  We provide a grid of PHOENIX spectra of red giant and supergiant stars, that
extend through optical and near-IR wavelengths. For the first time, models are
also provided with modified surface abundances of C, N and O, as a step towards
accounting for the changes known to occur due to convective dredge-up (and to
be enhanced in the case of rotation). The aims are (i) to assess how well
current models reproduce observed spectra, (ii) to quantify the effects of the
abundance changes on the spectra, and (iii) to determine how these changes
affect estimates of fundamental stellar parameters.
  Observed giant star spectra can be fitted very well at solar metallicity down
to about 3400K. Modified surface abundances are preferred in only a minority of
cases for luminosity class II, possibly indicating mixing in excess of standard
first dredge-up.
  Supergiant stars show a larger variety of near-IR spectra, and good fits are
currently obtained for about one third of the observations only. Modified
surface abundances help reproducing strong CN bands, but do not suffice to
resolve all the difficulties. The effect of the abundance changes on the
estimated Teff depends on the wavelength range of observation and can amount
several 100K. Reasons for the remaining discrepancies are discussed.
