Adaptive mesh refinement solution techniques for the multigroup SN transport equation using a higher-order discontinuous finite element method

In this dissertation, we develop Adaptive Mesh Refinement (AMR) techniques for the steady-state multigroup SN neutron transport equation using a higher-order Discontinuous Galerkin Finite Element Method (DGFEM). We propose two error estimations, a projection-based estimator and a jump-based indicator, both of which are shown to reliably drive the spatial discretization error down using h-type AMR. Algorithms to treat the mesh irregularity resulting from the local refinement are implemented in a matrix-free fashion. The DGFEM spatial discretization scheme employed in this research allows the easy use of adapted meshes and can, therefore, follow the physics tightly by generating group-dependent adapted meshes. Indeed, the spatial discretization error is controlled with AMR for the entire multigroup SN-transport simulation, resulting in group-dependent AMR meshes. The computing efforts, both in memory and CPU-time, are significantly reduced. While the convergence rates obtained using uniform mesh refinement are limited by the singularity index of transport solution (3/2 when the solution is continuous, 1/2 when it is discontinuous), the convergence rates achieved with mesh adaptivity are superior. The accuracy in the AMR solution reaches a level where the solution angular error (or ray effects) are highlighted by the mesh adaptivity process. The superiority of higher-order calculations based on a matrix-free scheme is verified on modern computing architectures.;A stable symmetric positive definite Diffusion Synthetic Acceleration (DSA) scheme is devised for the DGFEM-discretized transport equation using a variational argument. The Modified Interior Penalty (MIP) diffusion form used to accelerate the SN transport solves has been obtained directly from the DGFEM variational form of the S N equations. This MIP form is stable and compatible with AMR meshes. Because this MIP form is based on a DGFEM formulation as well, it avoids the costly continuity requirements of continuous finite elements. It has been used as a preconditioner for both the standard source iteration and the GMRes solution technique employed when solving the transport equation. The variational argument used in devising transport acceleration schemes is a powerful tool for obtaining transport-conforming diffusion schemes.;XUTHUS, a 2-D AMR transport code implementing these findings, has been developed for unstructured triangular meshes.