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In the present paper, we propose a preconditioned global approach as a new strategy to solve linear systems with several right-hand sides coming from saddle point problems. The preconditioner is obtained by replacing a (2,2)-block in the original saddle-point matrix A by another well-chosen block. We apply the global GMRES method to solve this new problem with several right-hand sides and give some convergence results. Moreover, we analyze the eigenvalue distribution and the eigenvectors of the proposed preconditioner when the first block is positive definite. We also compare different preconditioned global Krylov subspace algorithms (CG, MINRES, FGMRES, GMRES) with preconditioned block (CG, GMRES) algorithms. Numerical results show that our preconditioned global GMRES method is competitive with other preconditioned global Krylov subspace and preconditioned block Krylov subspace methods for solving saddle point problems with several right-hand sides.

Many applications in fluid mechanics require the numerical solution of sequences of linear systems typically issued from finite element discretization of the Stokes equations. The resulting matrices, then exhibit a saddle point structure and the iterative solution of such preconditioned linear systems is considered as challenging. The involved linear systems being large scale and ill-conditioned, effective linear solvers must be implemented. Here we proposes a direct approach with multiple right-hand sides, considering in particular the multi-frolintal massively parallel sparse solver (MUMPS), as a new strategy to improve the performance of Hermitian and skew-Hermitian precondi-tioner (P HSS) and accelerated Hermitian and skew-Hermitian preconditioner (P AHSS). The proposed preconditioners can be used as a preconditioner corresponding to the stationary iterative method or to accelerate the convergence of the generalized minimal residual method (GMRES). Several numerical experiments are run on a Linux cluster. We assess the performance of the preconditioned iterative solvers in terms of computational time and numbers of GMRES iterations. We find that the use of MUMPS to improve the preconconditioners P HSS and P AHSS is particularly efficient.

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