  We prove the consistency of Galerkin methods to solve Poisson equations where
the differential operator under consideration is the generator of the Langevin
dynamics. We show in particular how the hypocoercive nature of this operator
can be used at the discrete level to first prove the invertibility of the
rigidity matrix, and next provide error bounds on the approximation of the
solution of the Poisson equation. We present general convergence results in an
abstract setting, as well as explicit convergence rates for a simple
one-dimensional example discretized using a tensor basis. Our theoretical
findings are illustrated by numerical simulations.
