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?