Our DG formulation is obtained by applying for the Laplace equation posed in $\Omega_{I}$ the usual interior penalty finite element method, that can be traced back to [@arnoldIP], see also [@DiPietroErn] and the references cited therein for more details. In the conductor $\Omega_{C}$ we employ, as in [@HPSS05; @PS03], the interior penalty method corresponding to the Nédélec curl-conforming finite element space of the second kind. We point out that the introduction of discrete harmonic fields is necessary when considering domains of general topology. We prove the stability of the resulting combined DG scheme by exploiting the elliptic character of the problem. We also obtain, under adequate regularity assumptions, quasi-optimal asymptotic error estimates. It is worthwhile to notice that the implementation of the DG-method presented here only requires the use of standard shape functions. The curl-conforming finite elements, more precisely, the Nédélec finite elements of the second kind, are only needed for the theoretical convergence results in Section 5.

The outline of this paper is as follows. In Section 2 we derive the model problem used in the finite element approximation. We introduce our DG formulation in Section 3. Finally, Section 4 is devoted to the convergence analysis, and asymptotic error 