Bayon,

What's the most appropriate/easiest way to setup the matrix for such a 
> multi-domain problem in the dealii programming environment? 
>
 
Essentially, you are solving a Laplace problem  with discontinuous 
coefficients. In case, there are no interface conditions between the 
different materials,
you should be good to just look at step-6 (and the previous tutorial 
programs of course) to begin with. For periodic boundary consitionsyou 
might want to have a look at step-45. 
 

> In an actual electric machine, the air region is very thin, resulting a 
> large aspect ratio.  What's the most appropriate way to mesh this region?
>

We provide generators for a bunch of meshes in the GridGenerator namespace. 
In you case, I would make sure that each materials are not cut by cells.
You might want to merge triangulations via 
GridGenerator::merge_triangulations. With respect to the thin air region 
anisotropic refinement might be interesting (see 
https://www.dealii.org/current/doxygen/deal.II/step_30.html)
if you can't afford a suitably (isotropically) globally refined mesh.

Best,
Daniel

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