Preconditioners for generalized saddle-point problems

We propose and examine block-diagonal preconditioners and variants of indefinite preconditioners for block two-by-two generalized saddle-point problems. That is, we consider the nonsymmetric, nonsingular case where the ( 2,2) block is small in norm, and we are particularly concerned with the case where the ( 1,2) block is different from the transposed ( 2,1) block. We provide theoretical and experimental analyses of the convergence and eigenvalue distributions of the preconditioned matrices. We also extend the results of [ de Sturler and Liesen, SIAM J. Sci. Comput., 26 ( 2005), pp. 1598 - 1619] to matrices with nonzero ( 2,2) block and to the use of approximate Schur complements. To demonstrate the effectiveness of these preconditioners we show convergence results, spectra, and eigenvalue bounds for two model Navier - Stokes problems.