A priori error estimates and computational studies for a Fermi pencil-beam equation
Abstract
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 x=(x,y,z) and velocity v=(μ, η, ξ) 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 v0=(1,η, ξ). Hence the Fermi equation, stated in three dimensional spatial domain x=(x,y,z), depends only on two velocity variables v=(η, ξ). 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.