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.
Most of the problem solve independently, advisors can only advice you about the research that how to conduct it, he provides feedback, make critical analysis of your work, but evaluation of the research purely dependent upon your knowledge that you will provide to your supervisor. Certainly, Phd work is a kind of novel work and your supervisor also poses same knowledge that you have, that you will provide to him.
The problem started mid way 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 after meeting another professor in
this field he told me that it has been solved long before.Hence I had
no point working at that again.
Well in current age of science and information, you selected a topic that already covered Long before. Before selection of the topic, why did not you conduct a comprehensive study or search? Did you make any discussion with your adivsor? Most of the advisor can't do this job for you that they search novel topic for you. This is your mistake that you didn't make the selection of novel a topic. It is expected from a PhD student that he know preliminaries of the research.
He does listen to me when I say something about the problems but fails
to give me any critical feedback in my topic.
If this statement is true then, surely there is something wrong with the advisor. No matter that you don't know about a sub-topic of your field but you can make critical analysis of it. Even a common PhD student or researcher can do this in his field and this behavior expectation become increase for advisor. surely something wrong with advisor.
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.
You should need to do this work earlier before selecting the topic.
The problems are cropping up now after 1 year,Initially all was well
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
Internet, Google, Social Media, forums, are your friends, always ready to help you. Search expert of your fields on social media, make them friend/follow, get advice from them. Its easy now.
Has my decision to stay in my hometown and not go to reputed
institutes backfired on me?
You should travel to seek knowledge. This is common and very helpful also.
How can I come out of this situation?
Review your topic again, check its novelty and evaluate it yourself. If it has already solve then your methods can make it more efficient or can make any significant difference? Make questions, get technical feedback from advisor. If he satisfy you then, continue otherwise quit. BUT also tell him all the story that why are you quitting maybe he will advice you a lot better than this forum and my answer. He is advisor surely poses more knowledge and wisdom of his field than us.
ask is Is a PhD done in this way only without any help from the
No help of advisor is necessary but i don't know what are you getting from the meaning of "help". Mostly the adviosr don't bother for you to search novel topic and conduct any comprehensive study for you. you should do it yourself and get feedback from him.
Would the situation been better if I had chosen an experienced
certainly, but again its dependent on you also.
Note: I am not a PhD student but I conducted some of the research recently write a few papers and currently doing MS thesis in computer science. The answer based on my last 2/3 year experience in research and relationship with my supervisor. And the fact i get to know during this short period that you have to do everything yourself. Supervisor make critical and helpful comment on your work. And assist you if you are caught in any problem.