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.