In this study, we present an $hp$-multigrid preconditioner for a divergence-conforming HDG scheme for the generalized Stokes and the Navier-Stokes equations using an augmented Lagrangian formulation. Our method relies on conforming simplicial meshes in two- and three-dimensions. The $hp$-multigrid algorithm is a multiplicative auxiliary space preconditioner that employs the lowest-order space as the auxiliary space, and we developed a geometric multigrid method as the auxiliary space solver. For the generalized Stokes problem, the crucial ingredient of the geometric multigrid method is the equivalence between the condensed lowest-order divergence-conforming HDG scheme and a Crouzeix-Raviart discretization with a pressure-robust treatment as introduced in Linke and Merdon (Comput. Methods Appl. Mech. Engrg., 311 (2016)), which allows for the direct application of geometric multigrid theory on the Crouzeix-Raviart discretization. The numerical experiments demonstrate the robustness of the proposed $hp$-multigrid preconditioner with respect to mesh size and augmented Lagrangian parameter, with iteration counts insensitivity to polynomial order increase. Inspired by the works by Benzi & Olshanskii (SIAM J. Sci. Comput., 28(6) (2006)) and Farrell et al. (SIAM J. Sci. Comput., 41(5) (2019)), we further test the proposed preconditioner on the divergence-conforming HDG scheme for the Navier-Stokes equations. Numerical experiments show a mild increase in the iteration counts of the preconditioned GMRes solver with the rise in Reynolds number up to $10^3$.

翻译：本文针对广义Stokes方程和Navier-Stokes方程，基于增广拉格朗日形式，提出了一种用于满足散度相容条件的HDG格式的$hp$-多重网格预条件子。该方法依赖于二维和三维空间中的相容单纯形网格。$hp$-多重网格算法是一种乘性辅助空间预条件子，以最低阶空间作为辅助空间，并开发了几何多重网格方法作为辅助空间求解器。对于广义Stokes问题，几何多重网格方法的关键在于：凝聚后的最低阶散度相容HDG格式与Linke和Merdon（Comput. Methods Appl. Mech. Engrg., 311 (2016)）提出的带压力鲁棒处理的Crouzeix-Raviart离散格式之间存在等价关系，这允许将几何多重网格理论直接应用于Crouzeix-Raviart离散格式。数值实验表明，所提出的$hp$-多重网格预条件子对网格尺寸和增广拉格朗日参数具有鲁棒性，且迭代次数对多项式阶数的增加不敏感。受Benzi和Olshanskii（SIAM J. Sci. Comput., 28(6) (2006)）以及Farrell等（SIAM J. Sci. Comput., 41(5) (2019)）工作的启发，我们进一步将所提出的预条件子应用于Navier-Stokes方程的散度相容HDG格式。数值实验显示，当雷诺数升高至$10^3$时，预条件GMRes求解器的迭代次数仅有轻微增加。