Comparison of light transport models based on finite element and the discrete ordinates methods in view of optical tomography applications

Numerical tests are used to evaluate the accuracy of two finite element formulations associated with the discrete ordinates method for solving the radiative transfer equation: the Least Square and the Discontinuous Galerkin finite element formulations. The results show that the use of a penalization method to set the Dirichlet boundary conditions leads to a more accurate solution than the weakly type setting where the Least Square method is seen to be more sensitive. Convergence in mesh size shows that, while both methods give accurate results, the Discontinuous Galerkin formulation uses five times more degrees of freedom than the Least Square formulation, which may lead to large systems to handle when the number of mesh elements is large. The comparison of both methods using the S_n and the T_n angular quadratures has shown that the Discontinuous Galerkin gives more accurate solutions, as expected, for problems with strong discontinuities, but may exhibit some oscillations due to the Galerkin procedure. A last test featuring a collimated irradiation shows that both methods give the same accuracy due to the separation of the radiative intensity into transmitted and scattered components, which removes the discontinuities in the implementation of the boundary conditions.