Design the preconditioner for time-dependent Navier Stokes equation

1

Hi,
I tried to design the preconditioner for time-dependent Navier Stokes equation. In the tutorial about the stokes equation.
https://fenicsproject.org/docs/dolfin/latest/python/demos/stokes-iterative/demo_stokes-iterative.py.html
The preconditioner has a form like this

Form for use in constructing preconditioner matrix

b = inner(grad(u), grad(v))dx + pq*dx
However, it seems not appropriate for the generic time-dependent Navier Stokes equation. How can I extend this form to apply it to the time-dependent Navier Stokes equation?

2

See for instance this master thesis

3

It seems the calculated preconditioner matrix is time independent. It depends on test function and trial function you choose. So, you do not need to change it at every time step.