Ah poor you, I remember those awful first few weeks when you are going insane with the lack of sleep and worrying things are going wrong.
Firstly, it is a good sign she is feeding so well, so that is great news. And secondly, I would use GNs advice above - trying to make a difference between night and day so she kind of understands it. It takes a long time but will pay dividends later on. No need to be uber-strict, just say 11pm-7am is quiet, dark, dull. Daytime is fun! Most babies don't get their long stretch of sleep at night time until about 10 weeks at the very very earliest. (although someone will always say 'my baby slept through from 3 weeks' - just punch them ) Don't want to scare you just reassure that this is totally normal behaviour for a 3 week old. It will settle down, but it feels like forever in the mean time.
Early babies often take longer to settle into less frequent feedings and more sleep, not sure why (there are loads of studies about it, both my sister and i had early babies so have read and discussed it alot!).
Most importantly, just try and get some rest whenever you can, even if its lying next to her. I know it's boring but it really helps you at least, and your milk supply.
Good luck, and congratulations. This board has been a great support to me, hope you find the same!