I'm using classical $P^1$ finite elements to solve $- \Delta u = f$ with Dirichlet BC in a 2D domain $\Omega$.
I know from theory that the solution is not in $H^1$ for my particular choice of $f$, so all classical convergence estimates are useless. If I compute the $L^2$ error against the analytical solution indeed I don't see the correct slope, as expected. However, after I reach a certain number of DoFs the errors start increasing.
The rates are the following:
dofs L2 81 3.214e-02 289 1.735e-02 1089 4.777e-03 4225 2.633e-03 16641 1.330e-03 66049 9.378e-04 263169 1.256e-03 1050625 1.315e-03
As you see, the $L^2$ error at some point starts growing. Can this be justified by the low regularity of the solution?
If I take a smooth solution, I have correct rates as I refine the grid, and I don't observe this phenomenon.