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In standard finite element method, when we want to find $L^2$ error of a solution, we only find the error on Gaussian points that are inside of each triangle, for $DG$ method can we do this in a same way or we should add some boundary error term, because of the jump?

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Assuming that you're talking about the $L_2$ error in the functions themselves, and not doing something complicated with gradient reconstruction or anything, then restricting to the sum of the numerical integrals of the errors inside each element should be fine. Remember that the element boundaries are of zero measure in an integral over the domain and that the error at the jumps is finite.

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