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 7

Please hand in the solutions on Tuesday December13!

Exercise21 6Points

LetΩ

⊂

_R^dbe a polygonal domain with d

≤

_{3 and}

T

_h be a regular triangulation of Ω. Further, letV_h

⊂

H₀^1,2

(

_Ω

)

be a finite element space with P1

⊂

P, s.t. all elements^ˆ are affine equivalent to a reference element, and

I

_h be the Lagrange interpolation operator. Consider the bilinear forma : H₀^1,2

×

H₀^1,2

→

R, a

(

w,v

) =

R

Ω

∇

w

· ∇

vdx.

For f, f_h

∈

L²

(

_Ω

)

let u

∈

H₀^1,2

(

_Ω

)

be the solution of a

(

u,v

) =

R

Ω f

·

vdx for all v

∈

H₀^1,2

(

_Ω

)

, and u_h

∈

V_h be the discrete solution of a

(

u_h,v_h

) =

R

Ω f_h

·

v_hdx for all v_h

∈

V_h.

Prove the a posteriori error estimate

k

u

−

u_h

k

_0,2,Ω

≤

C

k

f

−

f_h

k

_0,2,Ω

+

C

∑

T∈T_h

µ²_T

!¹₂ ,

where

µ²_T :

= k

h

(

T

)

²

(

f_h

+

_∆u_h

)k

²_0,2,T

+ ∑

E∈E₀(T)

k

h

(

T

)

³²

[∇

u_h

·

n

]

_E

k

²_0,2,E.

Hint: Consider for w

=

u

−

u_h a dual solution ϕw s.t.

a

(

v,ϕw

) =

R

Ωwvdx for all v

∈

H₀^1,2

(

_Ω

)

.

Make use of the regularity estimate

k

ϕw

k

_2,2,Ω

≤

c

k

w

k

_0,2,Ω and the interpolation esti- mates

k

ϕw

− I

_hϕw

k

_0,2,T

≤

ch

(

T

)

²

k

ϕw

k

_2,2,T and

k

ϕw

− I

_hϕw

k

_0,2,E

≤

ch

(

T

)

³²

k

ϕw

k

_2,2,T (see Exercise 19).

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Exercise22 4+2+4=10Points Let I

= (

0, 1

)

be the unit interval. Consider the bilinear form a

(

u,v

) =

R

Iu⁰v⁰dx.

(i) Let for x

∈

I the Dirac distribution δx defined by

h

δx,φ

i

:

=

R

Iφ

(

y

)

dδx

(

y

)

:

=

φ

(

x

)

for all φ

∈

C_c^∞

(

I

)

.

Find a function Gx

∈

H₀¹

(

I

)

s.t.

a

(

v,Gx

) =

Z

Iv

(

y

)

dδx

(

y

)

for all v

∈

H₀¹

(

I

)

.

Hint: Consider the intervals

(

0,x

)

and

(

x, 1

)

and make the ansatz thatGx is piece-wise smooth.

(ii) Let

T

_h be an equidistant mesh on I with grid size h. Furthermore, letV_h be the space of piece-wise affine and continuous finite elements w.r.t.

T

_h_{, and let}

I

_h_{be the} corresponding Lagrange interpolation operator. Prove the following estimate for the function Gx:

k

_G_x

− I

_h_G_x

k

_0,1,I

≤

_ch²_.

Attention: Take into account that in generalx is not a multiple ofh.

(iii) For f, f_h

∈

L²

(

I

)

let u

∈

H₀^1,2

(

I

)

be the solution of a

(

u,v

) =

R

I f

·

vdx for all v

∈

H^1,2₀

(

I

)

, and u_h

∈

V_h be the discrete solution of a

(

u_h,v_h

) =

R

I f_h

·

v_hdx for all v_h

∈

V_h.

Prove the a posteriori error estimate

k

u

−

u_h

k

_0,∞,I

≤

C

k

f

−

f_h

k

_0,∞,I

+

Ch²

∑

T∈T_h

k(

f_h

+

_∆u_h

)k

_0,∞,T

! .

Hint: Use that a

(

u

−

u_h,Gx

) = (

u

−

u_h

)(

x

)

.