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I believe this might be a recurring topic, but i have not found a post that directly related to this issue.

I come from a finite volume background and my experience is more with predictor-corrector schemes (such as SIMPLE or PISO), so my knowledge of stabilized finite element (Galerkin Least Square (GLS) or Variable Multi-Scale (VMS) for instance) is limited.

My current understanding is this: P1-P1 (velocity-pressure) elements (lets say on tetrahedron) are known to be invalid for the incompressible Navier-Stokes equation because they do not respect the famous LBB condition. We can surpass this limitation by stabilizing the P1-P1 element using a GLS or VMS approach. This alters the saddle-point problem, notably adding term on the pressure diagonal of the pressure block matrix, and one can then solve the problem.

I have used such an approach and the results are coherent. Concretly, manufactured solutions for the velocity converge at the right order. My problem arises when I am trying to solve problem in a closed domain (such as a lid driven cavity, where all the normal velocities are equal to 0). In this case, I do not understand how the pressure level should be defined and it seems my global system is ill-posed (missing an equation). If I fix the pressure to constant a single node like I would in FVM, then I seem to instantly lose mass conservation. What is the right approach to "fix" the pressure level in closed simulations for stabilized method? Should one impose the average pressure to be zero? Would that not lead to a quasi-full line for this equation thus wrecking the band of the matrix?

This sounds like such a trivial question, but I am at loss here. Thanks!

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Using VMS doesn't affect how you deal with the pressure for continuous galerkin methods.

In the case of enclosed flow like you have with all dirichlet conditions on velocity, it is common to set the mean value of the pressure to 0. This can usually be done as a post processing step after each linear solve in your nonlinear solver.

Another possibility is to use a pressure penalization term like $ \varepsilon (p, q) $

In the case where you have outflow conditions (I.e. Channel flow), using the no normal stress condition is sufficient on the outflow.

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