Stable element pair for Navier - Stokes

1

Hi,

This question is a bit more theoretical, I hope it is ok. I have met some stability issues using Taylor-Hood elements in Fenics for Navier-Stokes with a small parameter.

The Fenics tutorial on Stokes equations using Taylor-Hood elements ( https://fenicsproject.org/docs/dolfin/1.4.0/python/demo/documented/stokes-taylor-hood/python/documentation.html) states that the Taylor-Hood element pair for the Stokes case is stable. On the other hand, the tutorial on Navier-Stokes also uses Taylor-Hood elements, but here in the nonlinear case there seems to be no comment made on whether P(2)-P(1) is a stable choice here.
Is the T-H element pair stable for the Navier-Stokes equations? Or it is used in the tutorial just as a “reasonably good” choice, but not necessarily a stable one?
Is there a better, proven, stable element choice for Navier-Stokes that, at least theoretically, guarantees well-posedness?

Thank you so much!

2

I recommend reading the following paper by Langtangen, Mardal, Winther.

It gives an overview over solution techniques for Navier-Stokes, and contains many good references.

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Thank you, it did have an estimate result that helped!