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.
What is a good way to solve the following linear system? (repeatedly)
linear algebra;linear solver;helmholtz equation