I'm reading the following paper
https://www.wias-berlin.de/people/john/ELECTRONIC_PAPERS/JMR06.CMAME.pdf
where calculates the solution of the time-dependent Navier--Stokes equations using different $\theta$-Schemes for the time, and finite element method in each time step (Taylor--Hood). My question is related with BWE (backward Euler) and CN (Crank--Nicolson). If we see the picture that I attach (Figure 2 of Section 4.1 of the paper) the error in the pressure using BWE is less than the error using CN, Is this possible? I understand that the CN method is better, so I would expect the pressure error to be lower using CN. I think it is ok, but I would like to know some explanation.
My second question is: BWE is order 1, and CN is order 2, but I see that in the pressure they have the same order. Why I can't see the order or the errors in these figures?
We can see the same behaviour in all the numerical tests, but I only attached one of them.
Typical error estimates for these algorithms are based on worst-case scenarios. However, on a specific problem, it's usually hard to predict which algorithm will perform the best.
As for CN behaving order 1 instead of order 2 -- my guess is that, if we check the list of assumptions in the theorem which guarantees order 2 for CN, one of them likely doesn't hold here.