Stability Estimates and Structural Spectral Properties of Saddle Point Problems

TL;DR

This paper derives sharp stability estimates for saddle point problems, analyzes spectral properties of related matrices, and demonstrates their application in optimal control problems with numerical experiments.

Contribution

It provides new sharp stability bounds and spectral analysis for saddle point problems, enhancing understanding and solution methods.

Findings

Sharp bounds for Babuška's inf-sup constants derived.

Spectral properties of preconditioned matrices analyzed.

Numerical experiments confirm theoretical results.

Abstract

For a general class of saddle point problems sharp estimates for Babu\v{s}ka's inf-sup stability constants are derived in terms of the constants in Brezzi's theory. In the finite-dimensional Hermitian case more detailed spectral properties of preconditioned saddle point matrices are presented, which are helpful for the convergence analysis of common Krylov subspace methods. The theoretical results are applied to two model problems from optimal control with time-periodic state equations. Numerical experiments with the preconditioned minimal residual method are reported.