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RESEARCH PAPERS: Solution Methods

An Efficient Sparse Finite Element Solver for the Radiative Transfer Equation

[+] Author and Article Information
Gisela Widmer

Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, Zurich CH-8092, Switzerlandgisela.widmer@math.ethz.ch

J. Heat Transfer 132(2), 023403 (Dec 02, 2009) (7 pages) doi:10.1115/1.4000190 History: Received October 30, 2008; Revised April 30, 2009; Published December 02, 2009; Online December 02, 2009

Abstract

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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Figures

Figure 1

(a)–(c) Examples of hierarchical hat basis functions up to level 2. The hierarchical hat basis functions of level 0 correspond to the vertices of the coarsest mesh (marked with circles in (d)), while the basis functions of levels 1 and 2 correspond to the vertices marked with diamonds and triangles, respectively.

Figure 2

The subspace VlD,lS contains all degrees of freedom up to level lD in physical space and up to level lS in solid angle. The figure shows the subspaces VlD,lS for L=3.

Figure 3

The blackbody intensity Ib(x) of example 2

Figure 4

The coarsest mesh TD0 in physical space

Figure 5

Number of CG-iterations with and without preconditioning for example 1

Figure 6

Number of CG-iterations with and without preconditioning for example 2

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