What if your advisor does not know anything about your research field? I  am a PhD student in Mathematics. I have found one advisor in my local University who does not know my research field at all, but  has allowed me to choose my own problems and solve them independently.
Initially, I was quite happy as I didn't need to switch places (which I don't want at all). The problem started when my advisor could not assist me in solving the problems I chose. It even happened that I chose a problem to work on and when I told him about it, he said that's an excellent problem and I should solve it.Later on after meeting another professor  in this field he told me that it has been solved long before and hence I had no point working on that again.
He does listen to me when I say something about the problems but fails to give me any critical feedback in my topic. I searched the web but there are so many directions of the topic I am working on, I am feeling puzzled which way to go. Moreover there is no research group existing in this field in my local University with whom I can discuss or get some inputs abut my problems.
I am unable to answer the following questions of mine which is affecting my research considerably:

1.Has my decision to stay in my hometown and not go to reputed institutes and talking to experienced professors backfired on me? How can I come out of this situation?
2.Is  PhD done in this way only without any help from anyone ?At least my advisor says so.

Any comments/advice will be helpful.
Well, then go and find another advisor. Do not get me wrong, I don't mean a new one, I just mean someone else to talk to. With a PhD, you are supposed to show that you are able to do independent research, that you are able to do more than just work on minor problems that dropped out of some paper/publication by your advisor (as it is often the case in Bachelor or Master thesises).

I think a PhD (at least in Mathematics, I don't know that much about other fields) where you only worked with a single advisor for multiple years is really uncommon. Go find other people, other professors, etc. to talk to.
Your advisor should help you with the following points in case you get stuck there (list might not be complete...):


How and where to find research material, maybe giving you access to institute libraries that are normally closed to regular students or access to the universities subscriptions to online journals.
Establishing contacts to other researchers on the field, giving you advice on who to talk to, maybe even recommending you for a research stay of a few months at another university for a project related to your research.
Administrative processes. That also includes questions on "How and where to best publish this result I came up with?" or "Which conferences should I visit, should I maybe even present something there?"


During a PhD, it is not the advisors job to do your research. You yourself have to decide what you look into, what you do, how much time you spend on one problem until you deem it too hard and move on, etc. If your professor could answer all your questions, then that would mean that your research is not really original and/or rather trivial and thus might not yield a PhD (or at least not an excellent one, and I'm sure you are aiming for that :) ).

So don't worry if the professor can't help you solve your problems. Maybe they will turn up to stay unsolved for centuries, who knows? Try to rephrase your questions, don't go and ask "How can I solve this?" but rather "Do you know of experts on this topic that might be willing to talk to me (maybe even at the same university)? Can you put us into contact?" or maybe asked more generally like "Do you have a suggestion on which direction to look from here on? Should I continue down this path or maybe try a different approach?".

So long story short: Even if the professor knew your research field, he would not be able to always assist you with your problems.

PS: As I am also writing a PhD in math just now I'm a little curious: What is your research field, for which there are no research groups and your professor has no idea about? :)