A Chebyshev-based two-stage iterative method as an alternative to the direct solution of linear systems

We consider the solution of ill-conditioned symmetric and positive definite large sparse linear systems of equations. These arise, for instance, when using some symmetrizable preconditioning technique for solving a general (possibly unsymmetric) ill-conditioned linear system, or in domain decomposition of a numerically difficult elliptic problem. We are also concerned with the consecutive solution of several linear systems with the same matrix and different right-hand sides. In such cases, the consecutive runs of some iterative methods like the conjugate gradient or the block conjugate gradient algorithms might be computationally prohibitive, and it might be preferable to use direct methods which are very well suited for this type of situation.