Stability Estimates and Structural Spectral Properties of Saddle Point Problems
Provides theoretical tools for analyzing stability and convergence of numerical methods for saddle point problems, relevant to researchers in numerical analysis and PDE-constrained optimization.
The paper derives sharp estimates for inf-sup stability constants in saddle point problems and presents spectral properties of preconditioned matrices, applied to optimal control problems with time-periodic equations. Numerical experiments with the preconditioned minimal residual method are reported.
For a general class of saddle point problems sharp estimates for Babuška'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.