Spectral properties of the Hermitian and skew-Hermitian splitting preconditioner for saddle point problems. (English) Zbl 1083.65047

The authors consider the saddle point problem \[ \begin{pmatrix} A & B^T\\ -B & 0\end{pmatrix} \begin{pmatrix} u\\ v\end{pmatrix}= \begin{pmatrix} f\\ -g\end{pmatrix}, \]
\[ \text{or}\qquad (H+S)x= b\quad\text{with}\quad H= \begin{pmatrix} A & 0\\ 0 & 0\end{pmatrix},\quad S= \begin{pmatrix} 0 & B^T\\ -B & 0\end{pmatrix}, \] where \(A\in\mathbb{R}^{n\times n}\) is symmetric positive semidefinite, \(B\in\mathbb{R}^{m\times n}\), \(f\in\mathbb{R}^n\), \(g\in\mathbb{R}^m\), \(\text{rank}(B)= m\leq n\), and \(A\) nonsingular. A Hermitian/skew-Hermitian preconditioner \(P= {1\over 2\alpha}(H+\alpha I)(S+\alpha I)\) is studied in a generalized case by M. Benzi and G. H. Golub [SIAM J. Matrix Anal. Appl. 26, No. 1, 20–41 (2004, Zbl 1082.65034)].
In the present paper bounds and clustering results on the eigenvalues of the preconditioned matrix that arise in \((H +S)= \eta Px\), \(\alpha> 0\), are provided. For small \(\alpha\) all the eigenvalues are real and are separated in two clusters, one near \(0\) and the other near \(2\). \(\alpha\) should be selected small enough in order to get clustering but not too small that the preconditioning matrix is close to being numerically singular. The results are an important property related to the convergence of preconditioned Krylov subspace methods. Numerical experiments demonstrate the quality of the bounds. It seem to be that when \(A\) is positive definite, \(\alpha\approx \lambda_n\) (\(\lambda_n\) smallest eigenvalue of \(A\)) is a good choice.


65F35 Numerical computation of matrix norms, conditioning, scaling
65F10 Iterative numerical methods for linear systems


Zbl 1082.65034
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