Numerical approximation of nematic liquid crystal flows governed by the Ericksen-Leslie equations
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 3, pp. 523-540.

Numerical approximation of the flow of liquid crystals governed by the Ericksen-Leslie equations is considered. Care is taken to develop numerical schemes which inherit the Hamiltonian structure of these equations and associated stability properties. For a large class of material parameters compactness of the discrete solutions is established which guarantees convergence.

DOI : 10.1051/m2an/2010065
Classification : 76A15, 65M12, 65M60, 76M10
Mots-clés : liquid crystal, Ericksen-Leslie equations, numerical approximation 
@article{M2AN_2011__45_3_523_0,
     author = {Walkington, Noel J.},
     title = {Numerical approximation of nematic liquid crystal flows governed by the {Ericksen-Leslie} equations},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {523--540},
     publisher = {EDP-Sciences},
     volume = {45},
     number = {3},
     year = {2011},
     doi = {10.1051/m2an/2010065},
     mrnumber = {2804649},
     zbl = {1267.76008},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2010065/}
}
TY  - JOUR
AU  - Walkington, Noel J.
TI  - Numerical approximation of nematic liquid crystal flows governed by the Ericksen-Leslie equations
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2011
SP  - 523
EP  - 540
VL  - 45
IS  - 3
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2010065/
DO  - 10.1051/m2an/2010065
LA  - en
ID  - M2AN_2011__45_3_523_0
ER  - 
%0 Journal Article
%A Walkington, Noel J.
%T Numerical approximation of nematic liquid crystal flows governed by the Ericksen-Leslie equations
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2011
%P 523-540
%V 45
%N 3
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2010065/
%R 10.1051/m2an/2010065
%G en
%F M2AN_2011__45_3_523_0
Walkington, Noel J. Numerical approximation of nematic liquid crystal flows governed by the Ericksen-Leslie equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 3, pp. 523-540. doi : 10.1051/m2an/2010065. http://archive.numdam.org/articles/10.1051/m2an/2010065/

[1] L. Baňas, A. Prohl and R. Schätzle, Finite element approximations of harmonic map heat flows and wave map into spheres of nonconstant radii. Numer. Math. 115 (2010) 395-432. | MR | Zbl

[2] J.W. Barrett, X. Feng and A. Prohl, Convergence of a fully discrete finite element method for a degenerate parabolic system modelling nematic liquid crystals with variable degree of orientation. Math. Model. Numer. Anal. 40 (2006) 175-199. | Numdam | 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] F. Bethuel, H. Brezis and F. Helein, Ginzburg-Landau Vorticies. Kluwer (1995). | MR | Zbl

[5] R. Cohen, R. Hardt, D. Kinderlehrer, S. Lin and M. Luskin, Minimum energy configurations for liquid crystals: Computational results, in Theory and Applications of Liquid Crystals, J.L. Ericksen and D. Kinderlehrer Eds., The IMA Volumes in Mathematics and its Applications 5, Springer-Verlag, New York (1987). | MR | Zbl

[6] Q. Du, B. Guo and J. Shen, Fourier spectral approximation to a dissipative system modeling the flow of liquid crystals. SIAM J. Numer. Anal. 39 (2001) 735-762. | MR | Zbl

[7] F. Duzaar, J. Kristensen and G. Mingione, The existence of regular boundary points for non-linear elliptic systems. J. Reine Angew. Math. 602 (2007) 17-58. | MR | Zbl

[8] J. Ericksen, Conservation laws for liquid crystals. Trans. Soc. Rheol. 5 (1961) 22-34. | MR

[9] J. Ericksen, Nilpotent energies in liquid crystal theory. Arch. Rational Mech. Anal. 10 (1962) 189-196. | MR | Zbl

[10] J. Ericksen, Continuum theory of nematic liquid crystals. Res. Mechanica 21 (1987) 381-392.

[11] F.C. Frank, On the theory of liquid crystals. Discuss. Faraday Soc. 25 (1958) 19-28.

[12] G.P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations I: Linearized steady problems, Springer Tracts in Natural Philosophy 38. Springer-Verlag, New York (1994). | MR | Zbl

[13] V. Girault and F. Guillén-González, Mixed formulation, approximation and decoupling algorithm for a penalized nematic liquid crystals model. Preprint (2009). | Zbl

[14] V. Girault, R.H. Nochetto and R. Scott, Maximum-norm stability of the finite element Stokes projection. J. Math. Pures Appl. 84 (2005) 279-330. | MR | Zbl

[15] R. Hardt and D. Kinderlehrer, Mathematical questions of liquid crystal theory, in Theory and Applications of Liquid Crystals, J.L. Ericksen and D. Kinderlehrer Eds., The IMA Volumes in Mathematics and its Applications 5, Springer-Verlag, New York (1987). | MR | Zbl

[16] R. Hardt and F.H. Lin, Stability of singularities of minimizing harmonic maps. J. Differential Geom. 29 (1989) 113-123. | MR | Zbl

[17] R. Hardt, D. Kinderlehrer and F.H. Lin, Existence and partial regularity of static liquid crystal configurations. Comm. Math. Phys. 105 (1986) 547-570. | MR | Zbl

[18] Q. Hu, X.-C. Tai and R. Winther, A saddle point approach to the computation of harmonic maps. SIAM J. Numer. Anal. 47 (2009) 1500-1523. | MR | Zbl

[19] R. Jerard and M. Soner, Dynamics of Ginzburg-Landau vortices. Arch. Rational Mech. Anal. 142 (1998) 99-125. | MR | Zbl

[20] F. Leslie, Some constitutive equations for liquid crystals. Arch. Rational Mech. Anal. 28 (1968) 265-283. | MR | Zbl

[21] F. Leslie, Theory of flow phenomenum in liquid crystals, in The Theory of Liquid Crystals 4, W. Brown Ed., Academic Press, New York (1979) 1-81.

[22] F.H. Lin, Mathematics theory of liquid crystals, in Applied Mathematics At The Turn Of Century: Lecture notes of the 1993 summer school, Universidat Complutense de Madrid (1995).

[23] F.H. Lin, Solutions of Ginzburg-Landau equations and critical points of renormalized energy. Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (1995) 599-622. | Numdam | MR | Zbl

[24] F.H. Lin, Some dynamic properties of Ginzburg-Landau vorticies. Comm. Pure Appl. Math. 49 (1996) 323-359. | MR | Zbl

[25] F.H. Lin and C. Liu, Nonparabolic dissipative systems, modeling the flow of liquid crystals. Comm. Pure Appl. Math. XLVIII (1995) 501-537. | MR | Zbl

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

[27] P. Lin, C. 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. | MR | Zbl

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

[29] C. Liu and N.J. Walkington, Mixed Methods for the Approximation of Liquid Crystal Flows. ESAIM: M2AN 36 (2002) 205-222. | Numdam | MR | Zbl

[30] G. Mingione, Regularity of minima: an invitation to the dark side of the calculus of variations. Appl. Math. 51 (2006) 355-426. | MR | Zbl

[31] C.W. Oseen, The theory of liquid crystals. Trans. Faraday Soc. 29 (1933) 883-889. | Zbl

[32] I.W. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals: a Mathematical Introduction. Taylor & Francis Inc., New York (2004).

[33] E.G. Virga, Variational theories for liquid crystals, Appl. Math. Math. Comput. 8. Chapman & Hall, London (1994). | MR | Zbl

[34] N.J. Walkington, Compactness properties of the DG and CG time stepping schemes for parabolic equations. SIAM J. Numer. Anal. 47 (2010) 4680-4710. | MR | Zbl

Cité par Sources :