# Convergence rate for not smooth solution with classical $P^1$ Lagrangian FEM

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.

• So, are you trying to obtain a function that is not $H^1$ using $H^1$ bases? Jul 31, 2022 at 18:55
• Uhm, yes. To be honest, my original problem was well posed in $H^1$. Then I tried with a not smooth solution, and I expected the error to decrease (slower, of course), not to increase. Jul 31, 2022 at 20:25
• I mean, I should not expect it to increase, right? Jul 31, 2022 at 20:25
• If $f\in L_2$, then $u\in H^1$. What kind of right hand sides are you considering that your solution is not in $H^1$? Aug 1, 2022 at 17:20
• I think in this 2D case, point evaluation is not well defined by the Sobolev embedding theorem. With standard P1 finite elements that employ point evaluation in the functionals, you should run into problems. If the regularity $u \in H^s$ fulfills $s > 1$, you should get a proper solution such that $u\in L_\infty$ and point evaluation is well defined.
– Pepe
Aug 3, 2022 at 23:33