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Ringvorlesung Wavephenomena (Winter Semester 2021/22)

Localized Orthogonal Decomposition for Helmholtz problems

Dr. Barbara Verfürth

In this series of lectures, the Localized Orthogonal Decomposition (LOD), a computational multiscale method, will be introduced and analyzed for the Helmholtz equation with heterogeneous coefficients and large wavenumber. The numerical simulation of this problem using standard discretization schemes (e.g., the finite element method) would require a very fine mesh resolution because of (i) the heterogeneity of the coefficients and (ii) the high-frequency regime. The LOD is based on splitting the solution space into a finescale component, characterized as the kernel of an interpolation operator, and an „orthogonal“ multiscale space. Using this (low-dimensional) multiscale space in a Galerkin method yields much better approximations than the finite element method on an a comparable coarse mesh.
We will first introduce the Helmholtz problem under consideration and collect some important analytical and numerical results. Then, we will introduce the LOD in detail and rigorously analyze its discretization error. In particular, we will explain how and why the results differ from those for the finite element method. If time permits, we will finally give an outlook on how the methodology can be extended to high-contrast coefficients and/or nonlinear problems.


The lectures take place on November 8, November 15, and November 22.


Boundary Integral Equation Methods for Time-Harmonic Scattering Problems

PD Dr. Tilo Arens

We consider the formulation of boundary integral equations for time-harmonic scattering problems in acoustics and electromagnetics, i.e. for the Helmholtz equation and the Maxwell system. Starting from the definitions and mapping properties of potentials and boundary operators in appropriate Sobolev spaces, we discuss existence and uniqueness of solution results for selected boundary integral equations. For the numerical solution of boundary integral equations of the first kind, operator preconditioning techniques derived from the Calderon projector are particularly useful and will be presented.

The lectures take place on November 29, December 12, and December 13.


Conserved energies for the one dimensional Gross-Pitaevskii equation

JProf. Dr. Xian Liao

In this minicourse I will present a family of conserved energies for the one dimensional Gross-Pitaevskii equation, which is the defocusing cubic nonlinear Schrödinger equation but with nonzero boundary condition at infinity. I will speak more precisely about

- The (generalized) energy spaces, in the first lecture,
- The Lax pair structure and the transmission coefficient, in the second lecture,
- The conserved energies and the conservation of the energy norms, in the last lecture.

Lecture Notes

The lectures take place on December 20, January 10, and January 17.


Wave packet analysis

Prof. Dr. Dorothee Frey

We give an introduction into aspects of wave packet analysis with applications to the well-posedness of wave equations with low regularity coefficients. In dispersive PDEs, the wave packet transform of Córdoba-Fefferman serves as a suitable replacement and refinement of Littlewood-Paley theory, realising a dyadic-parabolic decomposition in phase space. We will investigate how this can be used in the study of the wave equation.

Topics include wave packets and their interaction with Fourier integral operators, Hardy spaces as function spaces built from wave packets, Sobolev embeddings, and well-posedness results of wave equations on Hardy and L^p spaces.

The lectures take place on January 24, January 31, and February 7.


Schedule
Lecture: Monday 14:00-15:30 20.30 1. OG R. 1.66/ 1.67
Lecturers
Lecturer PD Dr. Tilo Arens
Office hours: Wednesday 11:00-12:00 Uhr
Room 1.047 Kollegiengebäude Mathematik (20.30)
Email: tilo.arens@kit.edu
Lecturer Prof. Dr. Dorothee Frey
Office hours: Tuesday, 10am - 11am, and by appointment
Room 2.042 Kollegiengebäude Mathematik (20.30)
Email: dorothee.frey@kit.edu
Lecturer JProf. Dr. Xian Liao
Office hours: by appointment
Room 3.027 Kollegiengebäude Mathematik (20.30)
Email: xian.liao@kit.edu
Lecturer JProf. Dr. Barbara Verfürth
Office hours: by appointment
Room 3.024 Kollegiengebäude Mathematik (20.30)
Email: barbara.verfuerth@kit.edu