Advances on a scaled least-squares method for the three-dimensional linear Boltzmann equation

In this dissertation, we study a multilevel algorithm for solving the 3-D linear Boltzmann equation that is posed as the minimization of a scaled least-squares functional. Past results demonstrate that the scaled least-squares method is a viable formulation of the 3-D equation; however, a scalable multilevel algorithm for solving the 3-D discrete system has not been proposed. In 1-D slab geometry, a multilevel algorithm has been previously defined which converges with a factor bounded by 0.1 for all parameter regimes. The extension of this algorithm to the 3-D case does not always lead to mesh-independent convergence.;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 we discretize spatially using trilinear elements and solve with a standard multilevel algorithm, then degradation in multilevel convergence is observed as the mesh size is refined and the parameters enter the notorious diffusive regime. This slow convergence is due to the system of equations for the first-order moments, which is similar to the 3-D equation I-grad div, except perturbed by a small epsilon-Laplacian term.;We present a new 3-D finite element space based on Raviart-Thomas finite elements in order to properly discretize the first-order moments. Using the new finite elements with the right multilevel algorithm, convergence rates for solving the scaled least-squares discrete system are observed to be 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, theoretical results are presented that imply 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.