Consider a $2\times 2$ block matrix and a linear system of equations associated to it:\begin{equation} \begin{pmatrix} - A & B \\ B^t & C \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} \phi \\ \psi \end{pmatrix}\end{equation}Assume that $A$ and $C$ are symmetric, and symmetric positive semi-definite, and that $A$ is even invertible. One can construct the Schur complement system$(C + B^t A^{-1} B ) y = \psi - B^t A^{-1} \phi$where $S = C + B^t A^{-1} B$ is symmetric positive-definite, by assumption.How do you precondition such a system in general? I have noted that preconditioners for $S$ are derived from block matrix preconditioners for the original block matrix. Is there a general consensus how such a block matrix preconditioner looks like?
Preconditioning symmetric Schur complement
preconditioning;block decomposition
You won't find any black-box scalable solutions because $S$ is typically dense and thus cannot be formed. If your problem comes from a mature research area, there might be experience in the literature demonstrating how to approximate Schur complements. One common technique is to use approximate commutator arguments. There are a quite a few papers on this topic by Elman and others for incompressible flow.  You can also see Benzi, Golub, and Liesen, Numerical Solution of Saddle Point Problems (2005) for a (slightly dated) broader review.