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doi: 10.3934/cpaa.2021116
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Longtime behavior of a second order finite element scheme simulating the kinematic effects in liquid crystal dynamics

1. 

Département de mathématiques, UFR des Sciences et Technologies, Université Assane Seck, BP 523 Ziguinchor, Sénégal

2. 

Svenska Handelsbanken, Capital Market, Blasieholmstorg 11,106 70 STOCKHOLM, Sweden

3. 

Institut Camille Jordan, Université Claude Bernard Lyon 1, Avenue Claude Bernard 69622 VILLEURBANNE cedex, France

* Corresponding author

Received  March 2021 Revised  June 2021 Early access July 2021

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.

Citation: Mouhamadou Samsidy Goudiaby, Ababacar Diagne, Leon Matar Tine. Longtime behavior of a second order finite element scheme simulating the kinematic effects in liquid crystal dynamics. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021116
References:
[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.  Google Scholar

[2]

J. M. Ball, Mathematics and liquid crystals, Molecular Crystals and Liquid Crystals, 647 (2017), 1-27.  doi: 10.1080/15421406.2017.1289425.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheology, 5 (1961), 23-34.  doi: 10.1122/1.548883.  Google Scholar

[8]

J. Ericksen, Continuum theory of nematic liquid crystals, Res. Mechanica, 21 (1987), 381-392.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

F. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283.  doi: 10.1007/BF00251810.  Google Scholar

[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.  Google Scholar

[20]

C. Liu and N. J. Walkington, Approximation of liquid crystal flows, SIAM J. Numer. Anal., 37 (2000), 725-741.  doi: 10.1137/S0036142997327282.  Google Scholar

[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.  Google Scholar

[22]

S. Lojasiewicz, Ensemble semi-analytiques, I. H. E. S. Notes, (1965), 87–89. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[2]

J. M. Ball, Mathematics and liquid crystals, Molecular Crystals and Liquid Crystals, 647 (2017), 1-27.  doi: 10.1080/15421406.2017.1289425.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheology, 5 (1961), 23-34.  doi: 10.1122/1.548883.  Google Scholar

[8]

J. Ericksen, Continuum theory of nematic liquid crystals, Res. Mechanica, 21 (1987), 381-392.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

F. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283.  doi: 10.1007/BF00251810.  Google Scholar

[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.  Google Scholar

[20]

C. Liu and N. J. Walkington, Approximation of liquid crystal flows, SIAM J. Numer. Anal., 37 (2000), 725-741.  doi: 10.1137/S0036142997327282.  Google Scholar

[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.  Google Scholar

[22]

S. Lojasiewicz, Ensemble semi-analytiques, I. H. E. S. Notes, (1965), 87–89. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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