Multigrid methods are popular solution algorithms for many discretized PDEs, either as standalone iterative solvers or as preconditioners, due to their high efficiency. However, the choice and optimization of multigrid components such as relaxation schemes and grid-transfer operators is crucial to the design of optimally efficient algorithms. It is well--known that local Fourier analysis (LFA) is a useful tool to predict and analyze the performance of these components. In this paper, we develop a local Fourier analysis of monolithic multigrid methods based on additive Vanka relaxation schemes for mixed finite-element discretizations of the Stokes equations. The analysis offers insight into the choice of "patches" for the Vanka relaxation, revealing that smaller patches offer more effective convergence per floating point operation. Parameters that minimize the two-grid convergence factor are proposed and numerical experiments are presented to validate the LFA predictions.

中文翻译：

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Stokes 方程加性 Vanka 弛豫的局部傅立叶分析

多重网格方法是许多离散化偏微分方程的流行求解算法，无论是作为独立的迭代求解器还是作为预处理器，由于它们的高效率。然而，多重网格组件（例如松弛方案和网格传输算子）的选择和优化对于设计最佳高效算法至关重要。众所周知，局部傅里叶分析 (LFA) 是预测和分析这些组件性能的有用工具。在本文中，我们基于 Stokes 方程的混合有限元离散化的加性 Vanka 松弛方案开发了整体多重网格方法的局部傅里叶分析。该分析提供了对 Vanka 松弛“补丁”选择的深入了解，揭示了较小的补丁为每个浮点运算提供了更有效的收敛。