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 article, 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弛豫的局部傅里叶分析

多重网格方法是许多离散PDE的流行解决方案算法，由于它们的高效率，它们既可以作为独立的迭代求解器，也可以作为预处理器。但是，选择和优化多网格组件（例如松弛方案和网格转移运算符）对于设计最佳有效算法至关重要。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，我们基于斯托克斯方程的混合有限元离散化，基于加性Vanka松弛方案，开发了整体式多重网格方法的局部傅里叶分析。该分析提供了Vanka松弛的“补丁”选择的见解，揭示了较小的补丁可为每个浮点运算提供更有效的收敛。