Winter 2016 / 17

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Exercises to Wissenschaftliches Rechnen I/Scientific Computing I (V3E1/F4E1)

Winter 2016 / 17

Prof. Dr. Martin Rumpf

Alexander Effland — Stefanie Heyden — Stefan Simon — Sascha Tölkes

Problem sheet 2

Please hand in the solutions in the lecture on Tuesday November 8 !

Remark: Exercise 4 was also contained in sheet 1. It is suggested to hand in the solution of this exercise with the other solutions of this sheet on November 8 !

Exercise 4 4 Points

Let a : ( 0, 1 ) → _R be the following function:

a ( x ) =







c ₁ if x ∈ ( 0, b ₁ ) , c 2 if x ∈ [ b 1 , b 2 ) , c ₃ if x ∈ [ b ₂ , 1 ) _,

for c ₁ , c ₂ , c ₃ ∈ _R _{and 0} < b ₁ < b ₂ < 1. Determine a weak solution u ∈ H ^1,2 ( _{0, 1} ) _of the boundary value problem

−( a ( x ) u ⁰ ( x )) ⁰ = 0 ∀ x ∈ ( 0, 1 ) , u ( 0 ) = 0 ,

u ( 1 ) = 1 .

Exercise 5 4 Points

Consider the function

u n ∈ H ^1,2 (( 0, 1 ) , R ) , x 7→ ¹ 4 q

1 4 + _n ¹

₂

− ( x − ¹ ₂ ) ² q

( x − ¹ ₂ ) ² + _n ¹

₂

.

Show that u n converges w.r.t. the H ^1,2 (( 0, 1 )) -norm to u ∈ H ^1,2 (( _{0, 1} ) _, _R ) _, x 7→ ¹

2 −

x − ¹ 2 .

Recall that for any f ∈ H ^1,2 (( 0, 1 )) the H ^1,2 (( 0, 1 )) -norm is defined as k f k ² _1,2 =

Z ₁

0 ( f ( x )) ² dx + Z ₁

0 ( f ⁰ ( x )) ² dx .

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Exercise 6 4 Points Let Ω ⊂ _R ² be a open and bounded domain with smooth boundary. Let Γ ⊂ _Ω be an injective smooth curve that separates Ω into two non-empty and connected sets Ω ⁺ and Ω ⁻ . Let f ∈ C ¹ ( _Ω ) be such that f

_Ω

₊

∈ C ^∞ ( _Ω ⁺ ) and f

_Ω

₋

∈ C ^∞ ( _Ω ⁻ ) . Show that f is twice weakly differentiable.

Exercise 7 4 Points

Let Ω ⊂ _R ⁿ be a bounded domain with Lipschitz boundary and a _ij , b _i , b ∈ L ^∞ ( _Ω ) with

∑ n i,j = 1

a i,j ξ i ξ j ≥ c 0 | ξ | ²

for all ξ ∈ _R ⁿ with c 0 > 0. Verify that the following operators are bounded and coercive bilinear forms on H ₀ ^1,2 ( _Ω ) :

i) L 1 ( u, v ) : = R

Ω ∑ i,j a _i,j ∂ _i u∂ _j v + buv dx under the assumption c ₀ > k b k _∞ C _P ( _Ω ) _. ii) L 2 ( u, v ) : = R

Ω ∑ i,j a _i,j ∂ _i u∂ _j v + _∑ _i b _i ∂ _i uv dx

under the assumption c 0 > ¹ ₂ ( _∑ _i k b _i k _∞ ( 1 + C P ( _Ω ))) . Hints:

Use the Poincaré inequality k u k ² ₂ ≤ C _P ( _Ω )k∇ u k ² ₂ with a constant C p ( _Ω ) depending solely on Ω.

For ii) use ∂ _i u · u ≤ ¹ ₂ ( ∂ _i u ) ² + ¹ ₂ u ² .

Winter 2016 / 17

Exercises to Wissenschaftliches Rechnen I/Scientific Computing I (V3E1/F4E1)

Winter 2016 / 17

Prof. Dr. Martin Rumpf

Alexander Effland — Stefanie Heyden — Stefan Simon — Sascha Tölkes

Problem sheet 2

Please hand in the solutions in the lecture on Tuesday November 8 !

Remark: Exercise 4 was also contained in sheet 1. It is suggested to hand in the solution of this exercise with the other solutions of this sheet on November 8 !

Exercise 4 4 Points

Let a : ( 0, 1 ) → R be the following function:

a ( x ) =







c 1 if x ∈ ( 0, b 1 ) , c 2 if x ∈ [ b 1 , b 2 ) , c 3 if x ∈ [ b 2 , 1 ) ,

for c 1 , c 2 , c 3 ∈ R and 0 < b 1 < b 2 < 1. Determine a weak solution u ∈ H 1,2 ( 0, 1 ) of the boundary value problem

−( a ( x ) u 0 ( x )) 0 = 0 ∀ x ∈ ( 0, 1 ) , u ( 0 ) = 0 ,

u ( 1 ) = 1 .

Exercise 5 4 Points

Consider the function

u n ∈ H 1,2 (( 0, 1 ) , R ) , x 7→ 1 4 q

1 4 + n 1

− ( x − 1 2 ) 2 q

( x − 1 2 ) 2 + n 1

.

Show that u n converges w.r.t. the H 1,2 (( 0, 1 )) -norm to u ∈ H 1,2 (( 0, 1 ) , R ) , x 7→ 1

2 −

x − 1 2 .

Recall that for any f ∈ H 1,2 (( 0, 1 )) the H 1,2 (( 0, 1 )) -norm is defined as k f k 2 1,2 =

Z 1

0

( f ( x )) 2 dx + Z 1

0

( f 0 ( x )) 2 dx .

Exercise 6 4 Points Let Ω ⊂ R 2 be a open and bounded domain with smooth boundary. Let Γ ⊂ Ω be an injective smooth curve that separates Ω into two non-empty and connected sets Ω + and Ω − . Let f ∈ C 1 ( Ω ) be such that f

Ω

∈ C ∞ ( Ω + ) and f

Ω

∈ C ∞ ( Ω − ) . Show that f is twice weakly differentiable.

Exercise 7 4 Points

Let Ω ⊂ R n be a bounded domain with Lipschitz boundary and a ij , b i , b ∈ L ∞ ( Ω ) with

∑ n i,j = 1

a i,j ξ i ξ j ≥ c 0 | ξ | 2

for all ξ ∈ R n with c 0 > 0. Verify that the following operators are bounded and coercive bilinear forms on H 0 1,2 ( Ω ) :

i) L 1 ( u, v ) : = R

Ω ∑ i,j a i,j ∂ i u∂ j v + buv dx under the assumption c 0 > k b k ∞ C P ( Ω ) . ii) L 2 ( u, v ) : = R

Ω ∑ i,j a i,j ∂ i u∂ j v + ∑ i b i ∂ i uv dx

under the assumption c 0 > 1 2 ( ∑ i k b i k ∞ ( 1 + C P ( Ω ))) . Hints:

Use the Poincaré inequality k u k 2 2 ≤ C P ( Ω )k∇ u k 2 2 with a constant C p ( Ω ) depending solely on Ω.

For ii) use ∂ i u · u ≤ 1 2 ( ∂ i u ) 2 + 1 2 u 2 .

Let a : ( 0, 1 ) → _R be the following function:

c ₁ if x ∈ ( 0, b ₁ ) , c 2 if x ∈ [ b 1 , b 2 ) , c ₃ if x ∈ [ b ₂ , 1 ) _,

for c ₁ , c ₂ , c ₃ ∈ _R _{and 0} < b ₁ < b ₂ < 1. Determine a weak solution u ∈ H ^1,2 ( _{0, 1} ) _of the boundary value problem

−( a ( x ) u ⁰ ( x )) ⁰ = 0 ∀ x ∈ ( 0, 1 ) , u ( 0 ) = 0 ,

u n ∈ H ^1,2 (( 0, 1 ) , R ) , x 7→ ¹ 4 q

1 4 + _n ¹

− ( x − ¹ ₂ ) ² q

( x − ¹ ₂ ) ² + _n ¹

Show that u n converges w.r.t. the H ^1,2 (( 0, 1 )) -norm to u ∈ H ^1,2 (( _{0, 1} ) _, _R ) _, x 7→ ¹

x − ¹ 2 .

Recall that for any f ∈ H ^1,2 (( 0, 1 )) the H ^1,2 (( 0, 1 )) -norm is defined as k f k ² _1,2 =

Z ₁

( f ( x )) ² dx + Z ₁

( f ⁰ ( x )) ² dx .

Exercise 6 4 Points Let Ω ⊂ _R ² be a open and bounded domain with smooth boundary. Let Γ ⊂ _Ω be an injective smooth curve that separates Ω into two non-empty and connected sets Ω ⁺ and Ω ⁻ . Let f ∈ C ¹ ( _Ω ) be such that f

_Ω

∈ C ^∞ ( _Ω ⁺ ) and f

_Ω

∈ C ^∞ ( _Ω ⁻ ) . Show that f is twice weakly differentiable.

Let Ω ⊂ _R ⁿ be a bounded domain with Lipschitz boundary and a _ij , b _i , b ∈ L ^∞ ( _Ω ) with

a i,j ξ i ξ j ≥ c 0 | ξ | ²

for all ξ ∈ _R ⁿ with c 0 > 0. Verify that the following operators are bounded and coercive bilinear forms on H ₀ ^1,2 ( _Ω ) :

Ω ∑ i,j a _i,j ∂ _i u∂ _j v + buv dx under the assumption c ₀ > k b k _∞ C _P ( _Ω ) _. ii) L 2 ( u, v ) : = R

Ω ∑ i,j a _i,j ∂ _i u∂ _j v + _∑ _i b _i ∂ _i uv dx

under the assumption c 0 > ¹ ₂ ( _∑ _i k b _i k _∞ ( 1 + C P ( _Ω ))) . Hints:

Use the Poincaré inequality k u k ² ₂ ≤ C _P ( _Ω )k∇ u k ² ₂ with a constant C p ( _Ω ) depending solely on Ω.

For ii) use ∂ _i u · u ≤ ¹ ₂ ( ∂ _i u ) ² + ¹ ₂ u ² .