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Scientific Computing II

Summer term 2018 Priv.-Doz. Dr. Christian Rieger

Christopher Kacwin

Sheet 3

Submission onTuesday, 8.5.18.

Exercise 1. (weak-* convergence)

LetY = (0,1)nand a∈L] (Y). For a(x) =a(x/), show that a*

Z

Y

a(y) dy weakly-* in (L1(Rn))0.

(4 points) Exercise 2. (two-scales convergence)

a) LetY = (0,1)n and a∈C0(Ω×Y) witha(x,·) Y-periodic anda(x) =a(x, x/).

Then one has

a * a in the two-scales sense and

a * Z

Y

a(·, y) dy

weakly inL2(Ω).

b) Ifu→u strongly inL2(Ω), then also u * u in the two-scales sense.

c) Ifu has an asymptotic expansion u(x) =X

i∈N

iui

x,x

withY-periodic functionsui ∈C0(Ω×Y), one hasu* u0 in the two-scales sense.

(6 points)

1

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