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