cordery@MENARD.MIT.EDU (Matthew Cordery) (06/17/89)
I am trying to solve a Stokes flow problem via the finite element algorithm of Bristeau, Glowinski, and Periaux ( Computer Physics Reports, 6, 1987, 73-187 ). In implementing their algorithm, I ran into a very strange and unfortunate problem which I am hoping some finite element god out there can help me solve. Basically, my problem is this: We have a constant viscosity fluid governed by the Stokes equation with no body forces acting. If we assume we know the pressure, then we can make a first guess at the velocity field. OK. Fine. Now, we discretize our problem by solving for velocities on a 9-node element mesh and pressure on a 4-node element mesh ( The elements are the same in each case, the pressure node corresponding to the corner nodes of the 9-node velocity elements ). Let us call the shape functions for these elements N4(x,y) and N9(x,y). The standard variational form of the pressure gradient term is integral( N*gradP ). If we integrate by parts this becomes integral( P*gradN ). Now we let the test function N = N9 and we interpolate P via N4. Therefore, our integral becomes integral( N4*gradN9 ). Our problem is this, if we let P = 1., then our pressure gradient term should be identically zero. However, we find that this does not happen. Some nodes are zero, some are not. If we assume that the pressure is also given on a nine-node grid, then everything works fine. ( Note, we are using standard Gaussian quadrature schemes to do the integrals. ) Does anybody have any idea why this happens and what I can do to fix it. This seems like such a stupid question by I have spent the better part of a week ruining my mental health over this. Trust me, I have tried calling these guys ( Glowinski ) in Houston, but it seems that everybody is in France for the summer and didn't leave any way to reach them ( I don't think that I would either. ) Many many thanks and a blessing on your first born if any one helps me out with this Matthew J. Cordery cordery@gilbert.mit.edu