Infinite elements for exterior Maxwell problems

In this project we develop and analyze highly accurate and fast solvers for electromagnetic scattering as well as resonance problems in open systems. Open systems means that some physical effects are non-local and physical quantities such as the electric or magnetic field exist on unbounded domains. Such problems occur for example in the modelling and the simulation of meta-materials, photonic cavities or plasmon resonances. For computational purposes, these unbounded domains are truncated to bounded domains using transparent boundary conditions at the artificial boundaries. There exist several numerical realizations of transparent boundary conditions. In this project we study new infinite element methods based on a radiation condition called pole condition, which characterizes radiating solutions via the singularities of their (partial) Laplace transforms. In numerical experiments these methods show a fast exponential convergence with respect to the number of additional degrees of freedom per degree of freedom on the artificial boundary. However, only a few one-dimensional convergence results for scalar problems are available. The project is subdivided into two parts: A numerical analysis part and an algorithmic part. In the analytic part of this project the convergence of the infinite element methods will be established for scattering as well as resonance problems. In the algorithmic part new iterative solvers will be combined with the infinite element methods to provide fast, highly accurate solvers for exterior Maxwell problems. For bounded domains, these solvers are based on domain decomposition preconditioners for a mixed hybrid discontinuous Galerkin formulation with non-standard penalty terms. Preliminary studies show small iteration numbers even without any coarse grid correction. Therefore they are well-suited for large scale problems.

The project dealt with the numerical simulation of electromagnetic waves. Such waves can be used to describe the propagation of light. Thereby, the color of light is determined by the frequencies of the waves, which form the complete signal. The numerical computation of these so-called resonance frequencies were the main target of the project. They can amongst others be used to construct lasers radiating light in a distinct color. Such resonance problems can be solved on powerful computers, if modern numerical methods are used. Unfortunately, the solutions of such a computation often contain not only the sought physical resonances but also artificial resonances. The latter are purely generated by the numerical method. The origin for these unwanted, artificial resonances are studied in this project. Moreover, new algorithms were developed to reduce artificial resonances and to distinguish them from physical relevant ones. The main difficulty in solving resonance problems numerically is the fact, that waves are non- local in the sense, that local perturbation may cause effects far away from the perturbation. Hence, actually large domains of propagation would have to be simulated, which is impossible due to the massive computational costs. Specialized numerical methods are needed to truncate the simulation domain while keeping the resulting errors small. We studied these errors with analytical tools and proved, that at least with sufficiently large effort correct numerical results are obtained. This ensures the validity of the used numerical methods. Nevertheless, even using these new methods the computations stay costly. Therefore, the software had to be improved using modern parallelization techniques and efficient third party solvers for some parts of the problem. Moreover, the usability of the software could be enhanced. A new interface in the programming language Python enables users to work with the code and the new numerical methods. This facilitates dissemination of the results of this project to non-mathematical users.