  The solution to the Poisson equation arising from the spectral element
discretization of the incompressible Navier-Stokes equation requires robust
preconditioning strategies. One such strategy is multigrid. To realize the
potential of multigrid methods, effective smoothing strategies are needed.
Chebyshev polynomial smoothing proves to be an effective smoother. However,
there are several improvements to be made, especially at the cost of symmetry.
For the same cost per iteration, a full V-cycle with $k$ order Chebyshev
polynomial smoothing may be substituted with a half V-cycle with order $2k$
Chebyshev polynomial smoothing, wherein the smoother is omitted on the up-leg
of the V-cycle. The choice of omitting the post-smoother in favor of higher
order Chebyshev pre-smoothing is shown to be advantageous in cases where the
multigrid approximation property constant, $C$, is large. Results utilizing
Lottes's fourth-kind Chebyshev polynomial smoother are shown. These methods
demonstrate substantial improvement over the standard Chebyshev polynomial
smoother. The authors demonstrate the effectiveness of this scheme in
$p$-geometric multigrid, as well as a 2D model problem with finite differences.
