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.
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://www.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://www.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://www.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://www.numdam.org/articles/10.1051/m2an/2013076/
[1] , A new algorithm for computing liquid crystal stable configurations: the harmonic mapping case. SIAM J. Numer. Anal. 34 (1997) 1708-1726. | MR | Zbl
[2] and , 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] , and , Finite element approximations of the Ericksen-Leslie model for nematic liquid crystal flow. SIAM J. Numer. Anal. 46 (2008) 1704-1731. | MR | Zbl
[4] , and , Reproductivity for a nematic liquid crystal model. Z. Angew. Math. Phys. (2006) 984-998. | MR | Zbl
[5] , The weak solutions to the evolution problems of harmonic maps. Math. Z. 201 (1989) 69-74. | MR | Zbl
[6] , The finite element method for elliptic problems. Amsterdam, North-Holland (1987). | MR | Zbl
[7] , and , The stability in Lq of the L2-projection into finite element function spaces. Numer. Math. 23 (1974/75) 193-197. | MR | Zbl
[8] and , Mixed formulation, approximation and decoupling algorithm for a nematic liquid crystals model. Math. Comput. 80 (2011) 781-819. | MR | Zbl
[9] , and , Estimates of the finite element Stokes projection in W1,∞. C. R. Math. Acad. Sci. Paris 338 (2004) 957-962. | MR | Zbl
[10] and , Two-grid finite-element schemes for the transient Navier-Stokes problem. ESAIM: M2AN 35 (2001) 945-980. | Numdam | MR | Zbl
[11] and . Finite element methods for Navier-Stokes equations: theory and algorithms. Springer-Verlag, Berlin (1986). | MR | Zbl
[12] and , 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] , Nonlinear theory of defects in nematic liquid crystals: phase transition and flow phenomena. Comm. Pure Appl. Math. 42 (1989) 789-814. | MR | Zbl
[14] and , Non-parabolic dissipative systems modelling the flow of liquid crystals. Comm. Pure Appl. Math. 48 (1995) 501-537. | Zbl
[15] and , Existence of solutions for the Ericksen-Leslie system. Arch. Rational. Mech. Anal. 154 (2000) 135-156. | MR | Zbl
[16] and , Simulations of singularity dynamics in liquid crystal flows: A C0 finite element approach. J. Comput. Phys. 215 (2006) 1411-1427. | MR | Zbl
[17] , and , 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] and , Approximation of liquid crystal flows. SIAM J. Numer. Anal. 37 (2000) 725-741. | MR | Zbl
[19] and , Mixed methods for the approximation of liquid crystal flows. ESAIM: M2AN 36 (2002) 205-222. | Numdam | MR | Zbl
[20] and , Vorticity and incompressible flows. Cambridge Texts in Applied Mathematics (2002). | MR | Zbl
[21] and , Finite element interpolation of non-smooth functions satisfying boundary conditions. Math. Comput. 54 (1990) 483-493. | MR | Zbl
[22] and , Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete Cont. Dyn. Sys. 28 (2010) 1669-1691. | MR | Zbl
[23] , Compact sets in the Space Lp(0,T;B). Ann. Mat. Pura Appl. 146 (1987) 65-97. | MR | Zbl
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