This article is devoted to the numerical simulation of the vibrations of an elastic mechanical structure equipped with several piezoelectric patches, with applications for the control, sensing and reduction of vibrations. At first, a finite element formulation of the coupled electromechanical problem is introduced, whose originality is that provided a set of non-restrictive assumptions, the system's electrical state is fully described by very few global discrete unknowns: only a couple of variables per piezoelectric patches, namely (1) the electric charge contained in the electrodes and (2) the voltage between the electrodes. The main advantages are (1) since the electrical state is fully discretized at the weak formulation step, any standard (elastic only) finite element formulation can be easily modified to include the piezoelectric patches (2) realistic electrical boundary conditions such that equipotentiality on the electrodes and prescribed global charges naturally appear (3) the global charge/voltage variables are intrinsically adapted to include any external electrical circuit into the electromechanical problem and to simulate shunted piezoelectric patches. The second part of the article is devoted to the introduction of a reduced-order model (ROM) of the problem, by means of a modal expansion. This leads to show that the classical efficient electromechanical coupling factors (EEMCF) naturally appear as the main parameters that master the electromechanical coupling in the ROM. Finally, all the above results are applied in the case of a cantilever beam whose vibrations are reduced by means of a resonant shunt. A finite element formulation of this problem is described. It enables to compute the system EEMCF as well as its frequency response, which are compared with experimental results, showing an excellent agreement.