A Deficiency Problem of the Least Squares Finite Element Method for Solving Radiative Transfer in Strongly Inhomogeneous Media

TL;DR

This paper analyzes the limitations of the least squares finite element method (LSFEM) for radiative transfer in strongly inhomogeneous media, revealing a deficiency that causes accuracy issues despite its stability advantages.

Contribution

It identifies and proves a deficiency in LSFEM when applied to inhomogeneous media, showing its equivalence to a second order central difference scheme and highlighting its limitations.

Findings

LSFEM is more stable than GFEM in homogeneous media.

LSFEM suffers from a deficiency in strongly inhomogeneous media.

The deficiency leads to severe accuracy degradation in LSFEM.

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.