  The gradient discretisation method (GDM) is a generic framework for designing
and analysing numerical schemes for diffusion models. In this paper, we study
the GDM for the porous medium equation, including fast diffusion and slow
diffusion models, and a concentration-dependent diffusion tensor. Using
discrete functional analysis techniques, we establish a strong
$L^2$-convergence of the approximate gradients and a uniform-in-time
convergence for the approximate solution, without assuming non-physical
regularity assumptions on the data or continuous solution. Being established in
the generic GDM framework, these results apply to a variety of numerical
methods, such as finite volume, (mass-lumped) finite elements, etc. The
theoretical results are illustrated, in both fast and slow diffusion regimes,
by numerical tests based on two methods that fit the GDM framework: mass-lumped
conforming $\mathbb{P}_1$ finite elements and the Hybrid Mimetic Mixed method.
