  We propose a new stabilised finite element method for the classical
Kolmogorov equation. The latter serves as a basic model problem for large
classes of kinetic-type equations and, crucially, is characterised by
degenerate diffusion. The stabilisation is constructed so that the resulting
method admits a \emph{numerical hypocoercivity} property, analogous to the
corresponding property of the PDE problem. More specifically, the stabilisation
is constructed so that spectral gap is possible in the resulting
``stronger-than-energy'' stabilisation norm, despite the degenerate nature of
the diffusion in Kolmogorov, thereby the method has a provably robust behaviour
as the ``time'' variable goes to infinity. We consider both a spatially
discrete version of the stabilised finite element method and a fully discrete
version, with the time discretisation realised by discontinuous Galerkin
timestepping. Both stability and a priori error bounds are proven in all cases.
Numerical experiments verify the theoretical findings.
