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

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 the performance of the LSFEM too much and makes it not a good choice to solve radiative transfer in strongly inhomogeneous media. It is also theoretically proved that the LSFEM using the one dimensional linear element is equivalent to a second order form of radiative transfer equation discretized by the central difference scheme.