[Jonas Holdeman]

When solving the incompressible Navier-Stokes equation by finite element mixed method using (discontinuous) P0 or P1 constraint functions, it seems that common practice is to solve the constraint equation as a separate step using a "pressure" Poisson equation. But when assembling element-by-element, one can solve the constraint by static condensation making the solution much simpler for computation. This can result in elements for interpolation which are divergence-free between the nodes. Solving via a PPE might result in nodal values that are "divergence-free" in some sense, but subsequent interpolation is not. Maybe I have missed something, but I would appreciate any discussion and/or references to using static condensation for the constraint. Thanks.

[Jonas Holdeman]

Earlier I wrote,

Originally Posted by Jonas Holdeman

When solving the incompressible Navier-Stokes equation by finite element mixed method using (discontinuous) P0 or P1 constraint functions, it seems that common practice is to solve the constraint equation as a separate step using a "pressure" Poisson equation. But when assembling element-by-element, one can solve the constraint by static condensation making the solution much simpler for computation. This can result in elements for interpolation which are divergence-free between the nodes. Solving via a PPE might result in nodal values that are "divergence-free" in some sense, but subsequent interpolation is not. Maybe I have missed something, but I would appreciate any discussion and/or references to using static condensation for the constraint. Thanks.

But have received no response.
The idea of using static condensation seems simple and useful enough that I would think that it has been used, so perhaps I should expand a bit. I am working on a paper describing the derivation of a family of divergence-free linear and quadratic Hermite finite elements in 3D, derived from Lagrange vector elements that have been augmented to have constant divergence. I take combinations in such a way as to produce Hermite elements with zero divergence (in the sense that the integral of the divergence squared is zero), and hence they are pointwise divergence-free. But the divergence of the augmented Lagrange element also vanishes in the same sense if the net flow into each element is zero. This is zero if the vector of divergences is orthogonal to the vector of velocities at the nodes. So one uses Graham-Schmidt to make this so, satisfying the continuity constraint.
I am attaching a pdf of the last two pages of my paper which explains this a little better, but not in detail as that is not the purpose of the paper. Please tell me if you know if this idea has been used elsewhere. Thanks.