where,

$$
\mathcal{H} = \left\lbrace f\in H^2([0,\ell]), \; f(0)=0, \; f'(0)=0 \right\rbrace.
$$ 

I can't impose the boundary conditions correctly. When I compile the code, I get the following error: 

 FiniteElement('Hermite', interval, 3), name=None, index=1, component=None), IndexedProxyFunctionSpace(<firedrake.mesh.MeshTopology object at 0x125a2ce80>, FiniteElement('Hermite', interval, 3), name=None, index=2, component=None), name='None_None_None'), Mesh(VectorElement(FiniteElement('Lagrange', interval, 1), dim=1), 0)), 9) defined on incompatible FunctionSpace!

Note the same problem was implemented using FEniCS with C0-interior penalty method, in the reference (sec. 3 eq 3.12-3.13): 

Enhanced models for the nonlinear bending of planar rods: localization phenomena and multistability
by:
Matteo Brunetti , Antonino Favata  and Stefano Vidoli.

I attached the codes to this email.

Thanks in advance

Smail MERABET