This document discusses homogenization theory for partial differential equations. Homogenization is used to derive a homogenized problem when there are multiple time or length scales involved. It works by treating the fast and slow scales independently. As an example, it goes through the process of homogenizing the steady state heat conduction equation. This involves a multiple scales expansion, deriving cell problems, and obtaining a homogenized equation with effective coefficients. The one dimensional case is examined in detail.

1. Homogenisation theory for
partial differential equations
G.A. Pavliotis – Homogenization theory for partial differential equations
http://www.ma.ic.ac.uk/~pavl/homogenization.html
→
Yves van Gennip, CASA Seminar Wednesday 26 January 2005
An introduction to homogenisation

2. • What is homogenisation?
• Homogenisation applied to steady state heat
conduction
• One dimensional case
• Some properties of the homogenised coefficients
Overview of my talk

3. What is homogenisation?
• Problem with two time or length scales:
slow/macroscopic and fast/microscopic
• Treat these scales as independent variables
• Derive a homogenised problem: depends only on
slow scale and still has the relevant macroscopic
structure

4. Steady heat conduction

5. Assumptions and ansatz

6. Treat x and y as independent

7. Insert expansion

8. Lemma

9. Sub problem (1)

10. Sub problem (2)

11. Sub problem (3)

12. Back to heat conducting

13. Solve equation

14. Some remarks at this point
• The cell problem satisfies the solvability
condition.
• Unique first order corrector field if we
demand zero average over Y.
• Function undetermined at this
point, but not needed here.

15. Third equation

16. Summary of homogenisation
• Multiple scales expansion ansatz
• Derive equations for ,
and .
• First equation independent of y.
• Second equation gives cell problem.
• Third equation gives homogenised
equation.

17. One dimensional case

18. Effective coefficient in 1D

19. Bounds

20. Recap
• Homogenised problem for heat
conduction.
• The effective coefficients in the one
dimensional case.
• Now: more general properties of the
coefficients.

21. Cell problem rewritten

22. Effective coefficients rewritten

23. Uniform ellipticity

24. Symmetry

25. Recap
• Variational formulation for cell problem and
effective coefficients rewritten.
• Homogenisation preserves positive
definiteness and symmetry.
• It does not preserve isotropy.

26. Conclusions
• Homogenisation: look at macro scale
structure.
• Get cell problem, homogenised equation
and effective coefficients.
• In one dimension we calculated the
coefficient.
• Homogenisation preserves some
properties, not all.