# Infinite elements for exterior Maxwell problems

## Address

Ass. Prof. Dr. Lothar Nannen

Institute for Analysis and Scientific Computing (Inst. E 101)

Vienna University of Technology

Wiedner Hauptstraße 8-10

1040 Wien, Austria

## Aims

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 modeling 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.

## Software

ngs-waves: Add-on to the finite element package ngsolve for acoustic and electromagnetic scattering and resonance problems.

## Project Members

The project was finished in the beginning of 2019. Former members of the project were

- Matthias Hochsteger
- Karl Hollaus
- Lothar Nannen (PI)
- Markus Schöbinger
- Markus Wess
- Thomas Heitzinger
- Michael Leumüller

## Funding Period

FWF grant P26252 from 08/2014 until 01/2019