This paper discusses our recent generalized optimal algebraic multigrid (AMG) convergence theory applied to the steady-state Stokes equations discretized using Taylor-Hood elements ($\pmb{ \mathbb{P}}_2/\mathbb{P}_{1}$). The generalized theory is founded on matrix-induced orthogonality of the left and right eigenvectors of a generalized eigenvalue problem involving the system matrix and relaxation operator. This framework establishes a rigorous lower bound on the spectral radius of the two-grid error-propagation operator, enabling precise predictions of the convergence rate for symmetric indefinite problems, such as those arising from saddle-point systems. We apply this theory to the recently developed monolithic smooth aggregation AMG (SA-AMG) solver for Stokes, constructed using evolution-based strength of connection, standard aggregation, and smoothed prolongation. The performance of these solvers is evaluated using additive and multiplicative Vanka relaxation strategies. Additive Vanka relaxation constructs patches algebraically on each level, resulting in a nonsymmetric relaxation operator due to the partition of unity being applied on one side of the block-diagonal matrix. Although symmetry can be restored by eliminating the partition of unity, this compromises convergence. Alternatively, multiplicative Vanka relaxation updates velocity and pressure sequentially within each patch, propagating updates multiplicatively across the domain and effectively addressing velocity-pressure coupling, ensuring a symmetric relaxation. We demonstrate that the generalized optimal AMG theory consistently provides accurate lower bounds on the convergence rate for SA-AMG applied to Stokes equations. These findings suggest potential avenues for further enhancement in AMG solver design for saddle-point systems.

翻译：本文讨论了我们近期提出的广义最优代数多重网格（AMG）收敛理论在采用Taylor-Hood元（$\pmb{ \mathbb{P}}_2/\mathbb{P}_{1}$）离散的稳态Stokes方程中的应用。该广义理论基于系统矩阵与松弛算子构成的广义特征值问题中左右特征向量的矩阵诱导正交性。该框架为两重网格误差传播算子的谱半径建立了严格下界，从而能够精确预测对称不定问题（如源于鞍点系统的问题）的收敛速率。我们将此理论应用于近期开发的Stokes方程整体式光滑聚合AMG（SA-AMG）求解器，该求解器采用基于演化的连接强度、标准聚合及光滑延拓算子构建。通过加性与乘性Vanka松弛策略评估这些求解器的性能。加性Vanka松弛在各层级上代数地构造补丁，由于单位分解仅作用于块对角矩阵的一侧，导致松弛算子非对称。虽然通过消除单位分解可恢复对称性，但这会损害收敛性。另一种方案是乘性Vanka松弛，它在每个补丁内顺序更新速度与压力，通过乘性方式在计算域内传播更新，有效处理速度-压力耦合并保证松弛算子的对称性。我们证明，广义最优AMG理论持续为应用于Stokes方程的SA-AMG提供收敛速率的精确下界。这些发现为鞍点系统AMG求解器的进一步优化设计指出了潜在路径。