Streamline diffusion methods for Fermi and Fokker-Planck equations

Abstract

We derive error estimates in certain weighted L 2-norms for the streamline diffusion and discontinuous Galerkin finite element methods for steady state, energy dependent, Fermi and Fokker-Planck equations in two space dimensions, giving error bounds of order O(hk+1/2), for the weighted current function J, as in the convection dominated convection-diffusion problems, with J ε H k+1(Ω) and h being the quasi-uniform mesh size in triangulation of our three dimensional phase-space domain Ω = Iz, × Iy × Iz , with z corresponding to the velocity variable. Our studies, in this paper, contain a priori error estimates for Fermi and Fokker-Planck equations with both piecewise continuous and piecewise discontinuous (in x and xy-directions) trial functions. The analyses are based on stability estimates which relay on an angular symmetry (not isotropy!) assumption. A continuation of this paper, the a posteriori error estimates for Fermi and Fokker-Planck equations, is the subject of a future work.