General Information
Short Course: "Introduction to Asymptotic Analysis and Homogenization Theory"
Instructor: SFB 1313 associated researcher Prof. Taras Mel'nyk (internal research project I-08) from the Taras Shevchenko National University of Kyiv (Ukraine)
Schedule: 10th to 12th February 2025
Duration: Three afternoon sessions
Location: Campus Vaihingen, location tba
ECTS: 3
Content of the course
Asymptotic analysis is a powerful mathematical tool for understanding the behavior of functions and solutions to problems as inputs become large or small. Asymptotic approaches are widely used in various fields, including physics, engineering, and computer science, to simplify complex equations and gain insights into the behavior of systems. Homogenization theory is a collections of asymptotic methods used to study the behavior of heterogeneous materials by averaging their properties to derive effective macroscopic models. The course consists of three lectures. The first lecture introduces the main concepts of asymptotic analysis, while the subsequent ones focus on the theory of homogenization. The presentation of the material will be brief but comprehensively to ensure a clear understanding of the topic, with rigorous mathematical proofs and justifications
provided.
More Information
- 10th February 2025: 2 - 3:30 pm CET
- 11th February 2025: 2 - 3:30 pm CET
- 12th February 2025: 2 - 3:30 pm CET
More information will be available soon
This course is organised by SFB 1313
More information will follow soon.
Lecture 1:”Asymptotic expansions” (Monday, 10th February; 14:00 - 15:30)
Asymptotic series, which often diverge, are the main concept in asymptotic analysis. In 1828, Niels Abel famously remarked: ”Divergent series are the invention of the devil.” In this lecture, we will explore the enigmatic nature of divergent series and uncover their properties. Additionally, we will introduce the concept of a formal asymptotic solution of a linear operator equation in a Banach space and prove a lemma on a general approach to justify the constructed asymptotics.
Lecture 2: ”Two-scale expansion method for boundary-value problems in thin perforated domains with rapidly variable thickness” (Tuesday, 11th February; 14:00 - 15:30)
The lecture will introduce the two-scale expansion technique, which is used to construct an asymptotic approximation for the solution to a mixed inhomogeneous boundary-value problem with rapidly oscillating coefficients in a thin perforated domain with rapidly changing thickness. The corresponding one-dimensional homogenised problem will be deduced, and asymptotic estimates for the difference between the solution of the original problem and the solution to the homogenised problem will be established.
Lecture 3: ”Inner-layer asymptotics for coupling in partially perforated domains” (Wednesday, 12th February; 14:00 - 15:30)
The lecture will examine the Poisson equation in a domain consisting of a porous medium and an imperforate region, with Neumann conditions prescribed at the perforation boundaries. Assuming the porous medium has a periodic structure with a small period, we explore the limiting behavior of the problem. Consequently, we determine the effective behavior in both the porous and imperforate regions. A crucial aspect of this study is deriving the correct coupling conditions at the common interface, which is achieved using inner-layer asymptotics. Finally, I’ll present a new, yet unpublished result: the construction and justification of an asymptotic expansion for the solution.
Samaneh Vahid Dastjerdi
Dr.-Ing.Postdoctoral Researcher, Management, Project MGK, Central Project Z