Add demo of stokes with various stable pairs of finite elements

[juliusgh]

There is already a demo of the Stokes equations for the lid driven cavity scenario with Taylor-Hood elements: https://docs.fenicsproject.org/dolfinx/main/python/demos/demo_stokes.html

Based on this, I have created a demo using various stable pairs of finite elements. So far these are:

1. $(\mathcal{P}_2, \mathcal{P}_1$): The Taylor-Hood element for $k=2$
2. $(\mathcal{P}_1 + \mathcal{B}_3, \mathcal{P}_1)$: The MINI element
3. $(\mathcal{P}_1^{\rm CR}, \mathcal{P}_0)$: The non-conforming Crouzeix-Raviart element
4. $(\mathcal{P}_4, \mathcal{P}_3^{-1})$: The Scott-Vogelius element for $k=4$

With the MINI element, an enriched finite element is also used in a demo now. Such a demo was requested in #2312.
I think this demo would be helpful for beginners who want to try new finite elements.
Since this is my first contribution to dolfinx I would appreciate any feedback.

[jhale]

I like this! My only comment would be that it would be good to show the lowest-order pressure-robust Scott-Vogelius finite element pair.

[jhale]

Thanks! The Scott-Vogelius element is only (provably) int-sup stable for k = 4 ( https://doi.org/10.1051/m2an/1985190101111) so I would propose using a higher-order for the demo.

[juliusgh]

Thank you for the feedback! I added the Scott-Vogelius element using the iterative procedure explained in the FEniCS book in chapter 20.3.
I am not completely sure about the reconstruction of the pressure field. Is there a built-in project function in DOLFINx like in DOLFIN? I could not find it so far and used the code from https://github.com/michalhabera/dolfiny/blob/master/dolfiny/projection.py instead.
In the meantime, I also noticed that the Scott-Vogelius element is not stable for $k<4$, since the iteration does not converge in this case.

[nate-sime]

Does this demo exhibit any dolfinx functionality that's not already shown in other demos?

E.g. https://github.com/FEniCS/dolfinx/blob/main/python/demo/demo_lagrange_variants.py and https://github.com/FEniCS/dolfinx/blob/main/python/demo/demo_stokes.py?

Also consider https://github.com/jorgensd/fenics22-tutorial/blob/main/comparing_elements.ipynb.

[juliusgh]

Does this demo exhibit any dolfinx functionality that's not already shown in other demos?

E.g. https://github.com/FEniCS/dolfinx/blob/main/python/demo/demo_lagrange_variants.py and https://github.com/FEniCS/dolfinx/blob/main/python/demo/demo_stokes.py?

The goal of this demo was to show an EnrichedElement in action, as requested in #2312. As far as I can tell, there is no demo that shows this yet. The MINI element for the Stokes equations is a simple example of this, which I think would be helpful for inexperienced DOLFINx users.

While the existing Stokes demo focuses on different algebraic solvers, different finite element types are compared here.

Also consider https://github.com/jorgensd/fenics22-tutorial/blob/main/comparing_elements.ipynb.

Thanks for the tip! I did not know this tutorial before. I think the convergence studies are very informative. I would love to extend this demo accordingly.

[nate-sime]

For every demo added, there's a burden of maintenance for the developers. This demo only shows element enrichment which has overloaded the + operator which at a glance is about 3 of 529 lines. I vote for incorporating this into one of the existing demos if it's really required.

Otherwise I think the rest of the demo is an interesting report of various numerical methods which would be better hosted as part of a tutorial or personal portfolio.

[garth-wells]

Remained open for some time, so closing.