> That's very insightful! I've double and triple checked the gradient of my
> exact solution and I'm 99.9% sure it's correct. Additionally, the error
> seems to go down in every norm which isn't the H^1 norm or H^1 seminorm..
> very strange.

If you can come up with a small example program demonstrating the problem, 
then we can take a look.

I can easily imagine situations where the broken H1 norm does not converge but 
the L2 norm does. For example, if you use piecewise constants to solve the 
Laplace equation using a DG formulation, then you can't expect the broken H1 
norm to converge (the gradient of the solution is zero on every cell). I don't 
know what your formulation is, however.

Best
 W.

-------------------------------------------------------------------------
Wolfgang Bangerth                email:            [email protected]
                                 www: http://www.math.tamu.edu/~bangerth/
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