The Inner-Element Subgrid Scale Finite Element Method for the Boltzmann Transport Equation

This paper presents a new multiscale radiation transport method based on a Galerkin finite element spatial discretization of the Boltzmann transport equation. The approach incorporates a discontinuous subgrid scale (SGS) solution within the continuous finite element representation of the spatial variables. While the conventional discontinuous Galerkin (DG) method provides accurate and numerically stable solutions that suppress unphysical oscillations, the number of unknowns is relatively high. The key advantage of the proposed SGS approach is that the solutions are represented within the continuous finite element space, and therefore, the number of unknowns compared with DG is relatively low. The applications of this method are explored using linear finite elements, and some of the advantages of this new discretization over standard Petrov-Galerkin methods are demonstrated. The numerical examples are chosen to be demanding steady-state mono-energetic radiation transport problems that are likely to form unphysical oscillations within numerical scalar flux solutions. The numerical examples also provide evidence that the SGS method has a thick diffusion limit. This method is designed to work under arbitrary angular discretizations, so solutions using both spherical harmonics and discrete ordinates are presented.