We present novel improvements in the context of symbol-based multigrid procedures for solving large block structured linear systems. We study the application of an aggregation-based grid transfer operator that transforms the symbol of a block Toeplitz matrix from matrix-valued to scalar-valued at the coarser level. Our convergence analysis of the Two-Grid Method (TGM) reveals the connection between the features of the scalar-valued symbol at the coarser level and the properties of the original matrix-valued one. This allows us to prove the convergence of a V-cycle multigrid with standard grid transfer operators for scalar Toeplitz systems at the coarser levels. Consequently, we extend the class of suitable smoothers for block Toeplitz matrices, focusing on the efficiency of block strategies, particularly the relaxed block Jacobi method. General conditions on smoothing parameters are derived, with emphasis on practical applications where these parameters can be calculated with negligible computational cost. We test the proposed strategies on linear systems stemming from the discretization of differential problems with $\mathbb{Q}_{d} $ Lagrangian FEM or B-spline with non-maximal regularity. The numerical results show in both cases computational advantages compared to existing methods for block structured linear systems.

翻译：我们提出了符号多重网格过程在求解大型块结构线性系统方面的创新改进。研究了基于聚合的网格传递算子，该算子将块Toeplitz矩阵的符号从矩阵值在粗网格层面转化为标量值。我们的双网格方法（TGM）收敛性分析揭示了粗网格层面标量符号特征与原始矩阵值符号性质之间的联系。这使我们能够证明使用标准网格传递算子的V-cycle多重网格在粗网格层面处理标量Toeplitz系统时的收敛性。因此，我们扩展了块Toeplitz矩阵的适用平滑器类别，重点研究了块策略的效率，特别是松弛块Jacobi方法。推导了平滑参数的一般条件，并重点关注这些参数可以在极低计算成本下计算的实用场景。我们将所提出的策略应用于来自$\mathbb{Q}_{d}$拉格朗日有限元或非最大正则性B样条离散微分问题产生的线性系统。数值结果表明，在两类情况下，该方法相比现有块结构线性系统的解法均具有计算优势。