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:由不压缩纳维耶-斯托克方程式的光谱元素分解产生的Poisson方程式的解决方案需要强有力的先决条件策略。 其中一个策略是多格化。 要实现多格化方法的潜力,需要有效的平滑战略。 Chebyshev 多元平滑证明是一种有效的平滑。 然而,需要做出一些改进, 特别是以对称成本为代价。 对于同样的循环成本, 一个全V周期, 以千元顺序Chebyshev 多边平滑, 可以用半V周期替换为2k$的 Chebyshev 多边平滑法, 其中平滑法在V- 周期的顶部省略了平滑法。 选择略去后一阵, 以更高排序 Chebyshev 预移动平滑法为优先。 在多格近度常值常值为$C$的大情况下, 将会做出一些有利的选择。 使用Lottes的第四类Chebyshev 多元平滑法的结果可以用半V- 半V 半周期取代, 将平滑法略法略法略法省省略法省在V- 模型上展示了这一模型, 的模型中展示了优度, 。</s>