Regularized HSS Iteration Method for Saddle - Point Linear Systems

We propose a class of regularized Hermitian and skew-Hermitian splitting methods for the solution of large, sparse linear systems in saddle-point form. These methods can be used as stationary iterative solvers or as preconditioners for Krylov subspace methods. We establish unconditional convergence of the stationary iterations and we examine the spectral properties of the corresponding preconditioned matrix, showing that the eigenvalues are clustered near 0 and 2 when the iteration parameter is close to 0. Inexact variants are also considered. Numerical results on saddle-point linear systems arising from the discretization of a Stokes problem and of a distributed control problem show that optimal convergence behavior can be achieved when using inexact variants of the proposed preconditioners.