Finite elements-based Padé approximants for Helmholtz frequency response problems

The focus of this project is the application of rational functions in the approximation of the solution map of a parametric Helmholtz problem with homogeneous Dirichlet boundary conditions. Such an approximation can be applied, for instance, in frequency response problems, where one wants to understand how the solution of the Helmholtz problem changes with respect to the wavenumber.
In this report it is proven that the Helmholtz equation, endowed with Dirichlet, Neumann or mixed Dirichlet-Neumann conditions, is meromorphic in $\mathbb{C}$. Moreover, the regularity of the scattering problem, i.e. the Helmholtz equation coupled with Bohr-Sommerfield radiation conditions, is discussed in some detail. The hypothesis that the solution map of such a problem may still be meromorphic is formulated after the analysis of a simple example.
A rational Padé approximant, which relies on the solution of an optimization problem, is defined for Hilbert space-valued meromorphic functions. Several original algorithms, based on slight variations of the original definition, are described in detail. Numerical tests are used to compare the different algorithms.