In basic lectures on finite elements theory, people always assume $H^2$ regularity of the solution in order to derive $O(h^k)$ a priori estimates in the norms $H^{2-k}$, $k=1,2$. For simplicity let's restrict to the usual Poisson equation.
I would be interested in a reference of what rates can be expected, in both norms, when the solution is just $H^1$ regular (or $H^s$ regular), and I am especially curious about approximation properties for the Ritz projection in this less smooth case.
Do you have any hints? Thanks!
Cross posted on CompSci