Modul MKMECH
Table of contents
Multiscale Analysis (Master-Modul: MKMECH)
The objective of mutliscale analysis are equations and models (e.g. ODEs, PDEs or variational problems) that feature different behaviors on different length and/or time scales. In many cases one is interested in effective properties, i.e. the behavior of the equation on the largest scale. The understanding of these properties is typically a difficult task, since effective properties of equations with multiple scales often emerge from a complex interplay of properties on smaller length scales. One prototypical problem of multiscale analysis is the following: Given a model with multiple scales, we want to find a so called effective model, i.e. a model that is simpler in the sense that it only resolves the macroscopic scale, but that is still a good approximation for the original model. In this course we study various analytical methods to approach this problem. Our main focus is
- the theory of homogenization of elliptic equations and systems,
- homogenization of equations with random coefficients,
- applications to the system of linearized elasticity.
Furthermore, if time permits we briefly discuss
- averaging of slow-fast dynamical systems,
- dimension reduction in variational models.
In the course we introduce and apply various methods and notions from PDE-theory and Functional Analysis; e.g. Sobolev spaces, existence theory for weak solutions to elliptic systems, weak and strong convergence, ergodic theory.
Course information
- Lecturer: Prof. Dr. Stefan Neukamm
- Place and time:
Thursday 11.10 - 12.40 am (3rd double period), WIL A221
Fri. 07.30 - 09.00 pm (1st double period), WIL A120
(tutorial is integrated) - Modul: Math-MA-MKMECH
- Literature:
- A. Bensoussan, J. L. Lions, G. Papanicolaou, Asymptotic analysis of periodic structures
- H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations
- L. C. Evans, Partial differential equations
- Language: German and English on demand
- Required Qualifications: Functional Analysis and Basic knowledge of PDE-theory (i.e. Math-Ba-HANA), the concept of Sobolev-Spaces will be introduced briefly, but not discussed in detail