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 spaceinterior penalty method: we use C0 continuous elements and seek the solution in the H1 spaceDoes 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. 
What's the difference between C0 penalty methods and Discontinuous Galerkin methods?
finite element;discontinuous galerkin
DG methods are a more general description, referring to finite element methods which represent the solution in a discontinuous manner (usually with coupling between elements through a numerical flux). The specific numerical flux tends to differ between type of equation (hyperbolic vs parabolic/elliptic) and formulation (2nd order equation vs first order system). The interior penalty DG method is one common choice of flux for elliptic equations (like diffusion or Laplace) in primal (2nd order) form.  