A Priori Error Estimates and Computational Studies for a Fermi Pencil-Beam Equation. Issue 1 (16th April 2018)

• Record Type: Journal Article
• Authors: Asadzadeh, M.; Beilina, L.; Naseer, M.; Standar, C.
• Is Part Of: Journal of computational and theoretical transport. Volume 47:Issue 1/3(2018)
• Page Start: 125
• Page End: 151
• Publication Date: 2018-04-16
• Subjects: Fermi and Fokker–Planck pencil-beam equations -- adaptive finite element method -- duality argument -- a priori error estimates -- efficiency -- reliability; Transport theory -- Periodicals; Statistical physics -- Periodicals; Statistical physics; Transport theory; Periodicals; 530.138
• DOI: 10.1080/23324309.2018.1496937
• Languages: English
• ISSNs: 2332-4309

Abstract

Abstract: 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 v ˜ = ( μ, η, ξ ) variables. For a constant transport cross-section, there is a closed form analytic solution available for the Fermi equation with a data as product of Dirac functions. Our objective is to study the case of nonconstant, nonincreasing transport cross-section. Therefore we start with a theoretical, that is, a priori, error analysis for a Fermi model with modified initial data in L 2 . Then we construct semi-streamline-diffusion and characteristic streamline-diffusion schemes and consider an adaptive algorithm for local mesh refinements. To derive the stability estimates, for simplicity, we rely on the assumption of nonincreasing transport cross-section. Different numerical examples, in two space dimensions are justifying the theoretical results. Implementations show significant reduction of the computational error by using such adaptive procedure.