topics in homogenization of differential equations Course - - PowerPoint PPT Presentation

topics in homogenization

• f differential equations

Course instructor: harsha hutridurga Email: h.hutridurga-ramaiah@imperial.ac.uk Lectures: fridays 11.00am – 01.00pm Office hours:

Fridays 02.00pm – 03.00pm (6M10) (or email + Skpye)

Register your interest: graduate.studies@maths.ox.ac.uk Lecture notes:

http://hutridurga.wordpress.com Harsha Hutridurga (Imperial) Homogenization course 12/05 Class III 1 / 6

references of interest

[BLP78] bensoussan, lions, papanicolaou

Asymptotic methods in periodic structures, Stud. Math. Appl., 5, (1978).

[S-P80] sanchez-palencia

Non homogeneous media and vibration theory, Lect. Notes Phy., 127, (1980).

[JKO94] jikov, kozlov, oleinik

Homogenization of Differential Equations and Integral Functionals, (1994).

[CD99] cioranescu, donato

An introduction to Homogenization, (1999).

[All02] allaire

Shape optimization by the homogenization method, Appl. Math. Sci., 146, (2002).

[PS08] pavliotis, stuart

Multiscale methods: averaging and homogenization, (2008).

[Tar09] tartar

The general theory of homogenization – A personalized introduction, (2009). Harsha Hutridurga (Imperial) Homogenization course 12/05 Class III 2 / 6

quick recap from last session Treated the one dimensional problem      − d dx

• aε(x)duε

dx

• = f(x)

in (ℓ1, ℓ2), uε(ℓ1) = uε(ℓ2) = 0. Assumption on the conductivity sequence: Entire sequence aε − − − − ⇀ a∗ weakly ∗ in L∞ which implies that the entire sequence 1 aε − − − − ⇀ a∗ weakly ∗ in L∞

Harsha Hutridurga (Imperial) Homogenization course 12/05 Class III 3 / 6

• ne dimensional homogenization result

Entire sequence uε is such that uε − − − − ⇀ u0 weakly in H1

0(Ω)

Accumulation point u0 solves      − d dx

• 1

a∗(x) du0 dx

• = f(x)

in (ℓ1, ℓ2), u0(ℓ1) = u0(ℓ2) = 0. In periodic setting, we have 1 aε − − − − ⇀

• Y

1 a(y) dy weakly ∗ in L∞ which implies that the homogenized conductivity 1 a∗ =

• Y

dy a(y) −1

Harsha Hutridurga (Imperial) Homogenization course 12/05 Class III 4 / 6

method of oscillatory test functions Elliptic boundary value problem for the unknown uε(x)    − div

• A

x ε

• ∇uε

= f in Ω, uε = 0

• n ∂Ω.

Weak formulation

• Ω

A x ε

• ∇uε(x) · ∇ϕ(x) dx =
• Ω

f(x)ϕ(x) dx Idea of Tartar: Replace ϕ(x) by a sequence ϕε(x) By the method of two-scale asymptotic expansions uε(x) = u0 (x) + ε

d

• i=1

ωi x ε ∂u0 ∂xi (x) ω1(y), . . . , ωd(y) are solutions to cell problems

Harsha Hutridurga (Imperial) Homogenization course 12/05 Class III 5 / 6

cell problems and their duals Cell problems:    −divy

• A (y)
• ∇yωi(y) + ei
• = 0

for y ∈ Y, y → ωi(y) is Y -periodic. Dual cell problems:    −divy

• A⊤(y)
• ∇yω∗

i (y) + ei

• = 0

for y ∈ Y, y → ω∗

i (y)

is Y -periodic. Sequence of test functions: ϕε(x) = ϕ(x) + ε

d

• i=1

ω∗

i

x ε ∂ϕ ∂xi (x)

Harsha Hutridurga (Imperial) Homogenization course 12/05 Class III 6 / 6