  A mathematical analysis is presented for a class of interior penalty (IP)
discontinuous Galerkin approximations of elliptic boundary value problems. In
the framework of the present theory one can derive some overpenalized IP
bilinear forms in a natural way avoiding any heuristic choice of fluxes and
penalty terms. The main idea is to start from bilinear forms for the local
average of discontinuous approximations which are rewritten using the theory of
distributions. It is pointed out that a class of overpenalized IP bilinear
forms can be obtained using a lower order perturbation of these. Also, error
estimations can be derived between the local averages of the discontinuous
approximations and the analytic solution in the $H^1$-seminorm. Using the local
averages, the analysis is performed in a conforming framework without any
assumption on extra smoothness for the solution of the original boundary value
problem.
