  Gradient schemes is a framework that enables the unified convergence analysis
of many numerical methods for elliptic and parabolic partial differential
equations: conforming and non-conforming Finite Element, Mixed Finite Element
and Finite Volume methods. We show here that this framework can be applied to a
family of degenerate non-linear parabolic equations (which contain in
particular the Richards', Stefan's and Leray--Lions' models), and we prove a
uniform-in-time strong-in-space convergence result for the gradient scheme
approximations of these equations. In order to establish this convergence, we
develop several discrete compactness tools for numerical approximations of
parabolic models, including a discontinuous Ascoli-Arzel\`a theorem and a
uniform-in-time weak-in-space discrete Aubin-Simon theorem. The model's
degeneracies, which occur both in the time and space derivatives, also requires
us to develop a discrete compensated compactness result.
