I don’t have legacy dolfin installed so I cant try this myself, but I notice two things:
To solve a fourth order problem with a mixed setting, you typically introduce only two fields. In your other post you state:
But you would typically not introduce a third space in your mixed space for this. You’d simply project onto a stand-alone space. Else, your’re creating unnecessarily big system matrices, with all potential issues involved.
Secondly, in the mixed setting you have direct access to the derivative on the boundary, so no need for Nitsche’s method:
For the dummy biharmonic equation:
w = \Delta u \\ \Delta w = f
(w, w_{test} )_\Omega + (\nabla u,\nabla w_{test} )_\Omega = (w_{test},\nabla \hat{u}\cdot n)_{\partial\Omega} \\ (\nabla w,\nabla u_{test} )_\Omega = (-f,u_{test})_\Omega + (u_{test},\nabla \hat{w}\cdot n)_{\partial\Omega}
On the right hand sides you have weak acces to \nabla u\cdot n and \nabla w\cdot n, so you can simply substitute your expression there. No need for Nitsche.
Also note those integrals elucidate the BCs that you can mix and match:
(w_{test},\nabla \hat{u}\cdot n)_{\partial\Omega} \Rightarrow Dirichlet on w=\Delta u, or natural for \nabla u\cdot n
(u_{test},\nabla \hat{w}\cdot n)_{\partial\Omega} \Rightarrow Dirichlet on u, or natural for \nabla w\cdot n = \nabla (\Delta u)\cdot n