I was taking a PHD research in engineering at a university and used an advanced method to obtain the results. But my supervisor didn’t accept my results and paper manuscripts. I then used another traditional method to complete my research. However,after graduation I immediately submitted my paper with that advanced method and it was successfully accepted and published by a famous international journal in only submission with a little bit wording revisions. My question is: would you suggest if I need to tell my supervisor about that new publication? If yes, how can I mention that? As the Christmas is coming I may send a seasonal greeting to that supervisor. Thank you for your help! Cheers, Donny.
If you're in the same field, I'm sure your previous supervisor will come across it anyways. I think relations with this colleague would be best if you proactively share it with them. Of course there are strong preferences and stylistic tendencies in analytical methods -- as long as your email doesn't take an "i told you so" tone, then I think it is a good idea to contact them.