Finite element discretization of Stokes problems can result in singular, inconsistent saddle point linear algebraic systems. This inconsistency can cause many iterative methods to fail to converge. In this work, we consider the lowest-order weak Galerkin finite element method to discretize Stokes flow problems and study a consistency enforcement by modifying the right-hand side of the resulting linear system. It is shown that the modification of the scheme does not affect the optimal-order convergence of the numerical solution. Moreover, inexact block diagonal and triangular Schur complement preconditioners and the minimal residual method (MINRES) and the generalized minimal residual method (GMRES) are studied for the iterative solution of the modified scheme. Bounds for the eigenvalues and the residual of MINRES/GMRES are established. Those bounds show that the convergence of MINRES and GMRES is independent of the viscosity parameter and mesh size. The convergence of the modified scheme and effectiveness of the preconditioners are verified using numerical examples in two and three dimensions.

翻译：Stokes问题的有限元离散化可能导致奇异、非一致鞍点线性代数系统。这种非一致性会导致许多迭代方法无法收敛。本文采用最低阶弱伽辽金有限元方法离散Stokes流动问题，通过修正所得线性系统的右侧项来研究一致性强制方法。研究表明，该修正方案不会影响数值解的最优阶收敛性。此外，针对修正方案的迭代求解，研究了非精确块对角与三角Schur补预条件子，以及最小残差法（MINRES）和广义最小残差法（GMRES）。建立了MINRES/GMRES的特征值与残差界。这些界表明MINRES和GMRES的收敛性与黏性参数和网格尺寸无关。通过二维和三维数值算例验证了修正方案的收敛性及预条件子的有效性。