Optimal Chebyshev Smoothers and One-sided V-cycles

TL;DR

This paper introduces optimized Chebyshev polynomial smoothers and one-sided V-cycles that improve multigrid methods for solving Poisson equations from spectral element discretizations, especially when the approximation property constant is large.

Contribution

It proposes a novel half V-cycle approach with higher order Chebyshev smoothing that enhances multigrid efficiency without increasing computational cost.

Findings

Half V-cycle with higher order Chebyshev smoothing outperforms standard methods.

Omitting post-smoothing benefits when the multigrid approximation constant is large.

Demonstrated effectiveness in p-geometric multigrid and 2D finite difference problems.

Abstract

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.