In the iRTG lecture series four members of CRC 1173 will talk about topics whithin the analysis and numerics of wave phenomena. The lectures are directed to Ph.D. students (in particular of the integrated research training group of CRC 1173) and to advanced master students with a solid background in partial differential equations. The series is organized by Willy Dörfler and Roland Schnaubelt.

Note that the final three lectures take place on Friday 7 July, 14 July, and 21 July at 09:45-11:15 in seminar room 1.067.

Carsten Rockstuhl

The lectures take place on 24 April, 8 May, and 15 May.

As a physicist in the SFB, the lectures I will concentrate on applied aspects of electromagnetic waves; but of course explored on mathematical grounds. Specifically, I will discuss the interaction of light with matter that has critical features at the nanoscale. I will emphasise the importance of exploiting resonances to notably enhance the optical response. These resonances lay the ground for various applications that are equally sketched. Three lectures are given on the following subjects.

In a first lecture, the notion of plasmonics is introduced and particularly localized surface plasmon polaritons are discussed. The latter being hybrid states of excitation where the electromagnetic field is coupled to the charge density oscillation in metals that are spatially confined to form nanoparticles. In the first lecture we concentrate on spherical particles for which an analytical solution to the scattering problem exists. Once a localized surface plasmon polariton is excited, a huge field enhancement close to the nanoparticle and a large scattering and absorption cross section in a narrow spectral region are observed.

In a second lecture, the further developments are discussed when not just spherical particles are considered but more complexly shaped objects. This is the wide field of optical nanoantennas. It eventually exploits the opportunity to induce a response in the nanoparticle that has different contributions in terms of electric and magnetic multipole moments. These multipole moments can be individually considered in the context of, e.g. meta-atoms as the basic building blocks of metamaterials. However, the interference among multiple multipole moments is equally important as it provides excellent control on the radiation characteristics of these nanoantennas.

In a third lecture, we discuss the possibility to observe many of the effects studied in plasmonics, that rely on metallic nanostructures, with dielectric structures only. This approach has the clear advantage that absorption is heavily suppressed at the expenses of a lower confinement of electromagnetic fields. However, using high-permittivity materials such as semiconductors, many interesting effects can be equally observed.

References.

• Stefan Maier: Plasmonics. Fundamentals and Applications.
• Lukas Novotny and Bert Hecht: Principles of Nano-Optics.
• Craig F. Bohren and Donald R. Huffman: Absorption and Scattering of Light by Small Particles.

Willy Dörfler

The lectures take place on 22 May, 29 May, and 12 June.

The lecture addresses the following two relevant topics for finite element computations: Is there a theoretical or practical reliable measure to estimate the error of the approximated solution? Can error estimation be used to significantly diminish the computational effort in an optimal way? The main techniques to control the approximation are local mesh-refinement, the (local) polynomial degree of the finite elements and the timestep size for time-dependent problems. We present an overview of main results for elliptic and parabolic equations.

References.

• M. Ainsworth and J. T. Oden: A Posteriori Error Estimation in Finite Element Analysis. John Wiley, New York, 2000.
• C. Schwab: p- and hp-finite element methods. Theory and applications in solid and fluid mechanics. Clarendon Press, Oxford, 1998.

Andreas Kirsch

The lectures take place on 19 June, 26 June, and 3 July.

A) Introduction and Basic Linear Theory
After an introduction into the notions of an inverse problem and the ill-posedness of a problem we will consider some examples, in particular formulated as integral equations of the first kind, and introduce the general regularization technique with filter functions.

(B) Particular Regularization Strategies
In this part we will study the classical Tikhonov regularization technique with a priori and a posteriori choices of the regularization parameter. Then we will introduce the Landweber method as an example of an iterative regularization strategy.

(C) The Problem of Impedance Tomography
This is an example of a nonlinear inverse problem. After a short repetition of the direct problem - an elliptic boundary value problem - We will discuss the Factorization Method to determine the support of the contrast and comment on other (iterative) techniques.

References. We will essentially follow the monograph:

• A. Kirsch: An Introduction to the Mathematical Theory of Inverse Problems (2nd Edition). Springer, 2011.

Jens Rottmann-Matthes

The lectures take place on Friday 7 July, 14 July, and 21 July at 09:45-11:15 in seminar room 1.067.

In this series of lectures we will consider patterns in equivariant evolution equations and their stability properties.

I) Equivariant evolution equations
We begin by introducing the abstract notion of equivariant evolution equations. Equivariance plays an important role for the appearance of patterns like traveling and rotating waves. We look at concrete examples which exhibit traveling or rotating waves and also generalize to other symmetry solutions.

II) Stability of traveling waves
We present a method of how to capture traveling waves and other patterns in equivariant evolution equations numerically. Then we show that spectral stability implies nonlinear stability of traveling waves in first order hyperbolic systems of partial differential equations.

III) The case of second order evolution equations
In the last lecture we will consider systems of semilinear wave equations. We show how the approach of capturing traveling and rotating waves generalizes to this case. Furthermore we present a stability traveling waves result and look at the spectrum of the linearized operators.

References.

• W.-J. Beyn, D. Otten, J. Rottmann-Matthes: Stability and Computation of Dynamic Patterns in PDEs.
• W.-J. Beyn, D. Otten, J. Rottmann-Matthes: Freezing traveling and rotating waves in second order evolution equations
• J. Rottmann-Matthes: Stability and freezing of nonlinear waves in first order hyperbolic PDEs.