akrivis, V. Dougalis, O. Karakashian (1991)
On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equationNumerische Mathematik, 59
H. Brezis (2003)
The interplay between analysis and topology in some nonlinear PDE problemsBulletin of the American Mathematical Society, 40
F. Lin, Chun Liu (1995)
Nonparabolic dissipative systems modeling the flow of liquid crystalsCommunications on Pure and Applied Mathematics, 48
Jia Zhao, Xiaofeng Yang, Jun Li, Qi Wang (2016)
Energy Stable Numerical Schemes for a Hydrodynamic Model of Nematic Liquid CrystalsSIAM J. Sci. Comput., 38
A. Rey, M. Denn (2002)
DYNAMICAL PHENOMENA IN LIQUID-CRYSTALLINE MATERIALSAnnual Review of Fluid Mechanics, 34
Jia Zhao, Xiaofeng Yang, Jie Shen, Qi Wang (2016)
A decoupled energy stable scheme for a hydrodynamic phase-field model of mixtures of nematic liquid crystals and viscous fluidsJ. Comput. Phys., 305
M. Akbas, Songul Kaya, L. Rebholz (2017)
On the stability at all times of linearly extrapolated BDF2 timestepping for multiphysics incompressible flow problemsNumerical Methods for Partial Differential Equations, 33
P. Lin, Chun Liu (2006)
Simulations of singularity dynamics in liquid crystal flows: A C0 finite element approachJ. Comput. Phys., 215
B. Ahmad, A. Alsaedi, F. Brezzi, L. Marini, A. Russo (2013)
Equivalent projectors for virtual element methodsComput. Math. Appl., 66
Xin Liu, Jian Li, Zhangxin Chen (2017)
A nonconforming virtual element method for the Stokes problem on general meshesComputer Methods in Applied Mechanics and Engineering, 320
Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations
G. Vacca, L. Veiga (2015)
Virtual element methods for parabolic problems on polygonal meshesNumerical Methods for Partial Differential Equations, 31
Chun Liu, Jie Shen, Xiaofeng Yang (2007)
Dynamics of Defect Motion in Nematic Liquid Crystal Flow: Modeling and Numerical SimulationCommunications in Computational Physics, 2
In this paper, we consider the numerical approximations of the Ericksen-Leslie system for nematic liquid crystal flows, which can be used to describe the dynamics of low molar-mass nematic liquid crystal in certain materials. The main numerical challenge to solve this system lies in how to discretize nonlinear terms so that the energy stability can be held at the discrete level. This paper address this numerical problem by constructing a fully discrete virtual element scheme with second-order temporal accuracy, which is achieved by combining the extrapolated Crank-Nicolson (C-N) time-stepping scheme for the nonlinear coupling terms and the convex splitting method for the Ginzburg-Landau term. The unconditional energy stability and unique solvability of the fully discrete scheme are rigorously proved, we further prove the optimal error estimates of the developed scheme. Finally, some numerical experiments are presented to demonstrate the accuracy, energy stability, and performance of the proposed numerical scheme.
Advances in Computational Mathematics – Springer Journals
Published: Jun 1, 2023
Keywords: Nematic liquid crystal flows; Virtual element method; Crank-Nicolson scheme; Unconditional energy stability; Error estimates; 35Q35; 65M12; 65M15; 65M60
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.