Eigenvalue bounds for preconditioned symmetric multiple saddle-point matrices

TL;DR

This paper derives eigenvalue bounds for symmetric block tridiagonal saddle-point matrices preconditioned with block diagonal matrices, extending previous results to arbitrary block numbers and providing practical bounds validated by numerical experiments.

Contribution

It generalizes eigenvalue bounds for preconditioned saddle-point matrices to any number of blocks, including cases with approximated Schur complements, and introduces practical bounds with numerical validation.

Findings

Eigenvalue bounds are established for arbitrary block saddle-point systems.

Results include bounds for systems with approximated Schur complements.

Numerical experiments confirm the theoretical bounds.

Abstract

We develop eigenvalue bounds for symmetric, block tridiagonal multiple saddle-point linear systems, preconditioned with block diagonal matrices. We extend known results for $3 \times 3$ block systems [Bradley and Greif, IMA J.\ Numer. Anal. 43 (2023)] and for $4 \times 4$ systems [Pearson and Potschka, IMA J. Numer. Anal. 44 (2024)] to an arbitrary number of blocks. Moreover, our results generalize the bounds in [Sogn and Zulehner, IMA J. Numer. Anal. 39 (2018)], developed for an arbitrary number of blocks with null diagonal blocks. Extension to the bounds when the Schur complements are approximated is also provided, using perturbation arguments. Practical bounds are also obtained for the double saddle-point linear system. Numerical experiments validate our findings.