  In this paper, we develop a Galerkin-type approximation, with quantitative
error estimates, for weak solutions to the Cauchy problem for kinetic
Fokker-Planck equations in the domain $(0, T) \times D \times \mathbb{R}^d$,
where $D$ is either $\mathbb{T}^d$ or $\mathbb{R}^d$. Our approach is based on
a Hermite expansion in the velocity variable only, with a hyperbolic system
that appears as the truncation of the Brinkman hierarchy, as well as ideas from
$\href{arXiv:1902.04037v2}{AAMN21}$ and additional energy-type estimates that
we have developed. We also establish the regularity of the solution based on
the regularity of the initial data and the source term.
