We give a sequence of operators approximating the Fokker-Planck-Landau
collision operator. This sequence is obtained by aplying the fast mul
tipole method based on the work by Greengard and Rokhlin [17], and ten
ds to the exact Fokker-Planck-Landau operator with an arbitrary accura
cy. These operators satisfy the physical properties such as the conser
vation of mass, momentum, energy and the decay of the entropy. Further
more, the quadratic structure due to the velocity coupling in the expr
ession of the Fokker-Planck-Landau operator is removed in the approxim
ating operators. This fact reduces seriously the computationnal cost o
f numerical simulations of the Fokker-Planck-Landau equation. Finally,
we give numerical conservative and entropy discretizations solving th
e homogeneous Fokker-Planck-Landau equation using the fast multipole m
ethod. In addition to the deterministic character of these approximati
ons, they give satisfactory results in terms of accuracy and CPU time.