  This work is concerned with the development of a family of Galerkin finite
element methods for the classical Kolmogorov's equation. Kolmogorov's equation
serves as a sufficiently rich, for our purposes, model problem for kinetic-type
equations and is characterised by diffusion in one of the two (or three)
spatial directions only. Nonetheless, its solution admits typically decay
properties to some long time equilibrium, depending on closure by suitable
boundary/decay-at-infinity conditions. A key attribute of the proposed family
of methods is that they also admit similar decay properties at the
(semi)discrete level for very general families of triangulations. The method
construction uses ideas by the general theory of hypocoercivity developed by
Villani [23], along with judicious choice of numerical flux functions. These
developments turn out to be sufficient to imply that the proposed finite
element methods admit a priori error bounds with constants independent of the
final time, despite Kolmogorov equation's degenerate diffusion nature. Thus,
the new methods provably allow for robust error analysis for final times
tending to infinity. The extension to three spatial dimensions is also briefly
discussed.

Numerical Analysis;Numerical Analysis