A note on $2\times 2$ block-diagonal preconditioning

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.