In nuclear engineering, the λ-modes associated with the neutron diffusion equation are
applied to study the criticality of reactors and to develop modal methods for the transient analysis.
The differential eigenvalue problem that needs to be solved is discretized using a finite element
method, obtaining a generalized algebraic eigenvalue problem whose associated matrices are large
and sparse. Then, efficient methods are needed to solve this problem. In this work, we used a
block generalized Newton method implemented with a matrix-free technique that does not store all
matrices explicitly. This technique reduces mainly the computational memory and, in some cases,
when the assembly of the matrices is an expensive task, the computational time. The main problem
is that the block Newton method requires solving linear systems, which need to be preconditioned.
The construction of preconditioners such as ILU or ICC based on a fully-assembled matrix is not
efficient in terms of the memory with the matrix-free implementation. As an alternative, several block
preconditioners are studied that only save a few block matrices in comparison with the full problem.
To test the performance of these methodologies, different reactor problems are studied.
