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I am trying to solve a "Navier-Stokes"-type problem where the viscosity is negative. Of course this renders the equation unstable and thus I add a fourth order term, so the entire equation becomes:

$$ \frac{dv}{dt} + (v\cdot\nabla)v = - \nabla p+\nu_1 \nabla^2 v - \nu_2 \nabla^4 v $$ with $\nu_1<0, \nu_2>0$.

Now the fourth order term should stabilize the solution with negative viscosity - I can show this by stability analysis - but I am not able to solve it numerically. I am using the finite element code Fenics.

I am able to solve the system with $\nu_1>0, \nu_2 = 0$ using a simple Picard technique, see e.g. http://fenicsproject.org/documentation/dolfin/dev/python/demo/pde/navier-stokes/python/documentation.html

However, I am not able to make the system stable with negative viscosity. I believe this is a problem with my Finite element approach, but I have no idea how to fix it.

Edit in response to comments: My implmentation is done via fenics. I have tried with both a Lagrange and a CG space. I've done only very simple implementations. I've tried doing a the fourth order derivative on the previous time step's solution. I've also tried doing partial integration twice.

By stable I mean that the field diverges. The velocity field quickly becomes infinite - just as it would if $\nu_2$ is set to zero. It basically behaves just as a Navier Stokes system with negative viscosity and not stabilized by the fourth order term.

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    $\begingroup$ Can you expand on what you actually see? You only say that you can't "solve" the equation and that you can't make it "stable", but these are broad terms. Can you give us more details how the solution fails, or how the instability looks? $\endgroup$ – Wolfgang Bangerth Oct 7 '13 at 14:39
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    $\begingroup$ To add to Wolfgang's comment, can you explain how you implement a 4th order viscosity? If you just integrate it by parts as with a second order viscosity, you can have numerical issues with piecewise continuous FEM. $\endgroup$ – Jesse Chan Oct 7 '13 at 17:10
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    $\begingroup$ Similar to JLC's comment, does your function space support nontrivial fourth derivatives? $\endgroup$ – Rhys Ulerich Oct 7 '13 at 20:46
  • $\begingroup$ Does my space support it? Not sure. I use a standard Lagrange function space. $\endgroup$ – Jesper Oct 10 '13 at 13:18
  • $\begingroup$ You either need $H^2$-conforming elements or you could implement the $C^0$-interior penalty method for your 4th order term. $\endgroup$ – Guido Kanschat Oct 10 '13 at 19:56

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