Solving sub-matrix preserving overall linear system I have a linear system
$$
\begin{pmatrix}
A_{11} \ A_{12} \ A_{13}\\
A_{21} \ A_{22} \ A_{23}\\
A_{31} \ A_{32} \ A_{33}\\
\end{pmatrix}
\cdot
\begin{pmatrix}
x_1 \\ x_2 \\ x_3
\end{pmatrix}
=
\begin{pmatrix}
b_1 \\ b_2 \\ b_3
\end{pmatrix}
$$
which comes from the discretization of a system of PDEs. The overall system is large, non-symmetric and contains a saddle point problem.
Let's assume I know something about the sub-matrix $A_{2,2}$, which I can use to solve the sub-matrix more efficient. For example $A_{2,2}$ might be a diagonal matrix or symmetric or I have a suitable pre-conditioner for it.
What can I do with this knowledge. Can I precondition or even solve $A_{2,2}$ (replacing $A_{2,2}$ by $I$ and adjusting $b_2$)? How does this affect the other parts of the matrix or the right hand side?
Edit: Additional ideas: $A_{ij}$ for $i\ne j$ (non-diagonal blocks) are very sparse and are only for coupling the systems of equations. Can there be made a connection to Block Jacobi methods / preconditioning which kind of tries the same?
 A: I know I'm giving you bad news, but what you in fact have is this (where obviously $A_{jk}$ are block matrices):
$$A_{11}x_{1}+A_{12}x_{2}+A_{13}x_{3}=b_{1}$$
$$A_{21}x_{1}+A_{22}x_{2}+A_{23}x_{3}=b_{2}$$
$$A_{31}x_{1}+A_{32}x_{2}+A_{33}x_{3}=b_{3}$$
Now...unfortunately if you don't have any other restriction I guess that's as far as you can get. If you other symmetries maybe you can go further. The fact that you solved the block matrix $A_{22}=b_2$ unfortunately doesn't help you much. Indeed is more or less like knowing that $y=7$ doesn't help you in find $x,z$ in the equation $$x+y+z=7$$
I'm sorry but I hope this answer was anyway useful telling to look other directions... 
A: The most basic preconditioner for 
$$A=
\begin{pmatrix}
A_{11} \ A_{12} \ A_{13}\\
A_{21} \ A_{22} \ A_{23}\\
A_{31} \ A_{32} \ A_{33}\\
\end{pmatrix}$$
is a Jacobi (or diagonal) preconditioner, i.e. $\mathrm{diag}(A)$. 
Now if you have suitable pre-conditioner for $A_{22}$, say $P_{22}$, then I would try to use the following preconditioner 
$$P=
\begin{pmatrix}
\mathrm{diag}(A_{11}) & 0 & 0\\
0 &P_{22} & 0\\
0 & 0 & \mathrm{diag}( A_{33})\\
\end{pmatrix}$$
If your matrix is block n-diagonal, then I would suggest to read the following thing. I didn't read it all, the name is block-matrix preconditioning, but at first pages looks like they talking about a block 3-diagonal. Perhaps later they generalize it. For a PDE discretization you get such matrices, so hopefully this is what you need.
