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:本文提出了一个统一的hp自适应框架,用于Friedrichs一类偏微分方程系统的混合化间断Galerkin(HDG)方法。特别地,我们针对抽象的一场和两场结构提出了统一的hp-HDG公式,并证明了其良好性质。为了处理非相容界面,我们简单地利用HDG内置的砂浆结构。通过分裂型砂浆和迹的逼近空间,可以通过Godunov方法导出数值通量,并且可以自然地在没有任何额外处理的情况下使用。因此,所提出的公式是无参数的。我们对时间独立和线性PDE进行了多项数值实验,包括椭圆型,双曲型和混合型,以验证所提出的统一的hp公式,并展示了hp适应技术的有效性。考虑了两种适应性标准:一种基于简单且快速的误差指示器,而另一种则使用基于伴随误差估计的严格但更昂贵的方法。数值实验表明,这两种方法在梯度强烈,不连续或奇异的问题中的收敛速度相当。