In this paper, we investigate a novel monolithic algebraic multigrid solver for the discrete Stokes problem discretized with stable mixed finite elements. The algorithm is based on the use of the low-order $\pmb{\mathbb{P}}_1 \text{iso}\kern1pt\pmb{ \mathbb{P}}_2/ \mathbb{P}_1$ discretization as a preconditioner for a higher-order discretization, such as $\pmb{\mathbb{P}}_2/\mathbb{P}_1$. Smoothed aggregation algebraic multigrid is used to construct independent coarsenings of the velocity and pressure fields for the low-order discretization, resulting in a purely algebraic preconditioner for the high-order discretization (i.e., using no geometric information). Furthermore, we incorporate a novel block LU factorization technique for Vanka patches, which balances computational efficiency with lower storage requirements. The effectiveness of the new method is verified for the $\pmb{\mathbb{P}}_2/\mathbb{P}_1$ (Taylor-Hood) discretization in two and three dimensions on both structured and unstructured meshes. Similarly, the approach is shown to be effective when applied to the $\pmb{\mathbb{P}}_2/\mathbb{P}_1^{disc}$ (Scott-Vogelius) discretization on 2D barycentrically refined meshes. This novel monolithic algebraic multigrid solver not only meets but frequently surpasses the performance of inexact Uzawa preconditioners, demonstrating the versatility and robust performance across a diverse spectrum of problem sets, even where inexact Uzawa preconditioners struggle to converge.

翻译：本文研究了一种针对稳定混合有限元离散的离散Stokes问题的整体代数多重网格求解器。该算法基于使用低阶$\pmb{\mathbb{P}}_1 \text{iso}\kern1pt\pmb{ \mathbb{P}}_2/ \mathbb{P}_1$离散作为高阶离散（如$\pmb{\mathbb{P}}_2/\mathbb{P}_1$）的预条件子。通过平滑聚合代数多重网格对低阶离散的速度场和压力场构建独立粗化，从而得到高阶离散的纯代数预条件子（即无需几何信息）。此外，我们引入了一种针对Vanka补丁的新型块LU分解技术，该技术在计算效率与低存储需求之间取得平衡。对于二维和三维结构化与非结构化网格上的$\pmb{\mathbb{P}}_2/\mathbb{P}_1$（Taylor-Hood）离散，新方法的有效性得到了验证。同样，该方法在二维重心细化网格上的$\pmb{\mathbb{P}}_2/\mathbb{P}_1^{disc}$（Scott-Vogelius）离散中也被证明是有效的。这种新型整体代数多重网格求解器不仅达到且常超越非精确Uzawa预条件子的性能，展现了其在各类问题集上的通用性与稳健表现，即使在非精确Uzawa预条件子难以收敛的情况下也表现出色。