I am trying to implement a finite element for the incompressible navier-stokes and i have trouble with the Pressure Poisson equation. the discrete laplace operator is as follows CT*ML-1*C , where C: gradient matrix and ML is the lumped mass, CT:divergence matrix the problem is, the boundary conditions coming from the velocity. I can form this matrix in two forms. first one: reducing the rows and columns of the known velocity boundary condition from the C and CT matrices and second without reducing these entries. In two cases the dimension of my laplace operator will be the same but they won't be the same. Which one should I implement?

I am not familar with projection FEM.

but according to my recent re-look at projection FDM, in the case of with pressure poisson equation, the projection procedure could easily cause the FD to lose one of its good properties: strong form.

however, this could be not a problem at all for FEM, since classical FEM is already in weak form and has nothing to lose.

Acusim, seeming the first FEM software using Galerkin/Least Square in a projection procedure, may have something for you. please check www.acusim.com

and there are a big bulk of papers on projection FEM.

and if you still have the smae question in summer, you can email me at kenn2003@yahoo.com, because i'll implement the projection version of my current mixed-formulation sometime in May.

The projection / fractional step method can be interpretated with the generalized block LU decomposition (see JB Perot, JCPH (1993), 108:51-58).

In fact, the pressure poisson equation is obtained using the LU decomposition. In this sense, it should contain the velocity BC from the momentum eqs. You may choose not to as in your 2nd case. The difference will show in a numerical boundary layer - reduced accuracy for pressure in time. You may find tons of paper out there for this issue.

Read the mentioned reference first.

Hope this helps.

Tony