We investigate the evolution of an almost flat membrane
driven by competition of the homogeneous, Frank, and
bending energies as well
as the coupling of the local order of the constituent molecules
of the membrane to its curvature.
We propose an alternative to
the model in [J.B. Fournier and P. Galatoa, J. Phys. II
7 (1997) 1509–1520; N. Uchida, Phys. Rev. E
66 (2002) 040902] which replaces
a Ginzburg-Landau penalization for the length of the
order parameter by a rigid constraint.
We introduce a fully discrete scheme, consisting of piecewise linear
finite elements, show that it is unconditionally stable for a large
range of the elastic moduli in the model, and prove its convergence
(up to subsequences) thereby proving the existence of a weak solution
to the continuous model.
Numerical simulations are included that examine typical model situations,
confirm our theory, and explore numerical predictions beyond that theory.