Preconditioner for diagonally dominant arrow matrix Consider the block matrix
$$
M=\begin{bmatrix}A & B\\
B^{\intercal} & C
\end{bmatrix}
$$
where $A\in\mathbb{R}^{n\times n}$ is a matrix of fixed bandwidth while $B\in\mathbb{R}^{n\times c}$ and $C\in\mathbb{R}^{c\times c}$ are dense matrices. Pictorially, the matrix has sparsity pattern
$$
M=\begin{bmatrix}\star &  &  &  & \star\\
 & \star &  &  & \star\\
 &  & \ddots &  & \star\\
 &  &  & \star & \star\\
\star & \star & \star & \star & \star
\end{bmatrix}.
$$
Moreover, $M$ is diagonally dominant. I am interested in solving systems of the form $Mx=b$ where the dimension $n$ may grow quickly but $c$ does not. 
As such, I am looking for references on effective preconditioners for this and similar systems.
 A: This reference may be a good jumping off point:
http://www.mathcs.emory.edu/~benzi/Web_papers/mor.pdf
This is a fairly common type of linear system that arises in systems with constrains (like Navier-Stokes ...) and as such there are lots of ways to go about preconditioning. Typically, however, the best preconditions come from looking at the overall problem you want to solve. For instance, with Navier-Stokes, the discrete system would have $A$ representing a momentum operator of some kind, while $B \equiv Grad$, $B^{T} \equiv Div$. In this case, it's best to precondition this by using a projection method (if you're not familiar, this is just a different numerical method to solve N-S through operator splitting) as an approximate inverse. While this is very application specific, the point I'm making is that this is good! You can really make big dents in the number of Krylov solver iterations by just looking for 'approximate solution methods' to your original (possibly continuum?) problem, rather than looking at linear algebra tricks for the resulting linear system. I could try to say more if you want to say more about where the problem comes from, but the reference I gave should be a good start. Happy hunting!
