Wellenausbreitung in periodischen Strukturen und Mechanismen negativer Brechung
Zusammenfassung der Projektergebnisse
This research project was very successful in different fields. With the common interest in wave propagation in heterogeneous media, two research teams could interact and collaborate, they could share useful methods and they succeeded to deepen today’s knowledge on wave propagation in periodic structures and negative refraction. Some specific questions turned out to be of particular interest and much of the research efforts have been directed to these fields: 1.) Homogenization of periodic structures, 2.) Bloch wave analysis, 3.) Multi-scale methods. In the first field, the first mathematically rigorous derivation of a meta-material with negative refractive index was obtained. Moreover, the influence of the microstructure’s geometry on the homogenization results was investigated. The second field considers the expansion of solutions in Bloch waves to obtain appropriate radiation conditions in photonic crystals, which is also implemented in a numerical scheme. The third field embraces the design and numerical analysis of multi-scale methods, in particular the convergence of the schemes. Approaches for general heterogeneous media based on the Localized Orthogonal Decomposition have also been examined. The scientific output of this project is large (and, as a consequence, the number of publications is considerable). We emphasize that most publications have been made possible by some more informal exchange between the partners in this research project.
Projektbezogene Publikationen (Auswahl)
- A negative index meta-material for Maxwell’s equations. SIAM J. Math. Anal., 48(6):4155–4174, 2016
A. Lamacz and B. Schweizer
(Siehe online unter https://doi.org/10.1137/16M1064246) - A new Heterogeneous Multiscale Method for time-harmonic Maxwell’s equations. SIAM J. Numer. Anal., 54(6):3493–3522, 2016
P. Henning, M. Ohlberger, and B. Verfürth
(Siehe online unter https://doi.org/10.1137/15M1039225) - Effective Maxwell’s equations in general periodic microstructures. Applicable Analysis, 97(13):2210–2230, 2017
B. Schweizer and M. Urban
(Siehe online unter https://doi.org/10.1080/00036811.2017.1359563) - Localized Orthogonal Decomposition for two-scale Helmholtz-type problems. AIMS Mathematics, 2(3):458–478, 2017
M. Ohlberger and B. Verfürth
(Siehe online unter https://doi.org/10.3934/Math.2017.2.458) - A Bloch wave numerical scheme for scattering problems in periodic waveguides. SIAM J. Numer. Anal., 56(3):1848–1870, 2018
T. Dohnal and B. Schweizer
(Siehe online unter https://doi.org/10.1137/17M1141643) - A new Heterogeneous Multiscale Method for the helmholtz equation with high contrast. Mulitscale Model. Simul., 16(1):385–411, 2018
M. Ohlberger and B. Verfürth
(Siehe online unter https://doi.org/10.1137/16M1108820) - Effective Maxwell’s equations for perfectly conducting split ring resonators. Arch. Ration. Mech. Anal., 229(3):1197–1221, 2018
R. Lipton and B. Schweizer
(Siehe online unter https://doi.org/10.1007/s00205-018-1237-1) - Numerical homogenization of H(curl)-problems. SIAM J. Numer. Anal., 56:1570–1596, 2018
D. Gallistl, P. Henning, and B. Verfürth
(Siehe online unter https://doi.org/10.1137/17M1133932) - Outgoing wave conditions in photonic crystals and transmission properties at interfaces. ESAIM: Math. Model. Numer. Anal., 2018
A. Lamacz and B. Schweizer
(Siehe online unter https://doi.org/10.1051/m2an/2018026)
