  This paper presents a 3D mesh adaptivity strategy on unstructured tetrahedral
meshes by a posteriori error estimates based on metrics, studied on the case of
a nonlinear finite element minimization scheme for the Landau-de Gennes free
energy functional of nematic liquid crystals. Newton's iteration for tensor
fields is employed with steepest descent method possibly stepping in.
  Aspects relating the driving of mesh adaptivity within the nonlinear scheme
are considered. The algorithmic performance is found to depend on at least two
factors: when to trigger each single mesh adaptation, and the precision of the
correlated remeshing. Each factor is represented by a parameter, with its
values possibly varying for every new mesh adaptation. We empirically show that
the time of the overall algorithm convergence can vary considerably when
different sequences of parameters are used, thus posing a question about
optimality.
  The extensive testings and debugging done within this work on the simulation
of systems of nematic colloids substantially contributed to the upgrade of an
open source finite element-oriented programming language to its 3D meshing
possibilities, as also to an outer 3D remeshing module.
