I'm simulating the time-dependent Navier--Stokes equations using a Taylor-Hood finite element scheme:
$$\dfrac{\partial u}{\partial t}-\nu\Delta u+(u\cdot\nabla)u+\nabla p=f$$ $$\nabla \cdot u=0$$
The example is in Section 4.1 of the following paper, with Re=1
https://www.wias-berlin.de/people/john/ELECTRONIC_PAPERS/JMR06.CMAME.pdf
I have not experience in time-dependent problems, and that is why I have problems interpreting the numerical results.
For short I do not show the exact solution here, but I'm using the one in a paper (velocity with divergente equal to zero and integral of the pressure equal to zero).
In the figures, from top to bottom, I show the pressure in the first time-step (t=0.1s), a middle time-step and last time-step (t=1s). The simulation starts in t=0s. In the right are the exact solution (pressure), and in the left is its corresponding numerical solution (also pressure).
As you can see, in the beginning (t=0.1s) of the simulation the pressure is far to the exact solution, but in the final time (T=1s, the bottom figure) the numerical result is very close to the exact solution.
The plot of the numerical velocity is very close to the exact solution, and that is why I do not show here. That is what disturbs me the most: the velocity graph is perfect at every instant of time, but the pressure graph starts far from what it should be but as the simulation progresses it seems to converge to the exact pressure.
My question is: Is it okay that the pressure is not very close to the exact one in the first moments of the simulation?
I tested with Newton and Picard method, and 4 types of theta-schemes for the time discretization (Backward Euler, Crank-Nicolson, FS0 and FS1) and I got the same numerical results.
In your experience, what may be happening?
Looking at the paper, the results seem to be similar, that is, as the "time step" increases the error decreases. I think this is because the "time step" on the x axis is a misnomer, since actually it is defined just before the plots that $\Delta t = 2^{-k}$ for $k=0,\ldots, 8$. That is, as k increases, the time increments are refined, so it makes sense that the error would decrease. I think the time step axis should be labelled k.