Consider the following linear PDE with Dirichlet boundary condition.
\begin{cases}
\mathcal{L} u = f \text{ in } \Omega \\
u = u_{D} \text{ on } \partial \Omega
\end{cases}
I want to solve this PDE many times for a ton of different sources $f_{1},f_{2},...,f_{N}$ for large $N$. An approximate Green's function to this would be nice then since then I would get a solution of
\begin{equation}
u_{n} = G * f_{n} \text{ for }n=1,...,N
\end{equation}
and where $*$ denotes the convolution operator below.
\begin{equation}
(f*g)(x) := \int_{\Omega} f(x-y) g(y) dy.
\end{equation}
Can I just solve the auxiliary problem below with finite differences, finite elements, etc. to solve for an approximate Green's function or does this run into issues?
\begin{cases}
\mathcal{L} G = \delta \approx \mathcal{L} \tilde{G} = \tilde{\delta} \text{ in } \Omega \\
u = u_{D} \text{ on } \partial \Omega
\end{cases}
where $G,\delta$ are the Green's function of interest and Dirac delta, respectively, and $\tilde{G}, \tilde{\delta}$ are their respective approximations (e.g., take a narrow Gaussian to approximation the Dirac delta).
(1) This approach seems natural to me to save computational cost, but I may be ignorant of literature, so please refer me to literature if this is a standard topic that I am simply not searching the right terms for.
(2) There may not be literature on this approach because what I describe may not be mathematically correct depending on the differential operator $\mathcal{L}$.
(2b) In particular, I am looking at $u_D=0$ and $\mathcal{L} = \nabla^2 + k^2(x)$, i.e., the Helmholtz equation. In actuality, my boundary condition is the Sommerfeld radiation condition, so I am in free space. Numerically, this can be approximated with a perfectly-matched layer, sponge layer, or some other absorbing boundary condition. In the case that $k^2$ is a constant, the Green's function can be readily constructed.
(2c) These notes by Laurent Demanet suggest that a Green's function does exist in the heterogeneous case, but there may be nuance I have missed between his explanation and my case. A few other places I looked are the Cauchy-Kowalevski theorem and Lewy's example (which I don't think necessarily is a counterexample to my case). Demanet's notes talk about using a traveltime function to approximate the Green's function, which is what made me interested in this question. Note that he is looking for an approximate solution for different velocities $k$, and I am interested in many right-hand sides for fixed $k$.