Let $n,m\in \mathbb{N}$ be such that $m\ge n$.
Let $M_1\in \mathbb{R}^{n\times n}$, $\{M_{2},M_{3} \} \subset \mathbb{R}^{m\times m}$ be symmetric positive definite and computationally cheap to invert. You can think of them as a diagonal matrix. Let $S\in \mathbb{R}^{n\times m}$ be full rank. Let $B\in \mathbb{R}^m$. I am interested in solving the following linear system over a range of $\omega \in \mathbb{R}$. $$ \begin{align*} \left\lbrack \begin{array}{cc} -i \omega M_1& S\\ -S^\mathrm{T} & -i\omega M_2+ M_3 \end{array} \right\rbrack \left\lbrack \begin{array}{c} U\\Q \end{array} \right\rbrack = \left\lbrack \begin{array}{c} 0\\B \end{array} \right\rbrack. \end{align*} $$
If $M_1$ and $M_2$ were both $0$ this would be a saddle point problem and if $M_3$ was zero this would be a Helmholtz problem. I am interested in solving this equation over a range of values of $\omega$, so reusing decompositions might be useful. This system comes from a Telegrapher's equation.
I am writing to inquire if there is a body of research on these types of problem. I know the research for the saddle point problems and Helmholtz problem is pretty extensive. Any ideas or suggestions are appreciated.
Edit.
This is what I am planning on doing, without getting a better idea from here.
Writing $U= U_R + i U_I$ and $Q= Q_R + i Q_I$ and rewriting the system by equating real and imaginary parts we get
$$ \left\lbrack \begin{array}{cccc} -\omega M_1 & 0 & 0 & S\\ 0 & \omega M_1& S&0\\ 0& -S^\mathrm{T}& -\omega M_2 & M_3\\ -S^{\mathrm{T}}&0&M_3& \omega M_2 \end{array} \right\rbrack \left\lbrack \begin{array}{c} U_R\\ U_I\\ Q_R\\ Q_I \end{array} \right\rbrack = \left\lbrack \begin{array}{c} 0\\ 0\\ 0\\ B \end{array} \right\rbrack. $$
Eliminating $U_I$ and $U_R$ from these we get
$$ \left\lbrack \begin{array}{cc} \omega M_3 & S^\mathrm{T} M_1^{-1} S - \omega^2 M_2\\ S^\mathrm{T} M_1^{-1} S - \omega^2 M_2& \omega M_3 \end{array} \right\rbrack \left\lbrack \begin{array}{c} Q_R\\ Q_I \end{array} \right\rbrack = \left\lbrack \begin{array}{c} \omega B\\0 \end{array} \right\rbrack. $$
Now we have a standard saddle point problem. I plan on solving it using a Schur complement (inner/outer CG solves) method. Note that I will use the $\omega M_3$ matrices as "pivots" because there exists $\omega$ for which $S^\mathrm{T} M_1^{-1} S - \omega^2 M_2$ is singular. Having obtained $Q_I$ and $Q_R$ we can use the eliminated equations to recover $U_I$ and $U_R$.
What I don't like about this approach is that there is no reuse. I mean the method doesn't achieve any economies of scale even though it will solve many closely related problems.