Dr. Robert Altmann
Prof. Dr. Volker Mehrmann
Technische Universität Berlin
Photonic crystals are special materials having a periodic structure that can be used for
trapping, filtering and guiding light. The key property of such materials is their ability
to prevent light waves with specific frequencies from propagating in any direction.
Because it is very challenging to build photonic crystals featuring a spectrum that can comply
with the specific requirements of applications, a mathematical description and analysis of the electromagnetic properties of photonic crystals is needed, in order to support engineers in finding suitable components as well as
optimal crystal geometries for new promising applications.
The goal of the project is to develop an efficient solver for parameter-dependent non-linear eigenvalue problems arising in the search of photonic band-gaps. This solver should combine, in a computing-time optimal way, adaptive finite element methods (AFEM) for PDE eigenvalue problems, numerical methods for nonlinear eigenvalue problems, and low-dimensional approximations for a parameter space. The free parameters needed for the design of photonic crystals describe, e. g., the geometry of the crystal or the electromagnetic properties of the material. In order to optimize the
properties of the photonic crystals over a given parameter set, we need to apply techniques from model order reduction. We plan to use approximations of the eigenfunctions, obtained by AFEM for several parameters in order to construct a reduced basis. These computations may be performed in parallel and, ideally, result in a set of eigenfunctions that contains good approximations of the eigenfunctions for all parameter values. We want to approximate the set of locally-expressed eigenfunctions with a low-dimensional non-local basis.
Moreover, having efficient computations in mind, we need rigorous error bounds in order to equilibrate the different kinds of errors introduced at every level of approximation.
Indeed, the total numerical error includes the discretization error arising from the AFEM, the algebraic error arising from the (iterative) solution of the nonlinear eigenvalue problems, and the model reduction error arising from the discretization of the parameter set. Since all these errors are normally measured in different norms, a unifying setting has to be developed in order to be able to compare all types of errors.