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Multigrid methods are popular for solving linear systems derived fromdiscretizing PDEs. Local Fourier Analysis (LFA) is a technique forinvestigating and tuning multigrid methods. P-multigrid is popular forhigh-order or spectral finite element methods, especially on unstructuredmeshes. In this paper, we introduce LFAToolkit.jl, a new Julia package for LFAof high-order finite element methods. LFAToolkit.jl analyzes preconditioningtechniques for arbitrary systems of second order PDEs and supports mixed finiteelement methods. Specifically, we develop LFA of p-multigrid with arbitrarysecond-order PDEs using high-order finite element discretizations and examinethe performance of Jacobi and Chebyshev smoothing for two-grid schemes withaggressive p-coarsening. A natural extension of this LFA framework is theanalysis of h-multigrid for finite element discretizations or finite differencediscretizations that can be represented in the language of finite elements.With this extension, we can replicate previous work on the LFA of h-multigridfor arbitrary order discretizations using a convenient and extensibleabstraction. Examples in one, two, and three dimensions are presented tovalidate our LFA of p-multigrid for the Laplacian and linear elasticity.

Local Fourier Analysis of P-Multigrid for High-Order Finite Element Operators