## Preconditioned global Krylov subspace methods for solving saddle point problems with multiple right-hand sides

### Abstract

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 $\mathcal{A}$ 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.

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### Key words

global Krylov subspace method, GMRES, MINRES, CG, preconditioner, saddle point problem

### AMS subject classifications

65F10, 65N22, 65F50

### Links to the cited ETNA articles

 [11] Vol. 16 (2003), pp. 129-142 A. El Guennouni, K. Jbilou, and H. Sadok: A block version of BiCGSTAB for linear systems with multiple right-hand sides [15] Vol. 20 (2005), pp. 119-138 K. Jbilou, H. Sadok, and A. Tinzefte: Oblique projection methods for linear systems with multiple right-hand sides

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