We introduce a new preconditioner for a recently developed pressure-robust hybridized discontinuous Galerkin (HDG) finite element discretization of the Stokes equations. A feature of HDG methods is the straightforward elimination of degrees-of-freedom defined on the interior of an element. In our previous work (J. Sci. Comput., 77(3):1936--1952, 2018) we introduced a preconditioner for the case in which only the degrees-of-freedom associated with the element velocity were eliminated via static condensation. In this work we introduce a preconditioner for the statically condensed system in which the element pressure degrees-of-freedom are also eliminated. In doing so the number of globally coupled degrees-of-freedom are reduced, but at the expense of a more difficult problem to analyse. We will show, however, that the Schur complement of the statically condensed system is spectrally equivalent to a simple trace pressure mass matrix. This result is used to formulate a new, provably optimal preconditioner. Through numerical examples in two- and three-dimensions we show that the new preconditioned iterative method converges in fewer iterations, has superior conservation properties for inexact solves, and is faster in CPU time when compared to our previous preconditioner.

中文翻译：

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对Stokes方程进行压力鲁棒的HDG离散化的预处理

我们为最近开发的Stokes方程的耐压杂交不连续Galerkin（HDG）有限元离散化引入了一种新的预处理器。HDG方法的一个特点是直接消除了在元素内部定义的自由度。在我们之前的工作（J.Sci.Comput。，77（3）：1936--1952，2018）中，我们介绍了一种预处理器，适用于通过静凝聚消除仅与单元速度相关的自由度的情况。在这项工作中，我们介绍了用于静态冷凝系统的预处理器，该系统中也消除了元件压力自由度。这样做可以减少全局耦合的自由度的数量，但要以分析更困难的问题为代价。但是，我们将显示 静态冷凝系统的Schur补谱在光谱上等效于简单的痕量压力质量矩阵。该结果用于配制新的，可证明的最佳预处理器。通过二维和三维中的数值示例，我们表明，与我们之前的预处理器相比，新的预处理迭代方法收敛的迭代次数更少，具有用于非精确求解的优异保留特性，并且CPU时间更快。