  This research note documents new developments regarding finite-element
discretizations of the relativistic Beliaev-Budker Coulomb collision operator
and the nonrelativistic Landau operator. Where energy conservation in a
finite-element approximation of the relativistic collision operator was
previously thought to be elusive, it is now achieved even with linear elements.
The same result applies to the nonrelativistic Landau operator for which the
energy conservation was thought to require at least quadratic elements. In both
cases, the momentum and density conservation are guaranteed as previously. The
new outcomes benefit from the findings reported in a recent
finite-difference-scheme paper [Shiroto & Sentoku, arXiv:1902.07866] which we
generalize to the finite-element method. This note focuses solely on the direct
discretization of the collision operator, leaving the discretization of the
underlying metriplectic formulation of the relativistic collision operator to
future publications.
