On Fully Discrete Schemes for the Fermi Pencil-Beam Equations.

Journal article, 2002

We consider standard Galerkin and streamline diffusion finite element methods in the two dimensional, bounded transversal domain combined with backward Euler, Crank-Nicolson and discontinuous Galerkin methods in the penetration variable. Assuming smooth solutions in the Sobolev space $H^{k+1}$ of functions with their partial derivatives up to order $k+1$ in $L_2$, we derive optimal, a priori, semi-discrete methods of order $O(h^k)$ and $O(h^{k+1/2})$, respectively. Numerical implementations are presented