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Semi-Algebraic Mode Analysis For Finite Element Discretisations Of The Heat Equation
A. Mesforush, N. Habibi, F. Gasper, C. Rodrigo
TL;DR
For finite element heat-equation discretizations, the paper studies how to predict convergence of multigrid waveform relaxation while accounting for the mass matrix. It extends SAMA to two and three dimensions and reports different two-grid bounds for mass-matrix and identity cases.
Problem
The paper addresses predicting convergence factors for time-dependent PDEs, for which Local Fourier Analysis has not been successful, in the finite element setting used for multigrid waveform relaxation.
Source: §1 · pp. 1–3Method
The analysis combines spatial Fourier-mode analysis with exact time analysis for multigrid waveform relaxation using finite element discretizations.
Source: §1 · p. 1, §3.1 · p. 11, §4 · p. 12Results
The two-dimensional two-grid convergence-factor bound is 0.3117 with the mass matrix and 0.0695 with the identity matrix over λ = τ/h² from 2^-12 to 2^12.
Source: §2.2 · p. 10Takeaways & Limitations
SAMA provides a systematic way to study multigrid waveform relaxation behavior for finite element discretizations.
Source: §4 · p. 13
Abstract
from arXiv · showhide
In this work, a semialgebraic mode analysis (SAMA) is proposed for investigating the convergence of a multigrid waveform relaxation method applied to the Finite Element (FE) discretization of the heat equation in two and three dimensions. This analysis for finite element methods is more involved and more general than that for Finite Difference (FD) discretizations, since mass matrix must be considered. The proposed analysis results in a very useful tool to study the behaviour of the multigrid waveform relaxation method depending on the parameters of the problem.
1. Introduction
The paper applies SAMA to multigrid waveform relaxation for finite element discretizations of the heat equation, where the mass matrix must be included. It motivates waveform relaxation as a time-parallel alternative to sequential time stepping.
1. Introduction
- SAMA combines spatial Local Fourier Analysis with exact analysis in time to predict convergence factors for time-dependent PDEs.The paper applies it to multigrid waveform relaxation for finite element discretizations of the heat equation.Source: §1 · p. 1
- The finite element model uses continuous piecewise bilinear or trilinear functions and yields an ODE system involving mass and stiffness matrices.The basis choice corresponds to two and three spatial dimensions, respectively.Source: §1 · p. 2
- Waveform relaxation offers a time-parallel alternative to sequential time stepping, and this work focuses on its multigrid extension.The method applies to time-dependent PDEs after spatial derivatives are discretized; its iteration splits the mass and stiffness matrices.Source: §1 · pp. 2–3
- The multigrid waveform relaxation method uses spatial grid coarsening with pre-smoothing, coarse-grid correction, and post-smoothing.The paper uses Gauss-Seidel waveform relaxation as the smoother.Source: §1 · pp. 3–4
2. SAMA in two dimensions
The two-dimensional analysis combines Fourier-mode smoothing and coarse-grid correction to estimate multigrid waveform relaxation convergence. For the tested parameter range, its predicted bound depends on whether the mass matrix or identity is used.
2.1. Theoretical framework.
- For each low frequency θ, the four 2h-harmonics span a space invariant under coarse-grid correction, whose representation on that space is a 4 × 4 matrix.Source: §2.1 · p. 7
- The two-grid operator combines Fourier smoothing analysis with coarse-grid correction, using Gauss-Seidel relaxation for pre- and post-smoothing.The analysis defines the two-grid convergence-factor estimate from the resulting operator.Source: §2.1 · p. 9
2.2. SAMA results in two dimensions.
- The model analysis applies bilinear finite elements on a uniform rectangular mesh and represents the mass and stiffness matrices by stencils.The reported results compare the mass-matrix case with an identity-matrix case while varying λ = τ/h².Source: §2.2 · pp. 9–10
- 0.3117 for the mass-matrix case and 0.0695 for the identity case bound the two-grid convergence factor over λ = τ/h² from 2^-12 to 2^12.The analysis uses two smoothing steps and M = 32 time steps; the asymptotic factor is reported as satisfactory for h very small (τ = 10^-5, λ = 2^12).Source: §2.2 · p. 10
3. SAMA in three dimensions
The three-dimensional SAMA combines spatial Fourier analysis with exact time-variable analysis for the multigrid waveform relaxation method. Using trilinear finite elements, it predicts convergence factors across values of λ and compares them with W(1,1)-cycle experiments.
3.1. Theoretical framework.
- The 3D framework represents smoothing, coarse-grid correction, and two-grid operators as 8M × 8M matrices, combining 3D spatial LFA with exact time-variable analysis.Source: §3.1 · p. 11
3.2. SAMA results in three dimensions.
- The 3D model uses trilinear finite elements, Gauss-Seidel waveform relaxation, and inter-grid transfer operators, with stencils specified for the mass and stiffness matrices.Source: §3.2 · pp. 11–12
- The 3D predicted upper bounds are 0.4040 for Bh as the mass matrix and 0.0881 for Bh as the identity matrix; Figure 3 pairs predictions with W(1,1)-cycle measurements.The results are reported for two smoothing steps and M = 32.Source: §3.2 · p. 12
- With Bh as the identity matrix, the results match very well with finite-difference results, and the asymptotic convergence factor is very satisfactory as h tends to zero.Source: §3.2 · p. 12
4. Conclusions
The paper presents two-grid SAMA for multigrid waveform relaxation with finite element discretizations in two and three spatial dimensions. With Bh as the identity matrix, its results are comparable to finite-difference results, and the analysis supports systematic study of method behavior.
- With Bh as the identity matrix, the results are comparable to finite-difference results; Figure 3 displays 3D predicted and experimentally computed factors across λ.Source: §4 · p. 12, Figure 3 · p. 13
- The analysis covers Crank-Nicolson time discretization and bilinear and trilinear finite elements in two and three dimensions, respectively.Source: §4 · p. 12
- The proposed SAMA analysis supports systematic study of multigrid waveform relaxation behavior for finite element discretizations.Source: §4 · p. 13