Adaptive Solution of the Neutron Diffusion Equation With Heterogeneous Coefficients Using the Mixed Finite Element Method on Structured Meshes

The neutron transport equation can be used to model the physics of the nuclear reactor core. Its solution depends on several variables and requires a lot of high precision computations. One can simplify this model to obtain the SP_N equation for a generalized eigenvalue problem. In order to solve this eigenvalue problem, we usually use the inverse power iteration by solving a source problem at each iteration. Classically, this problem can be recast in a mixed variational form, and then discretized by using the Raviart-Thomas-Nédélec Finite Element. In this article, we focus on the steady-state diffusion equation with heterogeneous coefficients discretized on Cartesian meshes. In this situation, it is expected that the solution has low regularity. Therefore, it is necessary to refine at the singular regions to get better accuracy. The Adaptive Mesh Refinement (AMR) is one of the most effective ways to do that and to improve the computational time. The main ingredient for the refinement techniques is the use of a posteriori error estimates, which gives a rigorous upper bound of the error between the exact and numerical solution. This indicator allows to refine the mesh in the regions where the error is large. In this work, some mesh refinement strategies are proposed on the Cartesian mesh for the source problem. Moreover, we numerically investigate an algorithm which combines the AMR process with the inverse power iteration to handle the generalized eigenvalue problem.