# 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?

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.
• 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... – Jesse Chan Nov 22 '14 at 2:41