Constrains in FEniCSx

General
1

HI all!

I would like your opinion on the following mathematical problem. Consider the Poisson Equation in 3D: {\nabla ^2}{\bf{u}} = 0 with Dirichlet Boundary Conditions but subject to “physical” constrains, i.e., {u_x} > 0 for any x in the geometry of interest or \nabla \cdot {\bf{u(x)}} = 0 etc.

Is there a way to implement such feature in a FEniCSx calculation? Are, for example, Lagrange multiplies or some Augmented Lagrangian Method implemented?

Thanks in advance!!!

2

This is heavily problem and discretisation dependent.
See, e.g. the stokes demo

3

Thanks for the link, I somehow missed the Stokes tutorial!

Correct if I am wrong though (I haven’t kept up with hydrodynamics for 15 years) the simple Stokes basic set of equations is 4, 3 coming from - {\nabla ^2}{\bf{u}} + \nabla p = {\bf{f}} and 1 from \nabla \cdot {\bf{u}} = 0 to be solved in a proper geometry and accompanied with boundary conditions with {\bf{u}} = (u_x,u_y,u_z), p the four unknown quantities.

I was trying something somewhat different to solve - {\nabla ^2}{\bf{u}} = 0 while the constraint \nabla \cdot {\bf{u}} = 0 (or i.e., |{\bf{u}}|>2 etc.) holds, therefore I have only 3 unkown quantities. Do you know if I can still apply the method presented in the tutorial.