  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.
