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Mixed Methods for the Approximation of Liquid Crystal Flows

Published online by Cambridge University Press:  15 May 2002

Chun Liu  and
Noel J. Walkington
Chun Liu
Affiliation:
Department of Mathematics, Pennsylvania State, University, University Park, PA 18601, USA.
Noel J. Walkington
Affiliation:
Department of Mathematics, Carnegie Mellon University, Pittsburgh, PA 15213, USA.
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Abstract

The numerical solution of the flow of a liquid crystal governedby a particular instance of the Ericksen–Leslie equations is considered.Convergence results for this system rely crucially upon energyestimates which involve H2 (Ω) norms of the director field. Weshow how a mixed method may be used to eliminate the need forHermite finite elements and establish convergence of the method.

Keywords

Liquid CrystalsMixed Finite Element ApproximationConvergence.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 36 , Issue 2 , March 2002 , pp. 205 - 222
Copyright
© EDP Sciences, SMAI, 2002

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References

Alouges, F., A new algorithm for computing liquid crystal stable configurations: The harmonic mapping case. SIAM J. Numer. Anal. 34 (1997) 1708-1726. CrossRef
Alouges, F. and Ghidaglia, J.M., Minimizing Oseen-Frank energy for nematic liquid crystals: algorithms and numerical results. Ann. Inst. H. Poincaré Phys. Théor. 66 (1997) 411-447.
I. Babuska and A.K. Aziz, 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.
Bethuel, F. and Brezis, H., Regularity of minimizers of relaxed problems for harmonic maps. J. Funct. Anal. 101 (1991) 145-161. CrossRef
F. Bethuel, H. Brezis and F. Helein, Ginzburg-Landau Vorticies. Klumer (1995).
Brezis, H., New developments on the ginzburg-landau model. Topol. Methods Nonlinear Anal. 4 (1994) 227-236. CrossRef
Brezis, H., Coron, J. and Lieb, E., Harmonic maps with defects. Comm. Math. Phys. 107 (1986) 649-705. CrossRef
F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, no. 15 in Computational Mathematics. Springer-Verlag (1991).
S. Chandrasekhar, Liquid Crystals. Cambridge (1992).
Chen, Y.M. and Struwe, M., Regularity for heat flow for harmonic maps. Math. Z. 201 (1989) 83-103. CrossRef
P.G. Ciarlet, The finite element method for elliptic problems. North-Holland (1978).
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., Vol. 5 of The IMA Volumes in Mathematics and its Applicatoins. Springer-Verlag, New York (1987).
Cohen, R., Lin, S. and Luskin, M., Relaxation and gradient methods for molecular orientation in liquid crystals. Comp. Phys. 53 (1989) 455-465.
Crouzeix, M. and Thomee, V., The stability in L p and W 1,p of the L 2 projection onto finite element function spaces. Math. Comp. 48 (1987) 521-532.
Davis, T. and Gartland, E., Finite element analsyis of the Landau-De Gennes minimization problem for liquid crystals. SIAM J. Numer. Anal. 35 (1998) 336-362. CrossRef
P.G. de Gennes, The Physics Of Liquid Crystals. Oxford (1974).
Deang, J., Du, Q., Gunzburger, M. and Peterson, J., Vortices in superconductors: modelling and computer simulations. Philos. Trans. Roy. Soc. London 355 (1997) 1957-1968. CrossRef
Du, Q. and Lin, F., Ginzburg-Landau vortices: dynamics, pinning, and hysteresis. SIAM J. Math. Anal. 28 (1997) 1265-1293.
Du, Q., Nicolaides, R.A. and Analysis, X. Wu and convergence of a covolume approximation of the Ginzburg-Landau model of superconductivity. SIAM J. Numer. Anal. 35 (1997) 1049-1072. CrossRef
Ericksen, J., Conservation laws for liquid crystals. Trans. Soc. Rheol. 5 (1961) 22-34.
Frank, F.C., On the theory of liquid crystals. Discuss. Faraday Soc. 28 (1958) 19-28. CrossRef
V. Girault and P.A. Raviart, Finite element approximation of the Navier-Stokes equations, no. 749 in Lecture Notes in Mathematics. Springer Verlag, Berlin, Heidelbert, New York (1979).
M. E. Gurtin, An introduction to continuum mechanics, no. 158 in Mathematics in Science and Engineering. Academic Press (1981).
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., Vol. 5 of The IMA Volumes in Mathematics and its Applicatoins. Springer-Verlag, New York (1987).
Hardt, R., Kinderlehrer, D. and Lin, F.H., Existence and partial regularity of static liquid crystal configurations. Comm. Math. Phys. 105 (1986) 547-570. CrossRef
Hardt, R. and Lin, F.H., Stability of singularities of minimizing harmonic maps. J. Differential Geom. 29 (1989) 113-123. CrossRef
Jerard, R. and Soner, M., Dynamics of Ginzburg-Landau vortices. Arch. Rational Mech. Anal. 142 (1998) 99-125.
J. Jost, Harmonic mapping between Riemannian surfaces. Vol. 14 of Proc. of the C.M.A., Australian National University (1983).
Leslie, F., Some constitutive equations for liquid crystals. Archive for Rational Mechanics and Analysis 28 (1968) 265-283.
F. Leslie, 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.
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).
Lin, F.H., Some dynamic properties of Ginzburg-Landau vorticies. Comm. Pure Appl. Math. 49 (1996) 323-359.
F.H. Lin and C. Liu, Nonparabolic dissipative systems, modeling the flow of liquid crystals. Comm. Pure Appl. Math. XLVIII (1995) 501-537.
F.H. Lin and C. Liu, Global existence of solutions for the Ericksen Leslie-system. Arch. Rational Mech. Anal. (1998).
Lin, S. and Luskin, M., Relaxation methods for liquid crystal problems. SIAM J. Numer. Anal. 26 (1989) 1310-1324. CrossRef
Liu, C., Dynamic theory for incompressible smectic-A liquid crystals: Existence and regularity. Discrete Contin. Dynam. Systems 6 (2000) 591-608. CrossRef
Liu, C. and Walkington, N.J., Approximation of liquid crystal flows. SIAM J. Numer. Anal. 37 (2000) 725-741. CrossRef
Oseen, C.W., The theory of liquid crystals. Trans. Faraday Soc. 29 (1933) 883-889. CrossRef
Rannacher, R. and Scott, R., Some optimal error estimates for piecewise linear finite element approximations. Math. Comp. 38 (1982) 437-445. CrossRef
Schatz, A.H. and Wahlbin, L.B., On the quasi-optimality in L ∞ of the H 1 0 projection into finite element spaces. Math. Comp. 38 (1982) 1-22.
Schoen, R. and Uhlenbeck, K., A regularity theory for harmonic maps. J. Differential Geom. 17 (1982) 307-335. CrossRef
Shatah, J., Weak solutions and development of singularities in su(2) σ-model. CPAM 41 (1988) 459-469.
Stenberg, R., On some three dimensional finite elements for incompressible materials. Comput. Methods Appl. Mech. Engrg. 63 (1987) 261-269. CrossRef
Stenberg, R., Error analysis of some finite element methods for the Stokes problem. Math. Comp. 54 (1990) 495-508. CrossRef
R. Temam, Navier-Stokes Equations. North Holland (1977).