Augmented Lagrangian preconditioners have successfully yielded Reynolds-robust preconditioners for the stationary incompressible Navier-Stokes equations, but only for specific discretizations. The discretizations for which these preconditioners have been designed possess error estimates which depend on the Reynolds number, with the discretization error deteriorating as the Reynolds number is increased. In this paper we present an augmented Lagrangian preconditioner for the Scott-Vogelius discretization on barycentrically-refined meshes. This achieves both Reynolds-robust performance and Reynolds-robust error estimates. A key consideration is the design of a suitable space decomposition that captures the kernel of the grad-div term added to control the Schur complement; the same barycentric refinement that guarantees inf-sup stability also provides a local decomposition of the kernel of the divergence. The robustness of the scheme is confirmed by numerical experiments in two and three dimensions.

翻译：Lagrangian 增强性能的附加条件成功地为固定的不压缩纳维-斯托克方程式产生了Reynolds-robust 的先决条件,但仅限于特定的离散性方程式。设计这些先决条件的离散性方程式拥有取决于Reynolds 号的误差估计值,随着Reynolds 号的增加,离散性错误也随之恶化。在本文中,我们提出了Scott-Voglogius 离散的强化的Lagrangian 先决条件。这既实现了Reynolds-robust 性能,也实现了Reynolds-robust 误差估计。一个关键的考虑因素是设计适当的空间分解位置,以捕捉控制Schur 补充的变异化-div 术语的内核门;同样的偏心式改进,即保证Sur 变异性能的稳定也提供了局部分解的内核。二、三个层面的数值实验证实了这一办法的坚固性。