Really? Can you expand on what parts are tricky to get right? I would have thought for simple 2D problems like this (with simple geometry and where O(N^3) methods are fine so that you don’t need iterative solvers or FMM) you can just mesh the domain yourself, setup the integral equation and discretize it brutally. At least that seems like a fun learning exercise (although less flexible for more complicated geometries, I agree).
(Also, since the Green function for the Poisson equation in the disk is known analytically, you can possibly even not have to mesh the circle?)
The Green’s function diverges at the origin, so you have to be careful with the discretization scheme (e.g. using specialized quadrature schemes), among other subtleties (e.g. choosing first- vs. second-kind formulations).
It’s all written out in great detail by many references for the Poisson case, of course, so if you follow the right textbook (I liked Bonnet’s book IIRC, though it is a bit old now) you should be fine. But even then there is a lot of material to digest.
Upon looking around I’ve encountered Home · Meshes.jl which seems like what I need for the geometry — Julia based, native julia plotting. I wonder if there is anyone who successfully used Meshes.jl with Gridap.jl.
Also, Ferrite.jl (JuAFEM.jl in the past) seems to be a good candidate for this sort of problem, though I can’t comprehend the docs as it clearly assumes some prior knowledge that I’m lacking