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A priori error estimates and computational studies for a Fermi pencil-beam equation

Mohammad Asadzadeh (Institutionen för matematiska vetenskaper, matematik) ; Larisa Beilina (Institutionen för matematiska vetenskaper, matematik) ; Muhammad Naseer (Institutionen för matematiska vetenskaper) ; Christoffer Standar (Institutionen för matematiska vetenskaper, matematik)
(2016)
[Preprint]

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.

Nyckelord: Fermi and Fokker-Planck pencil-beam equations, adaptive finite element method, duality argument, a priori error estimates, efficiency, reliability



Denna post skapades 2016-12-29. Senast ändrad 2017-01-16.
CPL Pubid: 246534

 

Institutioner (Chalmers)

Institutionen för matematiska vetenskaper, matematik (2005-2016)
Institutionen för matematiska vetenskaperInstitutionen för matematiska vetenskaper (GU)

Ämnesområden

Matematik

Chalmers infrastruktur