biharmonic demo uses discontinuous Galerkin?

Question

Hi, I'm trying to work through the demo files to learn Fenics and I am confused with the biharmonic demo.py file, which claims to use "discontinuous galerkin" but the Function space call is "CG":

mesh = UnitSquareMesh(32, 32)
V = FunctionSpace(mesh, "CG", 2)

https://fenicsproject.org/documentation/dolfin/2016.1.0/python/demo/documented/biharmonic/python/documentation.html

It would seem the solution will be continuous using continuous elements--am I missing something here? If so, what would it mean to use the function space:

V = FunctionSpace(mesh, "DG", 2)  

(this does not work if changed in the code).
Thanks for any help.

Comments

The function is continuous, but the gradient is not.

Answer 1

The method is called discontinuous Galerkin because the discrete space is not a subspace of the continuous space. It is also called the $C^0$-interior penalty method.

Classical Galerkin methods for fourth order problems uses piecewise polynomial basis functions with $C^1$ continuity (and even more in the vertices), for example Argyris and Bell triangles.

Comments

In addition to this answer, you can find analysis of the DG FEM formulation of the biharmonic equation in the work of Georgoulis and Houston http://eprints.nottingham.ac.uk/671/1/biharmonic_DG.pdf.