  We derive a priori error estimates for the standard Galerkin and streamline
diffusion finite element methods for the Fermi pencil-beam equation obtained
from a fully three dimensional Fokker-Planck equation in space ${\mathbf
x}=(x,y,z)$ and velocity $\tilde {\mathbf v}=(\mu, \eta, \xi)$ variables. The
Fokker-Planck term appears as a Laplace-Beltrami operator in the unit sphere.
The diffusion term in the Fermi equation is obtained as a projection of the FP
operator onto the tangent plane to the unit sphere at the pole $(1,0,0)$ and in
the direction of $ {\mathbf v}_0=(1,\eta, \xi)$. Hence the Fermi equation,
stated in three dimensional spatial domain ${\mathbf x}=(x,y,z)$, depends only
on two velocity variables ${\mathbf v}=(\eta, \xi)$. Since, for a certain
number of cross-sections, there is a closed form analytic solution available
for the Fermi equation, hence an a posteriori error estimate procedure is
unnecessary and in our adaptive algorithm for local mesh refinements we employ
the a priori approach. Different numerical examples, in two space dimensions
are justifying the theoretical results. Implementations show significant
reduction of the computational error by using our adaptive algorithm.
