  We propose a stable Petrov-Galerkin discretization of a kinetic Fokker-Planck
equation constructed in such a way that uniform inf-sup stability can be
inferred directly from the variational formulation. Inspired by well-posedness
results for parabolic equations, we derive a lower bound for the dual inf-sup
constant of the Fokker-Planck bilinear form by means of stable pairs of trial
and test functions. The trial function of such a pair is constructed by
applying the kinetic transport operator and the inverse velocity
Laplace-Beltrami operator to a given test function. For the Petrov-Galerkin
projection we choose an arbitrary discrete test space and then define the
discrete trial space using the same application of transport and inverse
Laplace-Beltrami operator. As a result, the spaces replicate the stable pairs
of the continuous level and we obtain a well-posed numerical method with a
discrete inf-sup constant identical to the inf-sup constant of the continuous
problem independently of the mesh size. We show how the specific basis
functions can be efficiently computed by low-dimensional elliptic problems, and
confirm the practicability and performance of the method with numerical
experiments.
