We perform molecular dynamics Simulations to study the dielectric relaxation of a head-spring model for a polymer melt in the bulk and in Supported films. By assigning dipole moments parallel and perpendicular to the backbone of all chains ill the completed simulation trajectories, we calculate the dielectric spectra of so-called type-A polymers which exhibit relaxation processes due to the local motion of chain segments ("segmental mode") and due to fluctuations of the end-to-end vector ("normal mode"). We investigate the dependence of both processes on film thickness and chain length and for the segmental mode also oil temperature T. We find that the relaxation of both modes is enhanced in the films relative to the bulk. For the segmental mode this difference between film and bulk dynamics increases oil cooling toward the glass transition. By a layer-resolved analysis of the segmental relaxation, we show that the acceleration of the average film dynamics is a consequence of a smooth gradient in relaxation, originating from both interfaces where the segmental dipoles relax much faster than in the bulk. Additionally. near the interfaces the segmental relaxation is more strongly stretched than in the center of the film where bulk behavior prevails. As the average film dynamics comprises contributions from all layers, the dielectric spectra of the films are broader than in the bulk at the same T. Finally. starting from the layer-resolved analysis which associates a dielectric function and so a capacitance with each layer. we suggest to think of a film as being a system of capacitors. The capacitors are arranged ill series, if the electric (E) field is perpendicular to the plane of the film (the usual experimental Situation), and in parallel. if the field is parallel to the plane. Because of these different arrangements of the capacitors, the resulting dielectric spectra depend on the direction of the E-field. For instance, we find that, although the segmental relaxation ill each layer is taken to be the same for both field directions. the average dielectric spectra differ because the layer-dependent dielectric strength and file limiting high-frequency permittivity enter into the average dielectric response in different ways for the E-field being perpendicular or parallel to the film plane.