The main focus of this paper is the study of efficient multigrid methods for large linear systems with a particular saddle-point structure. Indeed, when the system matrix is symmetric, but indefinite, the variational convergence theory that is usually used to prove multigrid convergence cannot be directly applied. However, different algebraic approaches analyze properly preconditioned saddle-point problems, proving convergence of the Two-Grid method. In particular, this is efficient when the blocks of the coefficient matrix possess a Toeplitz or circulant structure. Indeed, it is possible to derive sufficient conditions for convergence and provide optimal parameters for the preconditioning of the saddle-point problem in terms of the associated generating symbols. In this paper, we propose a symbol-based convergence analysis for problems that have a hidden block Toeplitz structure. Then, they can be investigated focusing on the properties of the associated generating function f, which consequently is a matrix-valued function with dimension depending on the block size of the problem. As numerical tests we focus on the matrix sequence stemming from the finite element approximation of the Stokes problem. We show the efficiency of the methods studying the hidden 9-by-9 block multilevel structure of the obtained matrix sequence. Moreover, we propose an efficient algebraic multigrid method with convergence rate independent of the matrix size. Finally, we present several numerical tests comparing the results with state-of-the-art strategies.

翻译：本文主要研究针对具有特殊鞍点结构的大型线性系统的高效多重网格方法。当系统矩阵对称但不定时，常用于证明多重网格收敛性的变分理论无法直接应用。然而，不同的代数方法可分析经过适当预处理的鞍点问题，并证明双层网格方法的收敛性。当系数矩阵的块具有Toeplitz或循环结构时，该方法尤为高效。实际上，可推导出收敛的充分条件，并根据相关生成符号为鞍点问题的预处理提供最优参数。本文针对具有隐藏块Toeplitz结构的问题，提出一种基于符号的收敛性分析。进而通过研究关联生成函数f的性质来探索该类问题，该函数为依赖于问题块大小的矩阵值函数。数值测试聚焦于Stokes问题有限元近似产生的矩阵序列，通过研究所得矩阵序列隐藏的9×9块多层结构，展示了方法的有效性。此外，我们提出一种收敛速率与矩阵规模无关的高效代数多重网格方法。最后，通过多项数值测试将结果与现有最优策略进行对比。