Advanced Search
Article Contents
Article Contents

Longtime behavior of a second order finite element scheme simulating the kinematic effects in liquid crystal dynamics

  • * Corresponding author

    * Corresponding author 
Abstract / Introduction Full Text(HTML) Related Papers Cited by
  • We consider an unconditional fully discrete finite element scheme for a nematic liquid crystal flow with different kinematic transport properties. We prove that the scheme converges towards a unique critical point of the elastic energy subject to the finite element subspace, when the number of time steps go to infinity while the time step and mesh size are fixed. A Lojasiewicz type inequality, which is the key for getting the time asymptotic convergence of the whole sequence furnished by the numerical scheme, is also derived.

    Mathematics Subject Classification: 35Q35, 35K55, 35B40, 65M60.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] P. F. AntoniettiB. MerletM. Pierre and M. Verani, Convergence to equilibrium for a second-order time semi-discretization of the Cahn-Hilliard equation, Z. Angew. Math. Phys, 1 (2016), 66-95.  doi: 10.3934/Math.2016.3.178.
    [2] J. M. Ball, Mathematics and liquid crystals, Molecular Crystals and Liquid Crystals, 647 (2017), 1-27.  doi: 10.1080/15421406.2017.1289425.
    [3] S. Bartels and A. Prohl, Constraint preserving implicit finite element discretization of harmonic map heat flow into spheres, Math. Comp, 76 (2007), 1847-1859.  doi: 10.1090/S0025-5718-07-02026-1.
    [4] R. BeckerX. Feng and A. Prohl, Finite element approximations of the Ericksen-Leslie model for nematic liquid crystal flow, SIAM J. Numer. Anal., 46 (2008), 1704-1731.  doi: 10.1137/07068254X.
    [5] B. Climent-Ezquerra and F. Guillén-González, Convergence to equilibrium for smectic-A liquid crystals in domains without constraints for the viscosity, Nonlinear Anal., 102 (2014), 208-219.  doi: 10.1016/j.na.2014.02.014.
    [6] B. Climent-Ezquerra and F. Guillén-González, Convergence to equilibrium of global weak solutions for a Cahn-Hilliard-Navier-Stokes vesicle model, Z. Angew. Math. Phys, 70 (2019), 27 pp. doi: 10.1007/s00033-019-1168-1.
    [7] J. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheology, 5 (1961), 23-34.  doi: 10.1122/1.548883.
    [8] J. Ericksen, Continuum theory of nematic liquid crystals, Res. Mechanica, 21 (1987), 381-392. 
    [9] J. Fan and F. Jiang, Large-time behavior of liquid crystal flows with a trigonometric condition in two dimensions, Commun. Pure Appl. Anal., 15 (2015), 73-90.  doi: 10.3934/cpaa.2016.15.73.
    [10] V. Girault and F. Gillén-Gonzàlez, Mixed formulation, approximation and decoupling algorithm for a penalized nematic liquid crystal model, Math. Comput., 80 (2011), 781-819.  doi: 10.1090/S0025-5718-2010-02429-9.
    [11] V. Girault and R. A. Raviart, Finite Element Approximation for Navier-Stokes Equations: Theory and Algorithms, Springer, Berlin, Heidelberg, New York, 1981. doi: 10.1007/978-3-642-61623-5.
    [12] M. Grasselli and H. Wu, Finite-dimensional global attractor for a system modeling the 2D nematic liquid crystal, Z. Angew. Math. Phys., 62 (2011), 979-992.  doi: 10.1007/s00033-011-0157-9.
    [13] M. GrasselliH. Wu and S. Zheng, Convergence to equilibrium for parabolic-hyperbolic time-dependent Ginzburg-Landau-Maxwell equation, SIAM J. Math. Anal., 40 (2008), 2007-2033.  doi: 10.1137/080717833.
    [14] F. Guillén-González and M. S. Goudiaby, Stability and convergence at infinite time of several fully discrete schemes for a Ginzburg-Landau model for nematic liquid crystal flows, Discrete Continuous Dynam. Syst., 32 (2012), 4229-4246.  doi: 10.3934/dcds.2012.32.4229.
    [15] F. Guillen-Gonzalez and J. V. Gutierrez-Santacreu, A linear mixed finite element scheme for a nematic Ericksen–Leslie liquid crystal model, Math. Model. Numer. Anal., 47 (2013), 1433-1464.  doi: 10.1051/m2an/2013076.
    [16] Q. HuangX. Yang and X. He, Numerical approximations for a smectic-A liquid crystal flow model: First-order, linear, decoupled and energy stable schemes, Discrete Contin. Dynam. Systems - B, 23 (2018), 2177-2192.  doi: 10.3934/dcdsb.2018230.
    [17] F. Leslie, Theory of flow phenomena in liquid crystals, Advances in Liquid Crystals, G. Brown, ed., Academic Press, New York, 4 (1979), 1-81.  doi: 10.1016/B978-0-12-025004-2.50008-9.
    [18] F. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283.  doi: 10.1007/BF00251810.
    [19] P. LinC. Liu and H. Zhang, An energy law preserving C0 finite element scheme for simulating the kinematic effects in liquid crystal dynamics, J. Comput. Phys., 227 (2007), 1411-1427.  doi: 10.1016/j.jcp.2007.09.005.
    [20] C. Liu and N. J. Walkington, Approximation of liquid crystal flows, SIAM J. Numer. Anal., 37 (2000), 725-741.  doi: 10.1137/S0036142997327282.
    [21] S. Lojasiewicz, Une propriété topologique des sous-ensembles analytiques réels, In "Les Equations aux Dérivées Partielles (Paris 1962)" Editions du Centre National de la Recherche Scientifique., (1963), 87–89.
    [22] S. Lojasiewicz, Ensemble semi-analytiques, I. H. E. S. Notes, (1965), 87–89.
    [23] B. Merlet and M. Pierre, Convergence to equilibrium for the backward Euler Scheme and applications, Commun. Pure Appl. Anal., 9 (2010), 685-702.  doi: 10.3934/cpaa.2010.9.685.
    [24] L. Simon, Asymptotics for a class of non-linear evolution equations with applications to geometric problems, Ann. Math., 118 (1983), 525-571.  doi: 10.2307/2006981.
    [25] H. WuX. Xu and C. Liu, Asymptotic behavior for a nematic liquid crystal model with different kinematic transport properties, Calc. Var. Partial Differ. Equ., 45 (2012), 319-345.  doi: 10.1007/s00526-011-0460-5.
  • 加载中
Open Access Under a Creative Commons license

Article Metrics

HTML views(1655) PDF downloads(294) Cited by(0)

Access History



    DownLoad:  Full-Size Img  PowerPoint