Demo 18 - Are there enough boundary conditions?

Question

Demo 18 illustrates how to solve an incompressible flow problem in a mixed formulation, introducing pressure as another variable which is solved simultaneously with velocity.

Although another variable must be solved, the number of boundary conditions does not change from the compressible version of the problem.

Is this a well-posed problem? Isn't it necessary to pin pressure at one point, at least?

My best guess is that the second equation of the problem, div(u) = 0, stays in strong form, so in a sense it acts as a Neumann BC.

Comments

You have Neumann conditions at the outflow (x0 = 0), this is enough to ensure uniqueness for the pressure.

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If Gamma_D is the whole boundary, then the pressure has to have zero mean-value but if Gamma_D is not the whole boundary, the condition is not needed. A simple way to see it is to set all u0, f, g as zero, and prove that u=p=0. The resulted system is a saddle point problem and it is well-known that the inf-sup condition for Stokes equation holds in this system, therefore it is well-posed. I refer to the well-known book of Girault and Raviart for details.

By the way, I think the demo is not valid from the physical viewpoint because the traction boundary condition (on Gamma_N) does not use the Cauchy stress tensor (i.e., it uses gradient of velocity instead of the symmetric part of it).