  This work proposes a unified $hp$-adaptivity framework for hybridized
discontinuous Galerkin (HDG) method for a large class of partial differential
equations (PDEs) of Friedrichs' type. In particular, we present unified
$hp$-HDG formulations for abstract one-field and two-field structures and prove
their well-posedness. In order to handle non-conforming interfaces we simply
take advantage of HDG built-in mortar structures. With split-type mortars and
the approximation space of trace, a numerical flux can be derived via Godunov
approach and be naturally employed without any additional treatment. As a
consequence, the proposed formulations are parameter-free. We perform several
numerical experiments for time-independent and linear PDEs including elliptic,
hyperbolic, and mixed-type to verify the proposed unified $hp$-formulations and
demonstrate the effectiveness of $hp$-adaptation. Two adaptivity criteria are
considered: one is based on a simple and fast error indicator, while the other
is rigorous but more expensive using an adjoint-based error estimate. The
numerical results show that these two approaches are comparable in terms of
convergence rate even for problems with strong gradients, discontinuities, or
singularities.
