  The properties of liquid crystals can be modelled using an order parameter
which describes the variability of the local orientation of rod-like molecules.
Defects in the director field can arise due to external factors such as applied
electric or magnetic fields, or the constraining geometry of the cell
containing the liquid crystal material. Understanding the formation and
dynamics of defects is important in the design and control of liquid crystal
devices, and poses significant challenges for numerical modelling. In this
paper we consider the numerical solution of a $\bf{Q}$-tensor model of a
nematic liquid crystal, where defects arise through rapid changes in the
$\bf{Q}$-tensor over a very small physical region in relation to the dimensions
of the liquid crystal device. The efficient solution of the resulting six
coupled partial differential equations is achieved using a finite element based
adaptive moving mesh approach, where an unstructured triangular mesh is adapted
towards high activity regions, including those around defects. Spatial
convergence studies are presented using a stationary defect as a model test
case, and the adaptive method is shown to be optimally convergent using
quadratic triangular finite elements. The full effectiveness of the method is
then demonstrated using a challenging two-dimensional dynamic Pi-cell problem
involving the creation, movement, and annihilation of defects.
