If the Stokes equations are properly discretized, it is well-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 and Krylov-Uzawa algorithms. However, in the case of complex geometries, the Schur complement matrix can become arbitrarily ill-conditioned having a significant portion of non-unit eigenvalues, which makes the established Uzawa preconditioner inefficient. In this article, we study the Schur complement formulation for the staggered finite-difference discretization of the Stokes problem in 3D CT images and synthetic 2D geometries. We numerically investigate the performance of the CG iterative method with the Uzawa and SIMPLE preconditioners and draw several conclusions. First, we show that in the case of low porosity, CG with the SIMPLE preconditioner converges faster to the discrete pressure and provides a more accurate calculation of sample permeability. Second, we show that an increase in the surface-to-volume ratio leads to an increase in the condition number of the Schur complement matrix, while the dependence is inverse for the Schur complement matrix preconditioned with the SIMPLE. As an explanation, we conjecture that the no-slip boundary conditions are the reason for non-unit eigenvalues of the Schur complement.

翻译：若Stokes方程得到恰当离散化，众所周知Schur补矩阵谱等价于单位矩阵。此外，在简单几何情形中，通常观察到其大部分特征值等于1。这些事实构成了经典的Uzawa算法与Krylov-Uzawa算法的基础。然而，在复杂几何情形下，Schur补矩阵可能因含有大量非单位特征值而变得病态严重，导致成熟的Uzawa预条件子失效。本文针对三维CT图像与二维合成几何中交错有限差分离散的Stokes问题，研究了Schur补公式。我们通过数值实验探究了采用Uzawa与SIMPLE预条件子的共轭梯度法性能，并得出以下结论：首先，在低孔隙度情形下，采用SIMPLE预条件子的共轭梯度法对离散压力的收敛速度更快，且能更精确地计算样本渗透率；其次，我们证明表面体积比的增大会导致Schur补矩阵条件数的增加，而经SIMPLE预条件后Schur补矩阵的条件数与其呈反比关系。作为解释，我们推测Schur补矩阵非单位特征值的产生源于无滑移边界条件。