In this dissertation, we study a multilevel algorithm for solving the 3-D linear Boltzmann equation, posed as the minimization of a scaled least-squares functional. Past results suggest the scaled least-squares method is a viable formulation of the 3-D equation; however, no scalable multilevel algorithm for solving the 3-D discrete system has been presented. For the scaled least-squares formulation in 1-D, there exists a multilevel algorithm that converges with a factor bounded by 0.1 for all parameter regimes. Unacceptable convergence results are observed when we extend this algorithm to 3-D.
The discrete system for the 3-D equation is generated using the variational form followed by a discretization of the directional variable using spherical harmonics and the spatial moments using finite elements. If the spatial moments are discretized with trilinear elements and a standard multilevel algorithm is used to solve the system, convergence significantly degrades as the mesh size is refined and the parameters enter the diffusive regime, a notorious trouble area. This slow convergence is due to the system of equations for the first-order moments, which is similar to the 3-D equation [ I - ∇ ∇ · ] , except perturbed by a small epsilon-Laplacian term.
We present a new 3-D finite element space for the first-order moments based on Raviart-Thomas finite elements. Using the new finite elements with the right multilevel algorithm, we get convergence rates for solving the scaled least-squares discrete system that are independent of mesh size for all parameter regimes. We also reveal a 2-D version of the 3-D finite element space. For the 2-D finite element space and multilevel algorithm, we provide theoretical results that imply that the 2-D algorithm converges independent of the mesh size for the worst case of epsilon equal to zero.
In addition to an improvement in the multilevel algorithm for the full scaled least-squares system, we extend ellipticity results for the functional to include anisotropic scattering. These new results imply the scaled least-squares method can be successfully applied to the anisotropic linear Boltzmann equation. We also remark on the effect of anisotropic scattering in the multilevel algorithm.