Multigrid methods are popular for solving linear systems derived from discretizing PDEs. Local Fourier Analysis (LFA) is a technique for investigating and tuning multigrid methods. P-multigrid is popular for high-order or spectral finite element methods, especially on unstructured meshes. In this paper, we introduce LFAToolkit.jl, a new Julia package for LFA of high-order finite element methods. LFAToolkit.jl analyzes preconditioning techniques for arbitrary systems of second order PDEs and supports mixed finite element methods. Specifically, we develop LFA of p-multigrid with arbitrary second-order PDEs using high-order finite element discretizations and examine the performance of Jacobi and Chebyshev smoothing for two-grid schemes with aggressive p-coarsening. A natural extension of this LFA framework is the analysis of h-multigrid for finite element discretizations or finite difference discretizations that can be represented in the language of finite elements. With this extension, we can replicate previous work on the LFA of h-multigrid for arbitrary order discretizations using a convenient and extensible abstraction. Examples in one, two, and three dimensions are presented to validate our LFA of p-multigrid for the Laplacian and linear elasticity.

翻译：本地 Fourier Alyer 分析(LFA) 是一种调查和调整多格方法的技术。 P- Multigrid 使用高阶或光谱限制元素方法,特别是在无结构的模件上,很受欢迎。在本文件中,我们为LFA引进了LFAToolkit.jl,这是一个新的高阶限制元素方法的Julia软件包。LFAToolkit.jl 分析二阶PDE系统任意系统的先决条件技术,并支持混合的有限元素方法。具体地说,我们利用高阶限制元素分解和光谱元素方法开发P-二阶PDes 的P-MlFA,并研究Jacobi 和Chebyshev 的两格化功能,且具有强势微缩缩缩缩图的两维功能的性。LAFA框架的自然延伸是对可用有限元素语言表示的h-mulgrid grid 或有限的分解法的分析。通过这一扩展,我们可以复制过去关于两个 h-mligridridridridal dical 3 和直线性FAFA的示例,利用一个方便和可扩展的样本的示例的样本和图像的复制和模型的复制。