The spectral properties of the Hermitian and skew-Hermitian splitting preconditioner for generalized saddle point problems

In this paper, we consider the Hermitian and skew-Hermitian splitting (HSS) preconditioner for generalized saddle point problems with nonzero (2, 2) blocks. The spectral property of the preconditioned matrix is studied in detail. Under certain conditions, all eigenvalues of the preconditioned matrix with the original system being non-Hermitian will form two tight clusters, one is near (0, 0) and the other is near (2, 0) as the iteration parameter approaches to zero from above, so do all eigenvalues of the preconditioned matrix with the original system being Hermitian. Numerical experiments are given to demonstrate the results.