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.