reference request: elliptical regularity in compact manifold without limit

Leave $ (M, g) $ be a compact Riemannian collector without limit, and $ Delta $ is the operator of Laplace-Beltrami in $ M $. Is there any result in elliptical regularity like this:

For any $ u in H ^ 1 (M) $Y $ f in L ^ 2 (M) $ such that $ Delta u = f $ (in the sense of distributions), then $ u in H ^ 2 (M) $.
If there is a good reference for that result of regularity, it would be good.