Eigenvalue Estimates for Preconditioned Nonsymmetric Saddle Point Matrices

Abstract

We present a valid lower bound for the real eigenvalues of a nonsymmetric saddle point matrix even if its $(2,2)$ block is singular. This bound, together with bounds introduced by Benzi and Simoncini in [Numer. Math., 103 (2006), pp. 173–196], is applied for the analysis of symmetric indefinite preconditioners and primal-based penalty preconditioners. Eigenvalue estimates for the Hermitian and skew-Hermitian splitting preconditioned saddle point matrices are also derived. The model problem of Stokes equations is used to illustrate the presented theoretical results.