A deficiency problem of the least squares finite element method for solving radiative transfer in strongly inhomogeneous media

Abstract

The accuracy and stability of the least squares finite element method (LSFEM) and the Galerkin finite element method (GFEM) for solving radiative transfer in homogeneous and inhomogeneous media are studied theoretically via a frequency domain technique. The theoretical result confirms the traditional understanding of the superior stability of the LSFEM as compared to the GFEM. However, it is demonstrated numerically and proved theoretically that the LSFEM will suffer a deficiency problem for solving radiative transfer in media with strong inhomogeneity. This deficiency problem of the LSFEM will cause a severe accuracy degradation, which compromises too much of the performance of the LSFEM and makes it not a good choice to solve radiative transfer in strongly inhomogeneous media. It is also theoretically proved that the LSFEM is equivalent to a second order form of radiative transfer equation discretized by the central difference scheme.

A second order radiative transfer equation and its solution by meshless method with application to strongly inhomogeneous media

“…In this section, the numerical characteristics of different radiative transfer equations discretizing by central difference schemes are studied in a unified analytical framework in frequency domain. As is known, the central difference discretization is equivalent to the Galerkin FEM in discretization of the convection terms [43][44][45]. The central difference scheme suffers stability problem for convection-dominated convection diffusion equations, which is also true for discretization of the RTE.…”

Massively parallel least squares finite element method with graphic processing unit

“…In the past three decades, Jiang 1 and Bochev and Gunzburger 2 have developed the least squares finite element method (LSFEM) by combining the least squares method and the finite element method. LSFEM has been applied initially in the field of incompressible flow by Ding and Tsang 3 and Tang and Sun, 4 thermodynamics by Zhao et al 5 and Luo et al 6 and fluid-structure interaction by Kayser-Herold and Matthies. 7 LSFEM has the advantages of good convergence, good universality, good robustness and high accuracy.…”