Scientific Computing II

Scientific Computing II

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

Christopher Kacwin

Sheet 2 Submission on Thursday, 3.5.18.

Let Ω ⊂ R

be an open domain and Y = (0, 1)

. Let f ∈ L

(Ω) and A ∈ A

(α, β, Ω, Y ), where A(x, y) = A(y). We consider the problem:

Find u

∈ H

(Ω) such that Z

Ω

A x

∇u

(x) · v(x) dx = Z

Ω

f (x)v(x) dx (1)

holds for all v ∈ H

(Ω).

Exercise 1. (asymptotic expansion I)

We assume that there exist smooth, Y -periodic functions u

(x, y), i ∈ N such that u

(x) = X

i∈N

u

x, x

.

Denote with div

, ∇

the corresponding differential operators with respect to the y- variable, as well as ¯ y =

.

a) Calculate ∇u

(x) in terms of the (u

)

s, ordered by powers of . b) Plug your result into equation (1) to obtain

[div

(A(¯ y)∇

u

(x, y))] ¯

+

[div

(A(¯ y)∇

u

(x, y)) + div ¯

(A(¯ y)(∇

u

(x, y) + ¯ ∇

u

(x, y)))] ¯ + [div

(A(¯ y)(∇

u

(x, y) + ¯ ∇

u

(x, y))) + div ¯

(A(¯ y)(∇

u

(x, y) + ¯ ∇

u

(x, y)))] ¯

+O() = −f (x) .

(4 points) Exercise 2. (asymptotic expansion II)

Assume that the equation from Exercise 1b) holds for general y ∈ Y and equate coeffi- cients to obtain the following differential equations:

•

− div

(A(y)∇

u

(x, y)) = 0

with u

(x, y) is Y -periodic. Conclude that u

(x, y) = u

(x) is not depending on y ∈ Y .

•

div

(A(y)(∇

u

(x) + ∇

u

(x, y))) = 0 with u

(x, y) is Y -periodic. Conclude that one can write

u

(x, y) = u

(x) + ∇

u

(x) · w(y) with w : Y → R

is Y -periodic, where w

(y) satisfies

div

(A(y)(∇

w

(y) + e

)) = 0 for i = 1, . . . , n (assume that such w

exist).

1

•

−f(x) = div

(A(y)(∇

u

(x, y) + ∇

u

(x, y))) + div

(A(y)(∇

u

(x) + ∇

u

(x, y))) Integrate this equation with respect to Y and show that the first summand on the right hand side vanishes due to a periodicity argument. For the second term, plug in the representation for u

(x, y) and conclude that

f (x) = − div

(A

∇

u

(x)) with

A

= Z

Y

A(y)(e

+ ∇

w

(y)) dy · e

.

(6 points) Exercise 3. (periodic boundary problem)

Let Y = (0, 1)

, f ∈ L

(Y ) and consider the problem:

Find u ∈ H ˜

(Y ) such that Z

Y

∇u(y)∇v(y) dy = Z

Y

f(y)v(y) dy for all v ∈ H ˜

(Y ).

Here,

H ˜

(Y ) = {v ∈ H

(Y ) | v has periodic boundary conditions and zero mean}

equipped with the usual H

-norm. Show that this problem has a unique solution u, which depends continuously on f . Show that if u ∈ C

(Y ), it solves the PDE −∆u = f in the strong sense.

(6 points)

2