In this paper, we employ local Fourier analysis (LFA) to analyze the convergence properties of multigrid methods for higher‐order finite‐element approximations to the Laplacian problem. We find that the classical LFA smoothing factor, where the coarse‐grid correction is assumed to be an ideal operator that annihilates the low‐frequency error components and leaves the high‐frequency components unchanged, fails to accurately predict the observed multigrid performance and, consequently, cannot be a reliable analysis tool to give good performance estimates of the two‐grid convergence factor. While two‐grid LFA still offers a reliable prediction, it leads to more complex symbols that are cumbersome to use to optimize parameters of the relaxation scheme, as is often needed for complex problems. For the purposes of this analytical optimization as well as to have simple predictive analysis, we propose a modification that is “between” two‐grid LFA and smoothing analysis, which yields reasonable predictions to help choose correct damping parameters for relaxation. This exploration may help us better understand multigrid performance for higher‐order finite element discretizations, including for Q₂‐Q₁ (Taylor‐Hood) elements for the Stokes equations. Finally, we present two‐grid and multigrid experiments, where the corrected parameter choice is shown to yield significant improvements in the resulting two‐grid and multigrid convergence factors.

中文翻译：

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拉普拉斯算子的高阶有限元离散化的多重网格两级傅里叶分析

在本文中，我们采用局部傅里叶分析（LFA）来分析多网格方法的收敛性，以解决Laplacian问题的高阶有限元近似问题。我们发现经典的LFA平滑因子（假设粗网格校正是消除低频误差分量并使高频分量保持不变的理想算子）无法准确预测观察到的多网格性能，因此， ，不能提供可靠的分析工具来对两网格收敛因子进行良好的性能估算。尽管两网格LFA仍可提供可靠的预测，但它会导致使用更复杂的符号来优化松弛方案的参数，这是复杂问题经常需要的。为了进行此分析优化以及进行简单的预测分析，我们提出了一种在“两网格LFA”和“平滑分析”之间进行的修改，该修改可产生合理的预测，以帮助选择正确的阻尼参数进行松弛。这项探索可能有助于我们更好地理解高阶有限元离散化的多重网格性能，包括Stokes方程的Q ₂ - Q ₁（Taylor-Hood）元素。最后，我们提出了两个网格和多个网格的实验，其中校正后的参数选择显示出对所得的两个网格和多个网格收敛因子的显着改善。