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This work proposes a unified $hp$-adaptivity framework for hybridizeddiscontinuous Galerkin (HDG) method for a large class of partial differentialequations (PDEs) of Friedrichs' type. In particular, we present unified$hp$-HDG formulations for abstract one-field and two-field structures and provetheir well-posedness. In order to handle non-conforming interfaces we simplytake advantage of HDG built-in mortar structures. With split-type mortars andthe approximation space of trace, a numerical flux can be derived via Godunovapproach and be naturally employed without any additional treatment. As aconsequence, the proposed formulations are parameter-free. We perform severalnumerical experiments for time-independent and linear PDEs including elliptic,hyperbolic, and mixed-type to verify the proposed unified $hp$-formulations anddemonstrate the effectiveness of $hp$-adaptation. Two adaptivity criteria areconsidered: one is based on a simple and fast error indicator, while the otheris rigorous but more expensive using an adjoint-based error estimate. Thenumerical results show that these two approaches are comparable in terms ofconvergence rate even for problems with strong gradients, discontinuities, orsingularities.

Unified $hp$-HDG Frameworks for Friedrichs' PDE systems