Singular function enriched finite element methods for multigroup neutron diffusion problems

Vita.The angular singularities of the multigroup neutron diffusion equations are studied. The solution of the diffusion equations may have unbounded derivatives at points where the boundary has a corner, where two material interfaces intersect, where an interface and the boundary intersect, or where the boundary conditions change discontinuously. At such points the solution is said to contain an angular singularity. The engineering importance of angular singularities is that they may reduce the accuracy of finite difference and finite element methods used to numerically solve the diffusion equations. In this study, the form of the angular singularities of the multigroup equations is determined for many of the singularities occurring in typical reactor calculations. Results from approximation theory are applied to ascertain how many terms of singularities must be included as test and trial functions (i.e., as singular functions) in order to restore the accuracy of finite element methods used to solve the multigroup equations. Special techniques are derived for accurately and efficiently computing inner products involving singular functions. Effects of singular functions on the usual fission source iteration are discussed, and methods for mitigating these effects are presented. Preliminary results are reported, and recommendations for future study are made