I am passionate about working on a particular problem and domain, and have an opportunity to complete a PhD that does not require department/university subsidy. However, the problem I am interested in is outside of the domain of the research faculty. I have been encouraged to find a small part of my problem space that fits with the research of available advisors. However, I feel like conducting my research at another institution, where the research faculty are actively working in the same domain as myself, may be a better choice.
What are the benefits and drawbacks may be to work with advisors directly related to one's own area of research, rather than trying to just fit one's research into the scope of available advisors' research interests?
There is some discussion of this regarding the master's thesis, but I feel that the doctoral level is a little different, in that the advisor/advisee relationship is perhaps more important.