Preprint, 2016

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 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; v2, v3). Hence the Fermi equation, stated in three dimensional spatial domain x = (x; y; z), depends only on two velocity variables v = (v2; v3). 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.

a priori error estimates

adaptive finite element method

efficiency

duality argument

Fermi and Fokker-Planck pencil-beam equations

reliability

Chalmers, Mathematical Sciences, Mathematics

University of Gothenburg

University of Gothenburg

Chalmers, Mathematical Sciences, Mathematics

Chalmers, Mathematical Sciences

University of Gothenburg

Chalmers, Mathematical Sciences, Mathematics

University of Gothenburg

Mathematics

Basic sciences