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)
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