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

TL;DR

This paper introduces LFAToolkit.jl, a Julia package for Local Fourier Analysis of high-order finite element methods, enabling analysis of p-multigrid and smoothing techniques for PDE discretizations across multiple dimensions.

Contribution

It develops a novel LFA framework for p-multigrid with high-order finite elements and extends existing h-multigrid analysis to arbitrary discretizations, validated through multiple examples.

Findings

LFAToolkit.jl effectively analyzes p-multigrid performance.

Jacobi and Chebyshev smoothers are evaluated for high-order discretizations.

Validation includes 1D, 2D, and 3D examples for Laplacian and elasticity problems.

Abstract

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.