A priori error estimates and computational studies for a Fermi pencil-beam equation

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.