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 1

Please hand in the solutions in the exercise classes on Wednesday/Thursday November 2 / 3 !

Exercise 1 4 Points

Let Ω ⊂ _R ² a polygonal domain and let T = ( T i ) _i = 1,...,I be a triangulation of Ω. Prove that for a continuous function u _h : Ω → R, which is piecewise affine w.r.t. T (i.e. u _h is affine on each T _i ), the surface area of the graph of u _h is given by

E[ u _h ] =

∑ I i = ₁

| T i | ^q 1 + |∇ u _h | _T

| ² .

Exercise 2 4 Points

Let Ω ⊂ _R ⁿ and p > 1. Consider the energy E[ u ] =

Z

Ω |∇ u | ^p dx .

(i) Compute the derivate D E[ u ]( v ) : = _ds ^d E [ u + sv ] _{s = 0} for a test function v. What are suitable function spaces for u and v?

(ii) Derive from D E [ u ]( v ) = 0 a partial differential equation for u.

Exercise 3 4 Points For any N ≥ 2 and h = _N ¹ let

x _i = i · h , i = 0, . . . , N ,

x i + ¹ 3

= x i + ¹ ₃ h , i = 0, . . . , N − 1 ,

x i + ² ₃ = x i + ² ₃ h , i = 0, . . . , N − 1 . Consider the finite element space

V _h ³ ([ 0, 1 ]) = ⁿ v ∈ C ⁰ ([ 0, 1 ] , R ) : v

[ x

,x

] ∈ P ₃ ([ x i , x i + 1 ]) ∀ i = 0, . . . , N − 1 o . Here, P ₃ ([ x i , x i + ₁ ]) is the set of all cubic polynomials on [ x i , x i + ₁ ] . Compute the set of base functions ( φ ⁱ _h ) _{i = 0,...,N} , ( φ ^{i +}

1 3

h ) _{i = 0,...,N − 1} , ( φ ^{i +}

2 3

h ) _{i = 0,...,N − 1} ⊂ V _h ³ ([ 0, 1 ]) , which satisfy φ ^{i +}

k 3

h ( x _{j +}

3

) = δ _i,j δ _k,l for k, l ∈ { 0, 1, 2 } .

Hint: Split the construction of the base functions on the different intervals [ x _j , x _j + 1 ] .

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 ( _{0, 1} ) = W ^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 .