I am working on a project involving steady solutions for the Navier-Stokes Equations. In the past I've only worked with the unsteady Navier-Stokes, so some of this is new to me.
In particular, at each iteration for some problems, one may need to resort to a relaxation parameter at each iteration, so letting $\tilde{\boldsymbol{u}}^n$ be the velocity field at an iteration $n$, we have:
$$ \boldsymbol{u}^n = \gamma \tilde{\boldsymbol{u}}^n + (1 - \gamma) \boldsymbol{u}^{n-1} $$ For some value of $\gamma$ between zero and one. This seems to work alright for my purposes, but I have a couple of questions:
- Is it necessary to relax both pressure and velocity? It seems for my purposes relaxing only velocity seems to be fine, and the pressure still converges to the desired physical solution. I do worry, however, that there are problems with consistency if the pressure is not also relaxed.
- If the pressure must also be relaxed, how does the pressure relxation parameter relate to the velocity relaxation parameter? Would they simply be the same value of $\gamma$?
Potentially relevant details:
- I am using a finite element code with P2 for velocity and P1 for pressure.
- I am prescribing a Dirichlet inflow with a parabolic profile and a Neumann outflow.
- I am solving the coupled monolithic system (and not using the pressure Schur Complement approach).
Thanks in advance to anyone who has any answers for me!