hp-Adaptive Discontinuous Galerkin Methods for Neutron Transport Criticality Problems

In this article we consider the application of high-order/$hp$-version adaptive discontinuous Galerkin finite element methods (DGFEMs) for the discretization of the $k_{eff}$-eigenvalue problem associated with the neutron transport equation. To this end, we exploit the dual weighted residual approach to derive a reliable and efficient a posteriori error estimate for the computed critical value of $k_{eff}$. Moreover, by exploiting the underlying block structure of the $hp$-version DGFEM, we propose and implement an efficient numerical solver based on exploiting Tarjan's strongly connected components algorithm to compute the inverse of the underlying transport operator; this is then utilized as an efficient preconditioner for the $k_{eff}$-eigenvalue problem. Finally, on the basis of the derived a posteriori error estimator we propose an $hp$-adaptive refinement algorithm which automatically refines both the angular and spatial domains. The performance of this adaptive strategy is demonstrated on a series ...