Vectorial finite elements for solving the radiative transfer equation

2018 ◽  
Vol 212 ◽  
pp. 59-74 ◽  
Author(s):  
M.A. Badri ◽  
P. Jolivet ◽  
B. Rousseau ◽  
S. Le Corre ◽  
H. Digonnet ◽  
...  
Keyword(s):  
Finite Elements ◽  
Radiative Transfer ◽  
Transfer Equation ◽  
Radiative Transfer Equation

An Efficient Sparse Finite Element Solver for the Radiative Transfer Equation

2009 ◽  
Vol 132 (2) ◽  
Author(s):  
Gisela Widmer
Keyword(s):  
Finite Elements ◽  
Linear System ◽  
Radiative Transfer ◽  
Transfer Equation ◽  
Degrees Of Freedom ◽  
Radiative Transfer Equation ◽  
Three Dimensional ◽  
Discrete Ordinates ◽  
Five Dimensions ◽  
Dimensional Domain

The stationary monochromatic radiative transfer equation is posed in five dimensions, with the intensity depending on both a position in a three-dimensional domain as well as a direction. For nonscattering radiative transfer, sparse finite elements [2007, “Sparse Finite Elements for Non-Scattering Radiative Transfer in Diffuse Regimes,” ICHMT Fifth International Symposium of Radiative Transfer, Bodrum, Turkey; 2008, “Sparse Adaptive Finite Elements for Radiative Transfer,” J. Comput. Phys., 227(12), pp. 6071–6105] have been shown to be an efficient discretization strategy if the intensity function is sufficiently smooth. Compared with 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 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.

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