Abstracts of talks given at an GRK-Event

Winter term 2014/15

Summer term 2014

20.08 - 22.08, 1C-03 - Erwan Faou

Introduction to Geometric Numerical Integration for PDEs

In this lecture, I will mainly consider the numerical integration of the nonlinear Schrödinger equation over long times and discuss in particular the numerical stability of solitons in the one dimensional case. I will explain the phenomenon of time and space resonances, and show that under some restriction between the time and space discretization parameter, we can prove some orbital stability result for numerical schemes.

Tuesday, 23.06, 16:30-17:15, 1C-04 - Maria Lopez-Fernandez

Runge-Kutta Generalized Convolution Quadrature

We present a generalization of Lubich's Convolution Quadrature based on Runge-Kutta methods which allows for variable time steps. The main application of our method is the time integration of retarded potentials arising in wave scattering problems. The algorithmic realization of the new method relies on contour integral techniques in the complex plane. Numerical experiments are provided to show the potential of our approach.

Tuesday, 13.05, 17:30-18:30, 1C-0x - Prof. Dr. Carsten Rockstuhl

Interaction of light with plasmonic nanostructure

The interaction of light with metallic nanostructure can be driven into resonance by exploiting surface plasmon polaritons (SPP). There, the electromagnetic field is coupled to the oscillation of the charge density in the metals. While distinguishing between propagating and localised SPPs, bound to an infinitely extended interface between a metal and a dielectric or bound to a small metallic nanoparticle, respectively, SPPs allow to localise light into tiny volumes, provide a spectral response in narrow frequency ranges, and provide unorthodox scattering characteristics for nanostructures with complicated shapes. These features are extremely appealing and can be beneficially used for various purposes. Examples thereof are optical metamaterials, for optical information processing at the nanoscale, for the photon management in solar cells, or to enhance chemical, nonlinear and quantum effects. The resonant character and the critical geometrical size of the nanostructures that is comparable to and in most cases even smaller than the wavelength of light, usually requires to solve Maxwell’s equations without further approximation to grasp all the properties. This is a major challenge which asks for contributions from multiple perspectives. In this talk, I will give an overview on my research in this field with emphasis on numerical and analytical aspects of light propagation in and its interaction with plasmonic nanostructures, where linear and nonlinear optical aspects are important.

Monday, 12.05, 15:45-16:30, 1C-04 - Prof. Anke-Susanne Müller and Markus Schwarz

Electromagnetic Wave Phenomena in Particle Accelerators

Accelerated charged particles emit electromagnetic radiation. At modern particle accelerators, this radiation is in the x-ray regime, but recent advances in accelerator physics made it possible to create intense THz-radiation as well. Their size ranges from 10m to 27km and applications range from material science to arts to cancer therapy and collision experiments that give insight into the structure of elementary particles. Usually up to a billion particles are accelerated simultaneously and the design and understanding of an accelerator requires solving the equations of motion for many particles in a given setup of electric and magnetic fields. A full understanding of the emitted radiation requires novel numerical tools for solutions to Maxwell equations. Besides the "wanted" radiation, the electrons create electromagnetic wake fields, propagating in a wave guide. The solution to the inverse problem would then yield information about important accelerator parameters.
This talk gives an introduction to accelerator physics and the different numerical methods used in modeling modern particle accelerators.

Winter term 2013/2014

Monday, 10.02, 16:30-17:15, 1C-04 - Martin Meyries (Universität Halle-Wittenberg)

A unified framework for parabolic equations with mixed boundary conditions and diffusion on interfaces

We investigate parabolic equations in a general non-smooth setting with emphasis on mixed interface and boundary conditions. In particular, we allow for dynamics and diffusion on a Lipschitz interface and on the boundary, where diffusion coefficients are only assumed to be bounded, measurable and positive semidefinite. In the bulk, we additionally take into account diffusion coefficients which may degenerate towards a Lipschitz surface. For this problem class, we introduce a unified functional analytic framework based on sesquilinear forms and show maximal regularity for the corresponding abstract Cauchy problem. This is joint work with K. Disser, A.F.M. ter Elst and J. Rehberg.

Summer term 2012

Monday, 09.07, 15:45-17:15, 1C-03 - Mike Botchev (Univ. Twente)

On efficient solution of large initial value problems by block Krylov projections

Exponential time integrators are a powerful tool for numerical solution of large time dependent problems. The actions of matrix functions on vectors, necessary for exponential integrators, an be efficiently computed by different elegant numerical techniques, such as Krylov subspaces. Unfortunately, in some situations the additional work required by exponential integrators per time step is not paid off because the time step can not be increased too much due to the accuracy restrictions. We consider one class of problems when we can get around this difficulty. This problem class is formed by linear systems of ordinary differential equations (ODEs) where the non-autonomous term lives in a small-dimensional subspace. We show that in this case the problem can be solved with a single block Krylov subspace and there is no time stepping process involved. We refer to this method as EBK, exponential block Krylov method. An extension of EBK to nonlinear ODE systems is an open problem and we discuss possible ways for such an extension.

Monday, 25.06, 15:45-17:15, 1C-03 - Andrii Khrabustovskyi (NAS of Ukraine)

Periodic differential operators with asymptotically preassigned spectrum

We will deal with the following operators in $\mathbb{R}^n$ : $\mathcal{A}=-\frac{1}{b(x)}\sum\limits_{k=1}^n\displaystyle{\frac{\partial}{\partial x_k}\left({a}(x)\frac{\partial}{\partial x_k}\right)$
Here ${a},{b}$ are bounded above and bounded away from zero $\mathbb{Z}^n$ -periodic functions. We denote by $\mathcal{L}_{\mathrm{per}}$ the set of such operators.

In the talk we will discuss the following result: for an arbitrary $L>0$ and for arbitrary pairwise disjoint intervals $(\alpha_j,\beta_j)\subset[0,L], j=1,\dots,m (m\in\mathbb{N})$ we construct the family of operators $\{\mathcal{A}^\varepsilon\in\mathcal{L}_{\mathrm{per}}\}_{\varepsilon>0}$ such that the spectrum of $\mathcal{A}^\varepsilon$ has exactly $m$ gaps in $[0,L]$ when $\varepsilon$ is small enough, and these gaps tend to the intervals $(\alpha_j,\beta_j)$ as $\varepsilon\to 0$ . The corresponding functions $a^\varepsilon,b^\varepsilon$ can be chosen in such a way that their ranges have at most $m+1$ values.
The idea how to construct the family $\left\{\mathcal{A}^\varepsilon\right\}_{\varepsilon>0}$ is based on methods of the homogenization theory.
Also we will discuss a similar result obtained for periodic Laplace-Beltrami operators.