Overlapping block smoothers efficiently damp the error contributions from highly oscillatory components within multigrid methods for the Stokes equations but they are computationally expensive. This paper is concentrated on the development and analysis of new block smoothers for the Stokes equations that are discretized on staggered grids. These smoothers are nonoverlapping and therefore desirable due to reduced computational costs. Traditional geometric multigrid methods are based on simple pointwise smoothers. However, using multigrid methods to efficiently solve more difficult problems such as the Stokes equations leads to computationally more expensive smoothers, for example, overlapping block smoothers. Nonoverlapping smoothers are less expensive, but have been considered less efficient in the literature. In this paper, we develop new nonoverlapping smoothers, the so-called triad-wise smoothers, and show their efficiency within multigrid methods to solve the Stokes equations. In addition, we compare overlapping and nonoverlapping smoothers by measuring their computational costs and analyzing their behavior by the use of local Fourier analysis.

中文翻译：

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斯托克斯方程的非重叠块平滑器

重叠块平滑器有效地抑制了斯托克斯方程多重网格方法中高度振荡分量的误差贡献，但它们的计算成本很高。本文主要针对交错网格上离散的 Stokes 方程的新块平滑器的开发和分析。这些平滑器是非重叠的，因此由于降低了计算成本而需要。传统的几何多重网格方法基于简单的逐点平滑器。然而，使用多重网格方法来有效地解决更困难的问题（例如斯托克斯方程）会导致计算成本更高的平滑器，例如重叠块平滑器。非重叠平滑器成本较低，但在文献中被认为效率较低。在本文中，我们开发了新的非重叠平滑器，即所谓的三元组平滑器，并展示了它们在求解斯托克斯方程的多重网格方法中的效率。此外，我们通过测量它们的计算成本并使用局部傅立叶分析来分析它们的行为来比较重叠和非重叠平滑器。