A linear mixed finite element scheme for a nematic Ericksen-Leslie liquid crystal model
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 5, p. 1433-1464

In this work we study a fully discrete mixed scheme, based on continuous finite elements in space and a linear semi-implicit first-order integration in time, approximating an Ericksen-Leslie nematic liquid crystal model by means of a Ginzburg-Landau penalized problem. Conditional stability of this scheme is proved via a discrete version of the energy law satisfied by the continuous problem, and conditional convergence towards generalized Young measure-valued solutions to the Ericksen-Leslie problem is showed when the discrete parameters (in time and space) and the penalty parameter go to zero at the same time. Finally, we will show some numerical experiences for a phenomenon of annihilation of singularities.

DOI : https://doi.org/10.1051/m2an/2013076
Classification:  35Q35,  65M12,  65M60
Keywords: liquid crystal, Navier-Stokes, stability, convergence, finite elements, penalization
@article{M2AN_2013__47_5_1433_0,
     author = {Guill\'en-Gonz\'alez, F. M. and Guti\'errez-Santacreu, J. V.},
     title = {A linear mixed finite element scheme for a nematic Ericksen-Leslie liquid crystal model},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {5},
     year = {2013},
     pages = {1433-1464},
     doi = {10.1051/m2an/2013076},
     zbl = {1290.82031},
     mrnumber = {3100770},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2013__47_5_1433_0}
}
Guillén-González, F. M.; Gutiérrez-Santacreu, J. V. A linear mixed finite element scheme for a nematic Ericksen-Leslie liquid crystal model. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 5, pp. 1433-1464. doi : 10.1051/m2an/2013076. http://www.numdam.org/item/M2AN_2013__47_5_1433_0/

[1] F. Alouges, A new algorithm for computing liquid crystal stable configurations: the harmonic mapping case. SIAM J. Numer. Anal. 34 (1997) 1708-1726. | MR 1472192 | Zbl 0886.35010

[2] P. Azérad and F. Guillen-González, Mathematical justification of the hydrostatic approximation in the primitive equations of geophysical fluid dynamics. SIAM J. Math. Anal. 33 (2001) 847-859. | MR 1884725 | Zbl 0999.35072

[3] R. Becker, X. 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. | MR 2399392 | Zbl 1187.82130

[4] B. Climent-Ezquerra, F. Guillén-González and M. Rojas-Medar, Reproductivity for a nematic liquid crystal model. Z. Angew. Math. Phys. (2006) 984-998. | MR 2279252 | Zbl 1106.35058

[5] Y.M. Chen, The weak solutions to the evolution problems of harmonic maps. Math. Z. 201 (1989) 69-74. | MR 990189 | Zbl 0685.58015

[6] P.G. Ciarlet, The finite element method for elliptic problems. Amsterdam, North-Holland (1987). | MR 520174 | Zbl 0383.65058

[7] J.J. Douglas, T. Dupont and L. Wahlbin, The stability in Lq of the L2-projection into finite element function spaces. Numer. Math. 23 (1974/75) 193-197. | MR 383789 | Zbl 0297.41022

[8] V. Girault and F. Guillén-González, Mixed formulation, approximation and decoupling algorithm for a nematic liquid crystals model. Math. Comput. 80 (2011) 781-819. | MR 2772096 | Zbl 1228.35176

[9] V. Girault, N. Nochetto and R. Scott, Estimates of the finite element Stokes projection in W1,∞. C. R. Math. Acad. Sci. Paris 338 (2004) 957-962. | MR 2066358 | Zbl 1049.65118

[10] V. Girault and J.L. Lions, Two-grid finite-element schemes for the transient Navier-Stokes problem. ESAIM: M2AN 35 (2001) 945-980. | Numdam | MR 1866277 | Zbl 1032.76032

[11] V. Girault and P.A. Raviart. Finite element methods for Navier-Stokes equations: theory and algorithms. Springer-Verlag, Berlin (1986). | MR 851383 | Zbl 0585.65077

[12] F. Guillén-González and J.V. Gutiérrez-Santacreu, Unconditional stability and convergence of a fully discrete scheme for 2D viscous fluids models with mass diffusion. Math. Comput. 77 (2008) 1495-1524. | Zbl 1193.35153

[13] F.H. Lin, Nonlinear theory of defects in nematic liquid crystals: phase transition and flow phenomena. Comm. Pure Appl. Math. 42 (1989) 789-814. | MR 1003435 | Zbl 0703.35173

[14] F.H. Lin and C. Liu, Non-parabolic dissipative systems modelling the flow of liquid crystals. Comm. Pure Appl. Math. 48 (1995) 501-537. | Zbl 0842.35084

[15] F. H. Lin and C. Liu, Existence of solutions for the Ericksen-Leslie system. Arch. Rational. Mech. Anal. 154 (2000) 135-156. | MR 1784963 | Zbl 0963.35158

[16] P. Lin and C. Liu, Simulations of singularity dynamics in liquid crystal flows: A C0 finite element approach. J. Comput. Phys. 215 (2006) 1411-1427. | MR 2215659 | Zbl 1101.82039

[17] P. Lin, C. Liu and H. Zhang, An energy law preserving C0 finite element scheme for simulating the kinematic effects in liquid crystal flow dynamics. J. Comput. Phys. 227 (2007) 348-362. | MR 2442400 | Zbl 1133.65077

[18] C. Liu and N.J. Walkington, Approximation of liquid crystal flows. SIAM J. Numer. Anal. 37 (2000) 725-741. | MR 1740379 | Zbl 1040.76036

[19] C. Liu and N.J. Walkington, Mixed methods for the approximation of liquid crystal flows. ESAIM: M2AN 36 (2002) 205-222. | Numdam | MR 1906815 | Zbl 1032.76035

[20] A.J. Majda and A.L. Bertozzi, Vorticity and incompressible flows. Cambridge Texts in Applied Mathematics (2002). | MR 1867882 | Zbl 0983.76001

[21] L. R. Scott and S. Zhang, Finite element interpolation of non-smooth functions satisfying boundary conditions. Math. Comput. 54 (1990) 483-493. | MR 1011446 | Zbl 0696.65007

[22] J. Shen and X. Yang, Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete Cont. Dyn. Sys. 28 (2010) 1669-1691. | MR 2679727 | Zbl 1201.65184

[23] J. Simon, Compact sets in the Space Lp(0,T;B). Ann. Mat. Pura Appl. 146 (1987) 65-97. | MR 916688 | Zbl 0629.46031