If the Stokes equations are properly discretized, it is known that the Schur complement matrix is spectrally equivalent to the identity matrix. Moreover, in the case of simple geometries, it is often observed that most of its eigenvalues are equal to one. These facts form the basis for the famous Uzawa algorithm. Despite recent progress in developing efficient iterative methods for solving the Stokes problem, the Uzawa algorithm remains popular in science and engineering, especially when accelerated by Krylov subspace methods. However, in complex geometries, the Schur complement matrix can become severely ill-conditioned, having a significant portion of non-unit eigenvalues. This makes the established Uzawa preconditioner inefficient. To explain this behaviour, we examine the Pressure Schur Complement formulation for the staggered finite-difference discretization of the Stokes equations. Firstly, we conjecture that the no-slip boundary conditions are the reason for non-unit eigenvalues of the Schur complement matrix. Secondly, we demonstrate that its condition number increases with increasing the surface-to-volume ratio of the flow domain. As an alternative to the Uzawa preconditioner, we propose using the diffusive SIMPLE preconditioner for geometries with a large surface-to-volume ratio. We show that the latter is much more fast and robust for such geometries. Furthermore, we show that the usage of the SIMPLE preconditioner leads to more accurate practical computation of the permeability of tight porous media. Keywords: Stokes problem, tight geometries, computing permeability, preconditioned Krylov subspace methods

翻译：若Stokes方程得到恰当离散化，已知其Schur补矩阵谱等价于单位矩阵。此外，在简单几何情形下，通常可观察到该矩阵的大部分特征值等于1。这些事实构成了著名的Uzawa算法的基础。尽管近年来在开发求解Stokes问题的迭代方法方面取得了进展，Uzawa算法在科学与工程领域依然广受欢迎，尤其当其通过Krylov子空间方法加速时。然而，在复杂几何中，Schur补矩阵可能变得严重病态，包含大量非单位特征值，这使得经典的Uzawa预处理子效率低下。为解释这一现象，本文研究了交错有限差分离散Stokes方程的压力Schur补形式。首先，我们推测无滑移边界条件是导致Schur补矩阵出现非单位特征值的原因。其次，我们证明其条件数随流动域表面积与体积之比的增大而增大。作为Uzawa预处理子的替代方案，我们建议在表面积体积比较大的几何中使用扩散SIMPLE预处理子。研究表明，对于此类几何，后者更加快速且鲁棒。此外，我们证明使用SIMPLE预处理子能更精确地计算致密多孔介质的渗透率。关键词：Stokes问题，狭窄几何，渗透率计算，预处理Krylov子空间方法