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Sparse adaptive finite elements for radiative transfer

by G. Widmer and R. Hiptmair and Ch. Schwab

(Report number 2007-01)

Abstract
The linear radiative transfer equation, a partial differential equation for the radiation intensity u(x,s), with independent variables x\in D \subset Rn in the physical domain D of dimension n=2,3, and in the angular variable s\inS2 := {y\in R3: |y|=1}, is solved in the n+2-dimensional computational domain D \times S2. We propose an adaptive multilevel Galerkin FEM for its numerical solution. Our approach is based on a) a stabilized variational formulation of the transport operator and b) on so-called sparse tensor products of two hierarchic families of Finite Element spaces in H1(D) and in L2(S2), respectively. An a-priori error analysis shows, under strong regularity assumptions on the solution, that the method converges with essentially optimal asymptotic rates while its complexity grows essentially only as that for a linear transport problem in Rn. Numerical experiments for n=2 on a set of example problems agree with the convergence and complexity analysis of the method and show its performance in terms of accuracy vs. number of degrees of freedom to be superior to the discrete ordinate method.

Keywords:

@Techreport{WHS07_362,
  author = {G. Widmer and R. Hiptmair and Ch. Schwab},
  title = {Sparse adaptive finite elements for radiative transfer},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2007-01},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2007/2007-01.pdf },
  year = {2007}
}

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