On fully discrete schemes for the Fermi pencil-beam equation Original Research Article

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. The model corresponds to a forward–backward degenerate, convection dominated, convection–diffusion problem. For this problem we study some fully discrete numerical schemes using the standard- and Petrov–Galerkin finite element methods, for discretizations of the transversal domain, combined with the backward Euler, Crank–Nicolson, and discontinuous Galerkin methods for discretizations in the penetration variable. We derive stability estimates for the semi-discrete problems. Further, assuming sufficiently smooth exact solution, we obtain optimal a priori error bounds in a triple norm. These estimates give rise to a priori error estimates in the L2-norm. Numerical implementations presented for some examples with the data approximating Dirac δ function, confirm the expected performance of the combined schemes.