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An efficient sparse finite element solver for the radiative transfer equation

by G. Widmer

(Report number 2009-11)

Abstract
The stationary monochromatic radiative transfer equation (RTE) is posed in five dimensions, with the intensity depending on both a position in a three-dimensional domain as well as a direction. For non-scattering radiative transfer, sparse finite elements [10, 11] have been shown to be an efficient discretization strategy if the intensity function is sufficiently smooth. Compared to the discrete ordinates method, they make it possible to significantly reduce the number of degrees of freedom N in the discretization with almost no loss of accuracy. However, using a direct solver to solve the resulting linear system requires O(N3) operations. In this paper, an efficient solver based on the conjugate gradient method (CG) with a subspace correction preconditioner is presented. Numerical experiments show that the linear system can be solved at computational costs that are nearly proportional to the number of degrees of freedom N in the discretization.

Keywords:

@Techreport{W09_398,
  author = {G. Widmer},
  title = {An efficient sparse finite element solver for the radiative transfer equation},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2009-11},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2009/2009-11.pdf },
  year = {2009}
}

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