A linear mixed finite element scheme for a nematic Ericksen-Leslie liquid crystal model
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 5, pp. 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 : 10.1051/m2an/2013076
Classification : 35Q35, 65M12, 65M60
Mots clés : 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 },
     pages = {1433--1464},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {5},
     year = {2013},
     doi = {10.1051/m2an/2013076},
     mrnumber = {3100770},
     zbl = {1290.82031},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2013076/}
}
TY  - JOUR
AU  - Guillén-González, F. M.
AU  - Gutiérrez-Santacreu, J. V.
TI  - A linear mixed finite element scheme for a nematic Ericksen-Leslie liquid crystal model
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2013
SP  - 1433
EP  - 1464
VL  - 47
IS  - 5
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2013076/
DO  - 10.1051/m2an/2013076
LA  - en
ID  - M2AN_2013__47_5_1433_0
ER  - 
%0 Journal Article
%A Guillén-González, F. M.
%A Gutiérrez-Santacreu, J. V.
%T A linear mixed finite element scheme for a nematic Ericksen-Leslie liquid crystal model
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2013
%P 1433-1464
%V 47
%N 5
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2013076/
%R 10.1051/m2an/2013076
%G en
%F 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 , Tome 47 (2013) no. 5, pp. 1433-1464. doi : 10.1051/m2an/2013076. http://archive.numdam.org/articles/10.1051/m2an/2013076/

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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

[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 | Zbl

[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

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

[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 | Zbl

[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 | Zbl

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

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

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

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

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

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

Cité par Sources :