| Author | Message | ||
     Bernd Markert (markert) New member Username: markert Post Number: 1 Registered: 01-2009 |
Hi, I'm new in this forum and currently playing around with the student version 5.1.0s of FlexPDE. My special interest is in solving strongly coupled PDE systems including some algebraic constraint, such as the continuity equation in incompressible fluid flow. It is well-known that mixed FE approximations can be used for the solution of such DAE systems, but stability is by no means guaranteed. In fact, the mixed finite elements have to fulfill the LBB condition or the patch test of Zienkiewicz. For instance, for a porous media problem with very low permeability that would mean that the solid displacements should be quadratic and the pore-fluid pressure should be linear. So, I was lucky as FlexPDE provides the SIMPLEX modifier which allows to choose a linear basis for distinct variables. However, FlexPDE tells me that "This Option is No Longer Supported." Is there another possibility to approximate some variables linearly? | ||
| Joseph Ribaudo (jribaudo) Member Username: jribaudo Post Number: 4 Registered: 11-2007 |
Not sure about version 5, but version 6 allows "order = 1" in the select section. | ||
| Robert G. Nelson (rgnelson) Moderator Username: rgnelson Post Number: 1214 Registered: 06-2003 |
The SIMPLEX option was installed some years ago because of the claims in the literature that the pressure should be approximated at lower order than the velocities. In practice, we never saw any advantage in the practice. Perhaps there was something lacking in our implementation. As a result, we have not maintained the option, and merely flag it as no longer supported. We have had good luck using the pressure equation P: div(grad(P)) = Big*div(u,v) This is related to some stabilization schemes that have been proposed in the literature, but we have never done a full stability analysis on it. It seems to work well for low Reynolds numbers. There is some "cut and try" necessary in choosing the value of "Big". We give a qualitative derivation of the method in the notes to the "viscous.pde" example. | ||
| Bernd Markert (markert) New member Username: markert Post Number: 2 Registered: 01-2009 |
Hi Robert, thanks for the prompt answer. However, the problem with algebraic constraints or at least equations yielding diagonal entries in the stiffness matrix which are governed by some very small parameter, say of magnitude 10^-10, cannot always be tackled by reformulation of the equation system. In case of the incompressible Stokes or Navier-Stokes equations this is quite simple. One just takes the divergence of the momentum balance and exploits div(v)=0 to get the new conditional equation for the pressure including div(grad(p)). Thus, this is no more an algebraic constraint as the variable p is in the equation. But for other coupled problems such manipulations are not that straight-forward. If in such cases equal-order approximations are used, one has to be aware of oscillations such as spurious pressure modes. In a monolithic solution procedure this can be overcome by use of stable mixed FE formulation like Taylor-Hood elements. So, it would be of considerable assistance if the interpolation order can individually be set for each variable or at least be restricted to linear (SIMPLEX). I will soon upload a pde example file where the instability problem occurs. If you need references on the problem, please let me know. |