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

A nonlinear time compactness result and applications to discretization of degenerate parabolic-elliptic PDEs