# Are additional penalty terms necessary to solve elliptic PDE's with DG-FEM?

After a cursory glance at several references to discontinuous galerkin finite element methods for the elliptic poisson PDE, i notice that all of them emphasize using penalty methods where an additional term in the weak form (involving the average and jump operators of the test function and trial functions) is penalized.

My colleagues seem to indicate that this is the way it is usually done, but can't really explain why. For hyperbolic PDE's involving conservation laws, no such additional penalty term is needed. So, i wonder if they are absolutely necessary. And if so, why?

That is, what exactly goes wrong if i use DG-FEM for the poisson problem and omit the penalty term? Does the method lose coercivity or boundedness, for example? How does the penalty term fix this problem?

The penalty term does indeed act as a stabilization - coercivity is usually shown using norms which depend on this penalty, and you typically need a "sufficiently large penalty" to show convergence of symmetric interior penalty DG.

I believe one exception is NIPG (non-symmetric interior penalty DG, they flip the sign of the numerical flux terms involving the average of gradient of $v$) you can use any positive value for the penalty. At the very least, the Nitsche's method analogue to NIPG for boundary conditions has this feature.

A difference with hyperbolic PDEs is that there's a clear directionality to the problem - you have characteristic directions in which information is transmitted, and in which you can choose upwind directions. Contrast this to an elliptic PDE, in which information is spread isotropically - there's no clear "upwind" direction.

One note - the idea behind LDG for Poisson is to write the heat equation in first order form, and then use upwind fluxes there (though the two fluxes are upwinded in opposite directions to mimic this isotropic spread of information).

• I think for the NIPG-approach you cannot omit the penalty term, since then you cannot obtain a coercivity bound in the DG-norm. Commented Nov 19, 2014 at 10:39
• I am pretty sure that with "any" they implicitly mean "any positive". And yes, the Aubin-Nitsche trick will not work for the NIPG. Moreover, despite all literature that observes its optimal convergence for odd order polynomials, there is actually a counter-example by Guzman and Rivière (2009). Commented Nov 20, 2014 at 9:28
• Could you expand on how "directionality" (or lack there of) affects the need for the penalty terms?
– Paul
Commented Nov 20, 2014 at 18:19
• Directionality is sort of a byproduct of first order PDEs - for example, 1D convection $cu' = f$ has a directionality from left to right. This means information flows naturally in this direction as well, which hints at an upwind flux, and upwind DG tends to be stable. These are all pretty loose intuitions only... Commented Nov 22, 2014 at 2:41