On Fully Discrete Schemes for the Fermi Pencil-beam Equation

We consider a Fermi pencil beam model in two space dimensions (x; y), where x is aligned with the beam's penetration direction and y together with the scaled angular variable z, correspond to a bounded symmetric , transversal cross section. We study some fully discrete numerical schemes using the standard Galerkin and streamline diiusion nite element methods for discretization of the transversal domain combined with backward Euler, Crank-Nicolson and discontinuous Galerkin methods for discretization in the penetration variable. We derive stability estimates for the semidiscrete problems and, assuming suuciently smooth exact solution, we show optimal a priori error estimates. Numerical examples presented in some canonical cases, with data approximating Dirac function, connrm the expected performance of the combined schemes. 1. introduction The Fermi pencil beam equation is derived from the Fokker-Planck equation through an asymptotic expansion. The Fokker-Planck equation itself is yet another asymptotic limit of the linear Boltzmann equation, see 6]. The assumption of forward-peaked scattering, in a transport process, comprises the backbone of the derivation of both equations. We study some fully discrete schemes for the numerical solution of a pencil beam