  For 2x2 block matrices, it is well-known that block-triangular or block-LDU
preconditioners with an exact Schur complement (inverse) converge in at most
two iterations for fixed-point or minimal-residual methods. Similarly, for
saddle-point matrices with a zero (2,2)-block, block-diagonal preconditioners
converge in at most three iterations for minimal-residual methods, although
they may diverge for fixed-point iterations. But, what happens for
non-saddle-point matrices and block-diagonal preconditioners with an exact
Schur complement? This note proves that minimal-residual methods applied to
general 2x2 block matrices, preconditioned with a block-diagonal
preconditioner, including an exact Schur complement, do not (necessarily)
converge in a fixed number of iterations. Furthermore, examples are constructed
where (i) block-diagonal preconditioning with an exact Schur complement
converges no faster than block-diagonal preconditioning using diagonal blocks
of the matrix, and (ii) block-diagonal preconditioning with an approximate
Schur complement converges as fast as the corresponding block-triangular
preconditioning. The paper concludes by discussing some practical applications
in neutral-particle transport, introducing one algorithm where block-triangular
or block-LDU preconditioning are superior to block-diagonal, and a second
algorithm where block-diagonal preconditioning is superior both in speed and
simplicity.
