  Discontinuous Galerkin (DG) discretizations with exact representation of the
geometry and local polynomial degree adaptivity are revisited. Hybridization
techniques are employed to reduce the computational cost of DG approximations
and devise the hybridizable discontinuous Galerkin (HDG) method. Exact geometry
described by non-uniform rational B-splines (NURBS) is integrated into HDG
using the framework of the NURBS-enhanced finite element method (NEFEM).
Moreover, optimal convergence and superconvergence properties of HDG-Voigt
formulation in presence of symmetric second-order tensors are exploited to
construct inexpensive error indicators and drive degree adaptive procedures.
Applications involving the numerical simulation of problems in electrostatics,
linear elasticity and incompressible viscous flows are presented. Moreover,
this is done for both high-order HDG approximations and the lowest-order
framework of face-centered finite volumes (FCFV).
