I am trying to understand the DG FEM methods, but I got lost in their definitions. In some papers I read that the "C0 penalty method" is one example of the DG method, but sometimes they are separated and handled as two different methods. Can someone explain what's the difference between them?
Edit: There is a paper dealing with three different finite element methods for solving the Monge-Ampere equation: C1 finite element method, C0 penalty method, discontinuous galerkin method. After I familiarized myself with the basics of the discontinuous galerkin methods I don't understand why the C0 penalty method is not a DG method (the C0 penalty method for the Monge-Ampere equations was developed in 2011, and this paper from 2014 considers itself as a first try to solve the Monge-Ampere equations with a DG method).
I am interested only about elliptic PDEs (second order and fourth order as well) and I would like to clarify the meanings and differences behind the names related to the topic. If I got it right, in case of a fourth order equation:
- continuous galerkin method: we need C1 continuity and seek the solution in the H2 space
- interior penalty method: we use C0 continuous elements and seek the solution in the H1 space
Does this mean that in the interior penalty method the elements are not fully discontinuous? Therefore we can define a truly discontinuous method, where we seek the solution in the L2 space.