# MASTER ''Mathematical Modelling''

# Ecole Polytechnique and University Paris 6

# M2 (second year): course by G. Allaire and F. Alouges (2013-2014)

# Homogenization

This course is an introduction to homogenization theory with a view
on multiscale modelling and numerical simulation. Mathematically,
homogenization can be defined as a theory for averaging partial
differential equations. It has many potential applications, including
the derivation of effective properties for heterogeneous media, the
rigorous definition of composite materials, the macroscopic modelling
of microscopic systems and the design of multiscale numerical algorithms.
We shall illustrate these issues by considering various examples from
continuum mechanics, physics or porous media engineering.

We first deal with the homogenization of periodic structures by
the method of two-scale asymptotic expansions which will be rigorously
justified by the notion of two-scale convergence. We shall discuss
issues related to correctors, boundary layers, error estimates as
well as some generalizations to the non-periodic case. We then use
this periodic homogenization theory as a modelling tool for deriving
macroscopic models for heterogeneous media. Finally we introduce
so-called multiscale finite element methods for performing
numerical homogenization.

#### Schedule (Thursday from 15H45 to 17H45 at the lecture room of CMAP)

Classes by G. Allaire on September 26, October 10, 17, 24 and 31,
November 7, December 5.

Classes by F. Alouges on October 3, November 14, 21 and 28, December 12 and 19.

#### Bibliography

Some **lecture notes** partly covering the topic of the course:

1) the theory of periodic homogenization,

2) homogenization in porous media,

3) numerical methods of homogenization.

A reference paper on the topic of the course:

Homogenization and two-scale convergence

Some **slides** partly covering the topic of the course:

1) introduction to periodic homogenization (Darcy's law, kinetic equations),

2) convergence theory,

3) transport in porous media.

Some **problems and exercises** for the students who want to practice
before the final exams (sorry, they are in French !):

homogenization in a porous medium,

homogenization of Stokes flows,

**Final exams:**

January 2011 (in French),

January 2011 (in English),

January 2012 (in French),

January 2012 (in English),

January 2013 (in French),

January 2013 (in English),