Substructuring preconditioners for the neutron diffusion equation

In order to analyse the steady state of a nuclear power reactor, the neutron diffusion equation in the approximation of several groups of energy is typically used. This problem corresponds to a differential generalized eigenvalue problem. For the spatial discretization of the neutron diffusion equation a high order finite element method is used, which makes use of Lagrange polynomials defined on Gauss-Lobato-Legendre quadrature points. These polynomials provide a natural partition of the shape functions into vertices, edges, faces and interior shape functions. The most expensive computation to solve the discretized problem consists of the calculation of the solution of linear systems of equations whose coefficient matrices are associated with each group of energy. These linear systems are large and sparse, and a preconditioned Krylov iterative method is typically used to approximate their solution. Classically, a preconditioner based on an incomplete factorization is used. Nevertheless, this kind of preconditioners is expensive from the point of view of the memory used. Using the natural partition defined by the finite element basis several alternative preconditioners are investigated in this work, which are cheaper in terms of the memory used. First, substructuring preconditioners are studied, where the coupling between the different types of shape functions is neglected. Also, the preconditioned Schur Complement method that algebraically decouples the interior degrees of freedom from the boundary ones is investigated. The performance of the different approaches is studied numerically using a three-dimensional model of a reactor core