A note on block diagonal and constraint preconditioners for non-symmetric indefinite linear systems

We study block diagonal preconditioners and an efficient variant of constraint preconditioners for general two-by-two block systems with a zero (2,2)-block. These block diagonal preconditioners are obtained by a sign change of the (2,2)-block of the block diagonal preconditioners constructed previously [De Sturler, E. and Liesen, J., 2005, Block diagonal and constraint preconditioners for nonsymmetric indefinite linear systems. Part one: Theory. SIAM Journal on Scientific Computing, 26, 1598-1619]. We analyse the properties of the corresponding preconditioned matrices, in particular the spectrum, and discuss the computational performance of preconditioned iterative methods. Numerical experiments of a model Navier-Stokes problem are presented.