TITLE: Elliptic regularity on compact manifold without boundary
QUESTION [9 upvotes]: Let $(M,g)$ be a Riemannian compact manifold without boundary, and $\Delta$ is the Laplace-Beltrami operator on $M$. Is there any result on the elliptic regularity like this:
For any $u\in H^1(M)$, and $f\in L^2(M)$  such that $\Delta u = f$ (in the sens of distributions), Then $u \in H^2(M)$.
If there is a nice reference for such regularity result It would be good.

REPLY [2 votes]: Expanding @Piotr Hajlasz's answer, the result it true, and does appear in Warner's  Foundations of differentiable manifolds and Lie groups. The result for the Laplacian appears as Theorem 6.32. Theorem 6.30 describes the local theory, then in 6.32 Warner describes how to apply the local theory to get a result for the whole manifold. 
Warner does the local theory for periodic functions in $\mathbb{R}^n$ (with period $2\pi$). Then he uses charts which map to subsets of the $2\pi$ cube, and extends the result to the whole cube using freezing coefficients (essentially set the coefficients in your linear operator equal to constants outside some open set). 
[Side notes: (1) Theorem 6.32 generalises easily to other elliptic second-order linear differential operators. (2) Warner doesn't discuss boundary points and boundary conditions, but Taylor does in Partial Differential Equations I, chapter 5.]