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：本文研究针对稳定混合有限元离散的Stokes问题的新型全代数多重网格求解器。该算法基于将低阶$\pmb{\mathbb{P}}_1 \text{iso}\kern1pt\pmb{ \mathbb{P}}_2/ \mathbb{P}_1$离散作为高阶离散（如$\pmb{\mathbb{P}}_2/\mathbb{P}_1$）的预处理子。采用光滑聚合代数多重网格对低阶离散的速度场和压力场进行独立粗化，从而构建纯代数形式的高阶离散预处理子（即无需几何信息）。此外，我们引入一种针对Vanka块的新型块LU分解技术，在降低存储需求的同时平衡计算效率。通过二维和三维结构网格与非结构网格上的$\pmb{\mathbb{P}}_2/\mathbb{P}_1$（Taylor-Hood）离散验证了该方法的有效性。同样地，该方法应用于二维重心加密网格上的$\pmb{\mathbb{P}}_2/\mathbb{P}_1^{disc}$（Scott-Vogelius）离散时也展现出优异性能。这种新型全代数多重网格求解器不仅能达到、且常超越非精确Uzawa预处理子的性能，在后者难以收敛的多种问题场景中均展现出广泛的适用性和鲁棒性。