  We consider the numerical investigation of surface bound orientational order
using unit tangential vector fields by means of a gradient-flow equation of a
weak surface Frank-Oseen energy. The energy is composed of intrinsic and
extrinsic contributions, as well as a penalization term to enforce the unity of
the vector field. Four different numerical discretizations, namely a discrete
exterior calculus approach, a method based on vector spherical harmonics, a
surface finite-element method, and an approach utilizing an implicit surface
description, the diffuse interface method, are described and compared with each
other for surfaces with Euler characteristic 2. We demonstrate the influence of
geometric properties on realizations of the Poincare-Hopf theorem and show
examples where the energy is decreased by introducing additional orientational
defects.
