NFMV016 Spring 2021 Introduction to Homogenization

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This page contains the program of the course: lectures, exercise sessions and computer labs. Other information, such as learning outcomes, teachers, literature and examination, are in a separate course PM.

Program

We are going to meet on Tuesdays 13:15-15 and Thursdays 10-11:45 in zoom

https://chalmers.zoom.us/j/62813626840 
Password: 719987  

Lectures

Date Reading Content To do Notes
25/03
  • Necas (Ch. 1),
  • Evans (Ch. 5)
 L^p spaces, Sobolev Spaces, embeddings, the Friedrichs and Poincaré inequalities Exercises week 1 F1-annotated
30/03, 1/04
  • Evans (Ch. 6),
  • Necas (Ch. 1),
  • [BeRy] Ch. 1.4, Ch. 2; 
  • [CiDo] Ch. 2 for oscillating functions; Ch. 4 for Lax-Milgram
 Trace theorem. The Lax-Milgram lemma. The du Bois-Reymond lemma. Rapidly oscillating functions and averaging lemma Exercises week 2

F2-annotated

F3-annotated
6/04, 8/04
  • [BeRy] Ch. 2 (1D), Ch. 4 (asymptotic expansions)
  • [CiDo] Ch. 2.3 Th. 2.6 (averaging lemma), Ch. 5.1-5.3 examples and 1D case; Ch. 7 asymptotic expansions
 Homogenization in 1D. Method of asymptotic expansions. Cell problem. Exercises week 3

F4-annotated

F5-annotated
13/04, 15/04
  • [BeRy] Ch. 4 (formal expansion).
  • [CiDo] Ch. 7 (formal expansion, homogenized matrix, error estimates)
 Asymptotic expansions cont.. Estimates for the homogenized matrix, error estimates No hand-in this week. Finish the exercises from week 1, 2 and 3. 

F6-annotated

F7-annotated
20/04, 22/04
  • [CiDo] Ch. 7 error estimates
  • [BeRy] Ch. 5 compensated compactness
 Error estimates (final) and the extension operator. Compensated compactness and oscillating test functions Exercise week 5

F8-annotated

F9-annotated
27/04, 29/04

 

Extension operator (cont.)

Two-scale convergence (stationary problems)

 No hand-in this week. Finish the previous hand-in.

F10_annotated

F11-annotated
4/05, 6/05
  • [Al] 2-scale in perforated domains (Neumann problem)
  • [CiMu97] Dirichlet problem in perforated domains
 Perforated domains and inclusions Start preparing the presentation

F12-annotated

F13-annotated
11/05
  • [Al] Highly heterogeneous media
 Some words about the capacity; spectral problems with weight; stationary double porosity Work on your presentation F14-annotated
18/05, 20/05
  • [CiDo] homogenization parabolic (oscillating functions)

 

Nonstationary problems

F15-annotated.pdf

F16-annotated.pdf
25/05, 27/05

Seminar presentations

25/05: Erik, Guillaume

27/05: Malin, Per

?: Morgan 

 

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Mandatory assignments

In order to take the exam you need to submit solutions to 50% of weakly problems. If there is an odd number n=2k+1 of problems, submit k+1. You will upload the solutions in canvas as (one) pdf-file. 

 

Reference literature:

  1. [CiDo] Cioranescu, D., and Donato, P., An Introduction to Homogenization. Vol. 17. Oxford: Oxford University Press, 1999.
  2. [BeRy] Berlyand, L. and Rybalko, V., Getting Acquainted with Homogenization and Multiscale. Springer International Publishing, 2018.
  3. [ChPiSha] Chechkin, G., A. Piatnitski, and A. Shamaev. ``Homogenization: Methods and Applications, Translations of Mathematical Monographs 234.'' Amer. Math. Soc., Providence (2007).
  4. [Al] Allaire, Grégoire. ``Homogenization and two-scale convergence.'' SIAM Journal on Mathematical Analysis 23.6 (1992): 1482-1518.
  5. [CiMu97] Cioranescu, D., & Murat, F. (1997). A strange term coming from nowhere. In Topics in the mathematical modelling of composite materials (pp. 45-93). Birkhäuser, Boston, MA.

Functional analysis, Sobolev Spaces, inequalities and other related topics:

5. Necas, J. (2011). Direct methods in the theory of elliptic equations. Springer Science & Business Media.

6. Evans, L. C. (1998). Partial differential equations. Graduate studies in mathematics, 19(2).

7. Adams, R. A., & Fournier, J. J. (2003). Sobolev spaces. Elsevier.

8. Allaire, G. (2007). Numerical analysis and optimization: an introduction to mathematical modelling and numerical simulation. Oxford university press.

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Course summary:

Course Summary
Date Details Due