Factorizing a saddle point operator

Question

I have a coupled pde which, after semi-discretizing in time, results in the solution of a sequence of continuous saddle point problems $Lu^t=b$. Written explicitly, the problem looks like this:

\begin{equation} \left[ \begin{array}{cc} -\nabla^2 & \nabla \\ \nabla^T & -\nabla^2 + \alpha I \end{array} \right] \left[ \begin{array}{c} u_1 \\ u_2 \end{array} \right]^t = \left[ \begin{array}{c} b_1 \\ b_2 \end{array} \right] \end{equation}

Here, $t$ is the discrete time level of the pde.

I would like to split my continuous operator into two operators $A$ and $B$ such that

1. $A$ and $B$ are non-singular
2. One of these two matrices is block lower triangular and the other is block upper triangular
3. $A\cdot B=L$

The motivation behind this is that I'd like to decoupled the saddle point operator into two uncoupled problems.

Are there any methods to creating this kind of splitting (factorizing) for a continuous saddle operator? Any suggestions for papers to read would be greatly appreciated.