Augmentation block preconditioners for saddle point‐type matrices with singular (1, 1) blocks

We consider the use of block preconditioners for the application of the preconditioned Krylov subspace iterative methods to the solution of large saddle point-type systems with singular (1, 1) blocks. Two block triangular preconditioners are introduced and the block diagonal preconditioner in Greif and Schotzau (Electron. Trans. Numer. Anal. 2006; 22:114–121) is extended to nonsymmetric saddle point systems. All these preconditioners are based on augmentation, using nonsingular weight matrices. If the nullity of the (1, 1) block takes its highest possible value, the preconditioned matrix with either block triangular preconditioner has precisely three distinct eigenvalues, and the preconditioned matrix with the block diagonal preconditioner has precisely two distinct eigenvalues, giving rise to immediate convergence of preconditioned GMRES. Finally, numerical experiments that validate the analysis are reported. Copyright © 2008 John Wiley & Sons, Ltd.

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