LEAST-SQUARES FINITE-ELEMENT SOLUTION OF THE NEUTRON-TRANSPORT EQUATION IN DIFFUSIVE REGIMES

A systematic solution approach for the neutron transport equation, bas ed on a least-squares finite-element discretization, is presented. Thi s approach includes the theory for the existence and uniqueness of the analytical as well as of the discrete solution, bounds for the discre tization error, and guidance for the development of an efficient multi grid solver for the resulting discrete problem. To guarantee the accur acy of the discrete solution for diffusive regimes, a scaling transfor mation is applied to the transport operator prior to the discretizatio n. The key result is the proof of the V-ellipticity and continuity of the scaled least-squares bilinear form with constants that are indepen dent of the total cross section and the absorption cross section. For a variety of least-squares finite-element discretizations this leads t o error bounds that remain valid in diffusive regimes. Moreover, for p roblems in slab geometry a full multigrid solver is presented with V ( 1, 1)-cycle convergence factors approximately equal to 0.1 independent of the size of the total cross section and the absorption cross secti on.