On the Spectral Clustering of a Class of Multigrid Preconditioners
Authors
Jose Pablo Lucero Lorca
Abstract
This paper studies a common two-level multigrid construction for block-structured linear systems and identifies a simple way to describe how its smoothing and coarse-grid components interact. By examining the method through a collection of small coupling modes, we show that its behavior can be captured by a single scalar quantity for each mode. The main result is an explicit choice of smoothing parameters that makes all modes respond in the same way, causing the nontrivial eigenvalues of the preconditioned operator to collapse to a single value. This gives a clear and self-contained description of the ideal version of the method and provides a concrete target for designing related schemes. Although the exact spectral collapse requires ideal components, we also show that the same construction naturally produces operators that resemble those used in practical discretizations. Examples from finite-difference and discontinuous Galerkin settings illustrate how the ideal parameters can be used in practice.
