How to enforce zero average pressure condition for HDG method with static condensation

NGSolve
1

Dear NGSolve community,

If we consider HDG methods for Stokes equations, where one needs to enforce the average free condition for pressure. If we use static condensation, how should we impose this average free condition such that the solution is correct?

Similar issues also arise for (pseudo)-stress velocity formulation for the Stokes, where one needs to enforce the trace-free condition for stress.

I would appreciate it if anyone can help me with this. Many thanks.

Best regards,
Lina

2

Hi,

Which HDG formulation are you refering to? If you are refering to “our” favourite formulations (or most other formulation), cf. e.g. 11.3. Normal-continuous HDG — NGSolve 24, you can simply use the NumberSpace to enforce the constraint as usual through a Lagrange multiplier, also with static condensation. The dof of the NumberSpace is a non-local dof and hence not considered when forming the Schur complement.
The trace-free condition is not a global property, but a local one and depends more crucially on your choice of method.

Best, Christoph

3

Dear Christoph,

I use the HDG formulation that is very similar to the normal continuous HDG, just with symmetric stress.

Thank you so much. It is working now.

Best regards,
Lina