TL;DR
This paper introduces a new Fourier analysis method for multigrid algorithms applied to 1D Poisson problems, demonstrating that with weighted-Jacobi smoothing, multigrid can serve as an exact solver.
Contribution
It presents an alternative, more understandable Fourier analysis that explicitly derives spectra and shows multigrid's exactness with weighted-Jacobi smoothing.
Findings
New Fourier analysis simplifies understanding of multigrid spectra.
Multigrid becomes an exact solver with weighted-Jacobi smoothing.
Analysis applies to 1D Poisson problems.
Abstract
We provide an alternative Fourier analysis for multigrid applied to the Poisson problem in 1D, based on explicit derivation of spectra of the iteration matrix. The new Fourier analysis has advantages over the existing one. It is easy to understand and enables us to write the error equation in terms of the eigenvector of the stiffness matrix. When weighted-Jacobi is used as a smoother with two different weights, multigrid is an exact solver.
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