Preconditioners for Generalized Saddle-Point Problems

\noindent 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.