Sharpened PCG Iteration Bound for High-Contrast Heterogeneous Scalar Elliptic PDEs

TL;DR

This paper introduces a refined iteration bound for the PCG method that better predicts convergence in high-contrast heterogeneous elliptic PDEs by considering spectral clusters, leading to improved preconditioner design.

Contribution

The work develops a new spectral cluster-based iteration bound for PCG, demonstrating its effectiveness on high-contrast PDEs and enabling better preconditioner selection.

Findings

The new bound is significantly sharper than classical bounds.

Simpler coarse spaces can be competitive for certain high-contrast problems.

The bound can be estimated early in PCG iterations using Ritz values.

Abstract

A new iteration bound for the preconditioned conjugate gradient (PCG) method is presented that more accurately captures convergence for systems with clustered eigenspectra, where the classical condition number-based bound is too pessimistic. By using the edge eigenvalues of each cluster in the spectral distribution, the bound is shown to be orders of magnitude sharper than the classical bound for certain examples. Its effectiveness is demonstrated on a high-contrast elliptic PDE preconditioned with a two-level overlapping Schwarz preconditioner, where the performance of different (algebraic) coarse spaces is successfully distinguished. A key contribution of this work is the observation that, for certain high-contrast problems, simpler coarse spaces can be made competitive in terms of PCG convergence. Conversely, more complex preconditioners are not always required. Finally, it is shown that the bound can be estimated effectively from Ritz values computed during early PCG iterations.