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.

翻译：由不可压缩Navier-Stokes方程的谱元离散产生的泊松方程的求解需要稳健的预处理策略，多重网格便是其中一种策略。为充分发挥多重网格方法的潜力，需要有效的平滑策略。切比雪夫多项式平滑被证明是一种有效的平滑器。然而，在对称性方面仍有改进空间，尤其体现在计算成本上。在每次迭代成本相同的情况下，采用k阶切比雪夫多项式平滑的完整V循环可被替换为采用2k阶切比雪夫多项式平滑的半V循环，其中在V循环的上升支中省略平滑操作。当多重网格近似性常数C较大时，选择省略后平滑器而采用更高阶切比雪夫预平滑器被证明更为有利。本文展示了利用Lottes第四类切比雪夫多项式平滑器的结果，这些方法相较于标准切比雪夫多项式平滑器有显著改进。作者在p型几何多重网格以及二维有限差分数值案例中验证了该方案的有效性。