 
Summary: A NEW ALGORITHM FOR COMPUTING LIQUID CRYSTAL
STABLE CONFIGURATIONS: THE HARMONIC MAPPING CASE
FRANC¸OIS ALOUGES
SIAM J. NUMER. ANAL. c 1997 Society for Industrial and Applied Mathematics
Vol. 34, No. 5, pp. 17081726, October 1997 003
Abstract. In this article, we propose a new algorithm for minimizing the energy of a nematic
liquid crystal. Based on the equal elastic constants OseenFrank model, the problem reduces to
finding harmonic minimizing maps that take values into the unit sphere of R3. The convergence of
this algorithm is proved in a continuous setting. Then, numerous numerical results that show its
efficiency are given.
Key words. nematic liquid crystals, harmonic maps, nonconvex optimization, finite differences,
relaxation method, Uzawa method, conjugate gradient method
AMS subject classifications. 35A40, 65C20, 65N30
PII. S0036142994264249
1. Introduction. In recent years, liquid crystals have been of constant interest
for mathematicians and physicists. Several models were proposed (harmonic maps,
OseenFrank model, Ericksen model) to explain the defects in the structure of ne
matics. Working with the OseenFrank model, nematic liquid crystals are naturally
represented by an S2
valued map, with which one associates an energy that depends
