What's a general way to solve Laplace's eq. on areas with holes?
I have trouble finding references that explain how to "minus" the holes from the model. Even if one knew how to solve the eq. on a "solid".
E.g while this might serve as an okay guide for rectangular regions:
http://faculty.wwu.edu/curgus/Courses/Math_pages/Math_430/Laplace_equation_rectangle.html
then it doesn't make me understand, what if the rectangle has holes?
Is it e.g. enough to minus the domains not containing material from the domain and then compute the solution just like for the full domain, but with reduced domain? But what about, if there are BCs applying to the boundaries of the holes?
You cannot ignore the holes, as they have boundaries, and you should take into account boundary conditions. If you are very lucky, the solution on solid without holes may happen to satisfy boundary condition on the internal boundary, but in practice this never happens.
If you want a theoretical approach, then you may try to use symmetries of the system to make the domain holeless. For example, if you have rectangle with 2 holes, you can reformulate your o problem on the quarter of the domain (which is solid).
Other approach is to find the conform map that will transform your domain to somewhat simpler, although it's a lot of work if there are many holes.
Finally, for practical purposes people use computational methods. FEM, the most popular of them, doesn't mind any topology of the domain, if you provide a good mesh.