Applied Mathematics and Plasma Physics

Travis Austin, *Advances on a Least-Squares Method for the 3D Linear
Boltzmann Equation*, PhD thesis, University of Colorado at Boulder, 2001

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.

@phdthesis{austin-2001-advances,

author = {Travis Austin},

title = {Advances on a Least-Squares Method for the 3D Linear
Boltzmann Equation},

year = {2001},

urlps = {http://math.lanl.gov/~austint/Papers/austin-2001-advances.ps.gz},

school = {University of Colorado at Boulder}

}