Suppose I have a linear system
$$ \left\lbrack \begin{array}{cc} M_1& S\\ S^{\mathrm{T}}& M_2 \end{array} \right\rbrack \left\lbrack \begin{array}{c} X\\ Y\end{array} \right\rbrack= \left\lbrack \begin{array}{c} F\\G\end{array}\right\rbrack, $$ where $M_1\in \mathrm{R}^{n\times n}$ is symmetric positive definite, $M_2\in \mathrm{R}^{m\times m}$ is symmetric positive semi-definite. $S$ is a full-rank matrix of compatible size. $X,Y,F,G$ are all vectors of appropriate size.
What factors are relevant in deciding between using GMRES on the big system and using a Schurr complement technique?
Factors that I would guess are relevant are how expensive $M_1$ is to invert, the condition number of $S$, and whether $M_2$ is the zero matrix.
Any links to the literature are appreciated.
Edit
I am asking this question because I need to solve the following system.
$$ \left\lbrack \begin{array}{cccc} M_1& S &0 &\Gamma^{\mathrm{T}} \\ -S^{\mathrm{T}}& 0 &\Xi^{\mathrm{T}}& 0\\ 0 & \Xi & 0 &0 \\ \Gamma&0&0&0 \end{array} \right\rbrack \left\lbrack \begin{array}{c} x_1\\x_2\\x_3\\x_4 \end{array} \right\rbrack= \left\lbrack \begin{array}{c} f_1\\f_2\\f_3\\f_4 \end{array} \right\rbrack $$
Right now, I solve this system using the Schurr complement method but it requires three iterative CG solves. I am wondering if it would be better to just "brute force it" and use GMRES.