  Local Fourier analysis is a commonly used tool to assess the quality and aid
in the construction of geometric multigrid methods for translationally
invariant operators. In this paper we automate the process of local Fourier
analysis and present a framework that can be applied to arbitrary, including
non-orthogonal, repetitive structures. To this end we introduce the notion of
crystal structures and a suitable definition of corresponding wave functions,
which allow for a natural representation of almost all translationally
invariant operators that are encountered in applications, e.g., discretizations
of systems of PDEs, tight-binding Hamiltonians of crystalline structures,
colored domain decomposition approaches and last but not least two- or
multigrid hierarchies. Based on this definition we are able to automate the
process of local Fourier analysis both with respect to spatial manipulations of
operators as well as the Fourier analysis back-end. This automation most
notably simplifies the user input by removing the necessity for compatible
representations of the involved operators. Each individual operator and its
corresponding structure can be provided in any representation chosen by the
user.
