  We propose a discrete functional analysis result suitable for proving
compactness in the framework of fully discrete approximations of strongly
degenerate parabolic problems. It is based on the original exploitation of a
result related to compensated compactness rather than on a classical estimate
on the space and time translates in the spirit of Simon (Ann. Mat. Pura Appl.
1987). Our approach allows to handle various numerical discretizations both in
the space variables and in the time variable. In particular, we can cope quite
easily with variable time steps and with multistep time differentiation methods
like, e.g., the backward differentiation formula of order 2 (BDF2) scheme. We
illustrate our approach by proving the convergence of a two-point flux Finite
Volume in space and BDF2 in time approximation of the porous medium equation.
