Non-overlapping block smoothers for the Stokes equations

TL;DR

This paper introduces and analyzes new non-overlapping triad-wise smoothers for multigrid methods solving the Stokes equations, aiming to reduce computational costs while maintaining efficiency.

Contribution

The paper develops and demonstrates the effectiveness of novel non-overlapping triad-wise smoothers for the Stokes equations, offering a computationally cheaper alternative to overlapping smoothers.

Findings

Triad-wise smoothers are efficient within multigrid methods.

Non-overlapping smoothers reduce computational costs.

Local Fourier analysis confirms their effectiveness.

Abstract

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 non-overlapping and therefore desirable due to reduced computational costs. Traditional geometric multigrid methods are based on simple pointwise smoothers. However, the efficiency of multigrid methods for solving more difficult problems such as the Stokes equations lead to computationally more expensive smoothers, e.g., overlapping block smoothers. Non-overlapping smoothers are less expensive, but have been considered less efficient in the literature. In this paper, we develop new non-overlapping 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 non-overlapping smoothers by measuring their computational costs and analyzing their behavior by the use of local Fourier analysis.