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.

翻译：本文针对一大类Friedrichs型偏微分方程，提出了一种用于混合间断伽辽金方法的统一$hp$自适应框架。具体而言，我们为抽象的单场和双场结构提出了统一的$hp$-HDG公式，并证明了其适定性。为处理非协调界面，我们直接利用HDG内置的mortar结构。通过分裂型mortar和迹的逼近空间，可基于Godunov方法导出数值通量，且无需任何额外处理即可直接使用。因此，所提出的公式无需参数。我们对包括椭圆型、双曲型和混合型在内的稳态线性偏微分方程进行了多项数值实验，以验证所提出的统一$hp$-公式并展示$hp$自适应的有效性。我们考虑了两种自适应准则：一种基于简单快速的误差指示器，另一种则基于伴随误差估计，更为严格但计算开销更大。数值结果表明，即使对于具有强梯度、间断性或奇异性的问题，这两种方法在收敛速度方面也具有可比性。