Universität Konstanz WS 11/12 Fachbereich Mathematik und Statistik

Universität Konstanz WS 11/12 Fachbereich Mathematik und Statistik

S. Volkwein, M. Gubisch, R. Mancini, S. Trenz

Übungen zu Theorie und Numerik partieller Differentialgleichungen

http://www.math.uni-konstanz.de/numerik/personen/volkwein/teaching/

Sheet 5 Deadline: 27.01.2012, 10:10 o’clock before the Lecture Exercise 13 (Quadratic ansatz functions) (4 Points) Let Ω = (a, b) ⊆ R and x ₀ , ..., x _n+1 ∈ Ω ¯ with x ₀ = a, x _n+1 = b, x _i < x _i+1 .

1. Define piecewise quadratic ansatz functions φ _i , i = 1, ..., n, and φ _i+

2

, i = 0, ..., n, in C ⁰ ( ¯ Ω) such that

φ _i (x _j ) = δ _ij & φ _i (x _j+

2

) = 0, φ _i+

2

(x _j ) = 0 & φ _i+

2

(x _j+

2

) = δ _ij .

Hint: A quadratic function is determined uniquely by its values on three interpo- lation points.

2. Calculate the derivatives φ _i , φ _i+

2

. 3. Draw φ _i , φ _i+

2

, φ ⁰ _i , φ ⁰ _i+

for one fixed i in different colours in one large plot. Don’t forget to label your axes.

4. Consider the ansatz

y(x) =

n

X

i=1

α _i φ _i (x) +

n

X

i=0

α _i+

2

φ _i+

2

(x) for arbitrary coefficients α _i , α _i+

2

∈ R . Explain why y in general is just of H ¹ - regularity (you don’t need to prove this).

Exercise 14 (Weak derivatives) (4 Points)

Let Ω = (−1, 1).

1. Let u ∈ L ² (Ω) Define: v ∈ L ² (Ω) is the weak derivative of u.

2. u(x) = abs(x). Show that the weak derivative v = u ⁰ ∈ L ² (Ω) of u is given by u ⁰ (x) =

−1 x ∈ [−1, 0]

1 x ∈ [0, 1] .

3. Prove that u ⁰ is not weakly differentiable in L ² (Ω).

Exercise 15 (Ansatz functions in two dimensions) (4 Points) Let Ω = (−1, 1) × (−1, 1). Consider the following triangulation of Ω:

1. Define the function S : ¯ Ω → R describing the canonical simplex on Ω, i.e. S _i = S| Ω ¯

is a plain with S _i (0, 0) = 1 and S _i (x, y) = 0 for all other grid points (x, y) ∈ Ω. ¯ 2. Draw the graph of S.

3. Assume that Ω is triangulized by n ² inner grid points x _i , y _j now. Let f ∈ C ⁰ ( ¯ Ω), f _ij = f (x _i , y _j ) and

F (x, y) =

n

X

i,j=1

f ij φ ij (x, y)

where φ _ij is the simplex with centre (x _i , y _j ).

Describe the graph of F .

4. Is F a good approximation for f ?