  An attempt at understanding downward overshooting in the convective envelopes
of post-main-sequence stars has been made on the basis of three-dimensional
large-eddy simulations, using artificially modified OPAL opacity and taking
into account radiation and ionization in the equation of state. Two types of
star, an intermediate-mass star and a massive star, were considered. To avoid a
long thermal relaxation time of the intermediate-mass star, we increased the
stellar energy flux artificially while trying to maintain a structure close to
the one given by a 1D stellar model. A parametric study of the flux factor was
performed. For the massive star, no such process was necessary. Numerical
results were analysed when the system reached the statistical steady state. It
was shown that the penetration distance in pressure scaleheights is of the
order of unity. The scaling relations between penetration distance, input flux
and vertical velocity fluctuations studied by Singh et al. were checked. The
anisotropy of the turbulent convection and the diffusion models of the
third-order moments representing the non-local transport were also
investigated. These models are dramatically affected by the velocity fields and
no universal constant parameters seem to exist. The limitations of the
numerical results were also discussed.
