  We extend the two leading methods for the \emph{ab initio} computational
descrip tion of phonon-mediated superconductors, namely Eliashberg theory and
density fu nctional theory for superconductors (SCDFT), to include plasmonic
effects. Furth ermore, we introduce a hybrid formalism in which the Eliashberg
approximation fo r the electron-phonon coupling is combined with the SCDFT
treatment of the dynam ically screened Coulomb interaction. The methods have
been tested on a set of we ll-known conventional superconductors by studying
how the plasmon contribution a ffects the phononic mechanism in determining the
critical temperature (\tc). Our simulations show that plasmonic SCDFT leads to
a good agreement between predict ed and measured \tc's, whereas Eliashberg
theory considerably overestimates the plasmon-mediated pairing and, therefore,
\tc. The hybrid approach, on the other hand, gives results close to SCDFT and
overall in excellent agreement with exper iments.
