Initially I thought that this is the kind of question which ought to have already been answered in the form of an example online, but so far I haven't found one. I will admit that I am very new to both FEniCS and the use of FEM software for solving PDEs in general, so my goal here in solving this problem is to a gain a better understanding of the boundary conditions (B.Cs.) used in FEM software (specifically FEniCS). Based on that I suppose my question can be broken down into the following parts:
- For isolated electrostatic systems (basically an object/objects with a non-zero charge) do we need to somehow include the B.C. that $v(r) \rightarrow 0$ as $r \rightarrow \infty$ ? If not, what B.Cs. do we choose to make sure that the FEM solution approximates the analytical solution?
- If we do need to pass the information that the potential goes to zero as $r$ goes to infinity how do we do so on a finite domain? What type of B.Cs. are we supposed to use and how do we implement them?
As far as answering this question I would appreciate anything from useful links to books/websites to explanations with equations. Ideally, I would like it if someone could show me how to implement this particular problem in FEniCS but I don't know how practical that is. Thank you for your time.
Edit: Is it possible to use something like the Boundary Element Method (BEM) to solve this type of problem more exactly? I found a software package that is written in Python that connects to FEniCS called Bempp (Link to Bempp website). A basic description of the BEM comes Wikipedia (Link to BEM Wikipedia page).