Least-Squares Finite-Element Solution of the Neutron Transport
                        Equation in Diffusive Regimes

                             Thomas A. Manteuffel
                       Program in Applied Mathematics,
                      University of Colorado at Boulder,
                               Campus Box 526,
                        Boulder, Colorado, 80309-0526,
                        tmanteuf@sobolev.colorado.edu

                                     and

                               Klaus J. Ressel
                    Center for Computational Mathematics.
                      University of Colorado at Denver,
                      Campus Box 170, P .O. Box 173364,
                        Denver, Colorado, 80217-3364,
                          kressel@tiger.cudenver.edu

A systematic solution approach for the single group, steady state isotropic
neutron transport equation is considered that is based on a Least-Squares
variational formulation and includes theory for the existence and uniqueness
of the analytical as well as for the discrete solution, bounds for the
discretization error and an efficient multigrid solver for the resulting
discrete system.

In particular, the solution of the transport equation for diffusive regimes is
studied.  In these regimes the numerical solution of the transport equation is
more difficult since the equation becomes singularly perturbed with a limit
solution that is a solution of a diffusion equation.  Therefore, to guarantee
an accurate discrete solution, a discretization of the transport operator is
needed that is at the same time a good approximation of a diffusion operator
in diffusive regimes.  Only few discretizations are known that have this
property.

A Least-Squares discretization converts the first-order transport equation in
to a self-adjoint variational problem.  However, in combination with piecewise
linear elements in space, this discretization fails to be accurate in
diffusive regimes.  For this reason a scaling transformation is applied to the
transport operator prior to the discretization, which is increasing the weight
for the important components of the solution in the Least-Squares functional
and guarantees thereby an accurate discrete solution in diffusive regimes even
for piecewise linear elements.

Not only for slab geometry but also for x-y-z geometry it is proven that the
scaled Least-Squares bilinear form is continuous and V-elliptic with constants
independent of the total cross section and the scattering cross section.  For
a variety of discrete spaces this leads to bounds for the discretization error
that stay also valid in diffusive regimes.  Thus, the Least-Squares approach
in combination with the scaling transformation represents a general framework
for the construction of discretizations that are accurate in diffusive
regimes.

For the discretization with piecewise linear elements in space and slab
geometry a multigrid solver was developed that gives V-cycle convergence rates
in the order of 0.1 independent of the size of the total cross section.
Therefore, one full multigrid cycle of this algorithm computes a solution with
an error in the order of the discretization error.