After the green bar reaches the end, the analysis stops. But why can't it go forever? (for more accuracy)!
Because, while local analysis uses your computer's resources, it still uses Lichess's engine, and it takes resources. If you want an infinite analysis using your computer, try downloading stockfish and getting a GUI such as Arena Chess.
Hatrix, you are wrong.... Local analysis uses Stockfish 7 ported to javascript, running locally. Have a look here if you're interested ;-)
fr.lichess.org/assets/vendor/stockfish7.js