The Search for Superconvergence in Spherical Harmonics Approximations

The occurrence of superconvergence in various first-order spherical harmonics approximations of the neutral particle transport equation is being investigated. Superconvergence refers to the added accuracy gained in evaluating the solution of the transport equation at optimally chosen base points of the finite element trial functions. It has been observed that this phenomenon is happening when primal and dual discretizations in space and angle lead to the same numerical result, a property also referred as primal-dual agreement. A systematic search is presented for primal-dual agreement on one-dimensional slab, tube, and spherical geometries and on Cartesian two-dimensional geometries based on complete and simplified P_n approximations. Primal-dual agreement was successfully obtained in all Cartesian geometries but not in tube and spherical geometries, due to the angular redistribution term.