The Landau-DeGennes Q-model of uniaxial nematic liquid crystals seeks a rank-one
traceless tensor Q that minimizes a Frank-type energy plus a double well potential
that confines the eigenvalues of Q to lie between -1/2 and 1. We propose a finite
element method (FEM) which preserves this basic structure and satisfies a discrete
form of the fundamental energy estimates. We prove that the discrete problem Gamma
converges to the continuous one as the meshsize tends to zero, and propose a discrete
gradient flow to compute discrete minimizers. Numerical experiments confirm the ability
of the scheme to approximate configurations with half-integer defects, and to deal with
colloidal and electric field effects. This work, joint with J.P. Borthagaray and S.
Walker, builds on our previous work for the Ericksen's model which we review briefly.
- Computational Mathematics and Applications Seminar