The numerical solution of the flow of a liquid crystal governed by a particular instance of the Ericksen-Leslie equations is considered. Convergence results for this system rely crucially upon energy estimates which involve norms of the director field. We show how a mixed method may be used to eliminate the need for Hermite finite elements and establish convergence of the method.
Mots clés : liquid crystals, mixed finite element approximation, convergence
@article{M2AN_2002__36_2_205_0, author = {Liu, Chun and Walkington, Noel J.}, title = {Mixed methods for the approximation of liquid crystal flows}, journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique}, pages = {205--222}, publisher = {EDP-Sciences}, volume = {36}, number = {2}, year = {2002}, doi = {10.1051/m2an:2002010}, mrnumber = {1906815}, zbl = {1032.76035}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an:2002010/} }
TY - JOUR AU - Liu, Chun AU - Walkington, Noel J. TI - Mixed methods for the approximation of liquid crystal flows JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2002 SP - 205 EP - 222 VL - 36 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2002010/ DO - 10.1051/m2an:2002010 LA - en ID - M2AN_2002__36_2_205_0 ER -
%0 Journal Article %A Liu, Chun %A Walkington, Noel J. %T Mixed methods for the approximation of liquid crystal flows %J ESAIM: Modélisation mathématique et analyse numérique %D 2002 %P 205-222 %V 36 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2002010/ %R 10.1051/m2an:2002010 %G en %F M2AN_2002__36_2_205_0
Liu, Chun; Walkington, Noel J. Mixed methods for the approximation of liquid crystal flows. ESAIM: Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 2, pp. 205-222. doi : 10.1051/m2an:2002010. http://www.numdam.org/articles/10.1051/m2an:2002010/
[1] A new algorithm for computing liquid crystal stable configurations: The harmonic mapping case. SIAM J. Numer. Anal. 34 (1997) 1708-1726. | Zbl
,[2] Minimizing Oseen-Frank energy for nematic liquid crystals: algorithms and numerical results. Ann. Inst. H. Poincaré Phys. Théor. 66 (1997) 411-447. | Numdam | Zbl
and ,[3] Survey lecutures on the mathematical foundations of the finite element method, in The mathematical foundations of the finite element method with applications to partial differential equations, A.K. Aziz Ed., New York (1972), Academic Press, 5-359. | Zbl
and ,[4] Regularity of minimizers of relaxed problems for harmonic maps. J. Funct. Anal. 101 (1991) 145-161. | Zbl
and ,[5] Ginzburg-Landau Vorticies. Klumer (1995).
, and ,[6] New developments on the ginzburg-landau model. Topol. Methods Nonlinear Anal. 4 (1994) 227-236. | Zbl
,[7] Harmonic maps with defects. Comm. Math. Phys. 107 (1986) 649-705. | Zbl
, and ,[8] Mixed and hybrid finite element methods, no. 15 in Computational Mathematics. Springer-Verlag (1991). | Zbl
and ,[9] Liquid Crystals. Cambridge (1992).
,[10] Regularity for heat flow for harmonic maps. Math. Z. 201 (1989) 83-103. | Zbl
and ,[11] The finite element method for elliptic problems. North-Holland (1978). | Zbl
,[12] Minimum energy configurations for liquid crystals: Computational results, in Theory and Applications of Liquid Crystals, J.L. Ericksen and D. Kinderlehrer, Eds., Vol. 5 of The IMA Volumes in Mathematics and its Applicatoins. Springer-Verlag, New York (1987). | MR | Zbl
, , , and ,[13] Relaxation and gradient methods for molecular orientation in liquid crystals. Comp. Phys. 53 (1989) 455-465.
, and ,[14] The stability in and of the projection onto finite element function spaces. Math. Comp. 48 (1987) 521-532. | Zbl
and ,[15] Finite element analsyis of the Landau-De Gennes minimization problem for liquid crystals. SIAM J. Numer. Anal. 35 (1998) 336-362. | Zbl
and ,[16] The Physics Of Liquid Crystals. Oxford (1974).
,[17] Vortices in superconductors: modelling and computer simulations. Philos. Trans. Roy. Soc. London 355 (1997) 1957-1968. | Zbl
, , and ,[18] Ginzburg-Landau vortices: dynamics, pinning, and hysteresis. SIAM J. Math. Anal. 28 (1997) 1265-1293. | Zbl
and ,[19] Analysis and convergence of a covolume approximation of the Ginzburg-Landau model of superconductivity. SIAM J. Numer. Anal. 35 (1997) 1049-1072. | Zbl
, and ,[20] Conservation laws for liquid crystals. Trans. Soc. Rheol. 5 (1961) 22-34. | MR
,[21] On the theory of liquid crystals. Discuss. Faraday Soc. 28 (1958) 19-28.
,[22] Finite element approximation of the Navier-Stokes equations, no. 749 in Lecture Notes in Mathematics. Springer Verlag, Berlin, Heidelbert, New York (1979). | MR | Zbl
and ,[23] An introduction to continuum mechanics, no. 158 in Mathematics in Science and Engineering. Academic Press (1981). | MR | Zbl
,[24] Mathematical questions of liquid crystal theory, in Theory and Applications of Liquid Crystals, J. L. Ericksen and D. Kinderlehrer Eds., Vol. 5 of The IMA Volumes in Mathematics and its Applicatoins. Springer-Verlag, New York (1987). | MR | Zbl
and ,[25] Existence and partial regularity of static liquid crystal configurations. Comm. Math. Phys. 105 (1986) 547-570. | Zbl
, and ,[26] Stability of singularities of minimizing harmonic maps. J. Differential Geom. 29 (1989) 113-123. | Zbl
and ,[27] Dynamics of Ginzburg-Landau vortices. Arch. Rational Mech. Anal. 142 (1998) 99-125. | Zbl
and ,[28] Harmonic mapping between Riemannian surfaces. Vol. 14 of Proc. of the C.M.A., Australian National University (1983).
,[29] Some constitutive equations for liquid crystals. Archive for Rational Mechanics and Analysis 28 (1968) 265-283. | Zbl
,[30] Some topics in equilibrium theory of liquid crystals, in Theory and Applications of Liquid Crystals, J.L. Ericksen and D. Kinderlehrer Eds., Vol. 5 of The IMA Volumes in Mathematics and its Applications. Springer-Verlag, New York (1987) 211-234.
,[31] 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).
,[32] Some dynamic properties of Ginzburg-Landau vorticies. Comm. Pure Appl. Math. 49 (1996) 323-359. | Zbl
,[33] Nonparabolic dissipative systems, modeling the flow of liquid crystals. Comm. Pure Appl. Math. XLVIII (1995) 501-537. | Zbl
and ,[34] Global existence of solutions for the Ericksen Leslie-system. Arch. Rational Mech. Anal. (1998). | Zbl
and ,[35] Relaxation methods for liquid crystal problems. SIAM J. Numer. Anal. 26 (1989) 1310-1324. | Zbl
and ,[36] Dynamic theory for incompressible smectic-A liquid crystals: Existence and regularity. Discrete Contin. Dynam. Systems 6 (2000) 591-608. | Zbl
,[37] Approximation of liquid crystal flows. SIAM J. Numer. Anal. 37 (2000) 725-741. | Zbl
and ,[38] The theory of liquid crystals. Trans. Faraday Soc. 29 (1933) 883-889. | Zbl
,[39] Some optimal error estimates for piecewise linear finite element approximations. Math. Comp. 38 (1982) 437-445. | Zbl
and ,[40] On the quasi-optimality in of the projection into finite element spaces. Math. Comp. 38 (1982) 1-22. | Zbl
and ,[41] A regularity theory for harmonic maps. J. Differential Geom. 17 (1982) 307-335. | Zbl
and ,[42] Weak solutions and development of singularities in su(2) -model. CPAM 41 (1988) 459-469. | Zbl
,[43] On some three dimensional finite elements for incompressible materials. Comput. Methods Appl. Mech. Engrg. 63 (1987) 261-269. | Zbl
,[44] Error analysis of some finite element methods for the Stokes problem. Math. Comp. 54 (1990) 495-508. | Zbl
,[45] Navier-Stokes Equations. North Holland (1977). | MR | Zbl
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