Hi Pascal: I would not interpret the convergence graphs to necessarily mean more accuracy. It just means the numerical algorithm has converged to the tolerance of the criterion that you set, and stopped going any further. Usually this is accomplished via mesh refinement or polynomial order (I'm assuming you're using the h- or p-adaptive methods for a static analysis). The accuracy really depends if the physics (such as material properties, manufacturing/machining tolerances, boundary conditions, and so on) are well-represented. If there is a stress singularity, there will be a problem in the final answer. If the loads and reactions don't exactly balance, there is an error. These errors can occur even if the algorithm has converged. As a global check, it is good to look at the total strain energy and see if that's equal to the work done by external loads.
In any case, I usually resort to conducting the convergence check by hand, comparing answers of run 1, to those for run 2, 3,...n, and so on, until I get answer (n-1) within 10% or so of answer n. Compare that answer to a rough hand calculation or textbook approximation. Then, if you can find or conduct a lab test: compare to that, too. Results in FEA are considered quite good if you can get within 5 or 10% of a laboratory test.
The take-away is not to focus on the automatic convergence algorithms, but just check if the results make sense according to any other references you can find.
As far as the normalization - I'm not sure I have seen that one. I have seen where they graph the change in something, such as total strain energy, or displacement, or stress, or estimated stress error. Perhaps they are normalizing with respect to the first value obtained (as a ratio).