# What's the difference between C0 penalty methods and Discontinuous Galerkin methods?

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.

• Your title asks about "C0 penalty" methods but the body just says "interior penalty". There is a class of (fairly exotic) "C0 interior penalty methods" used for fourth order problems. The answers so far address second order problems, which are more usual, but if you are really interested in C0 interior penalty methods for fourth order problems you should edit/clarify your question. – Andrew T. Barker Jun 27 '15 at 19:20
• For second order, continuous Galerkin methods have $C^0$ continuity and are $H^1$ conforming, while discontinuous methods have no continuity and are not $H^1$ conforming (if you wish to put it this way, they are $L^2$ conforming). For fourth order, you can have continuous non-conforming as well as discontinuous non-conforming methods. To minimize confusion, I would suggest to focus your question on a single, specific problem you are interest in -- either a fourth-problem or the Monge-Ampère problem (which is second-order, but nonlinear). – Christian Clason Jun 28 '15 at 21:43
• TL;DR: Your last paragraph after the edit exactly answers your question. (In case of the Monge-Ampère equation; for the more usual linear second-order equations both concepts coincide.) – Christian Clason Jun 28 '15 at 22:00

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.

From a mathematical perspective, the term 'DG method' by itself refers only to the class of methods that use a piecewise discontinuous basis. This choice of basis necessitates the introduction of some additional parameters enforcing the continuity requirements on the solution as desired (penalty parameters) as well as to control the numerical properties of the resulting linear system (s-form)

To put it simply, for the case of elliptic and parabolic equations, the choice of parameters dictates the type of flux introduced into the problem (as mentioned by Jesse). You mention the C0/IIPG method, which is what you get when one of the parameters is chosen to be zero. For elliptic equations, the choice of an appropriate sform and penalty parameter gives rise to SIPG, IIPG, NIPG methods. So ALL these methods are DG-methods, but their solutions differ from one another in that some would exhibit more numerical dispersion and/or smearing of solution fronts.

I suggest you read through Ch 1 of Rivière's Book if you can get a hold of it.

• I'm not sure what you mean by C0/IIPG. Are you using C0 to refer to the continuity or to some parameter in the method? – Andrew T. Barker Jun 27 '15 at 19:21
• Good point. I think there is some ambiguity in the wording of the question, as well as the wording of my answer for which I apologize. IIPG does select a sform=0, but by C0 I was referring to the choice of basis. – prussian_metal Jun 27 '15 at 19:32
• Follow up: I feel the choice of basis is inconsequential for the purposes of this discussion – prussian_metal Jun 27 '15 at 19:32