IWR – Universit¨at Heidelberg Prof. Dr. Guido Kanschat
Due date:8.11.2013
Homework No. 2
Numerical Methods for PDE, Winter 2013/14 Problem 2.1:
Given the sequence of functionsfn(x) = |x|3
|x2|+n1.
(a) Show thatfnis continuously differentiable.
(b) Show thatfn → |x|inH1(−1,1)asn→ ∞.
Hint:Use de l’Hˆopital’s rule for quotients of sequences which converge to infinity.
Problem 2.2:
LetΩ = (−1,1). Show that on the space of continuous functions onΩthe norms kfk∞= supx∈Ω
|f(x)| and kfk2= Z
Ω
|f(x)|2dx
are not equivalent.
Hint:Find a sequence which is bounded in one norm and tends to zero with respect to the other.
Problem 2.3: Friedrichs’ inequality
(a) Prove Friedrichs’ inequality
kukL2(Ω)≤cku0kL2(Ω), with c= (b−a)2 forΩ = (a, b)and functionsu∈C01(Ω).
(b) Generalize the proof for functions inH01(Ω), using that each function inH01(Ω)is the limit of a sequence inC01(Ω).
Problem 2.4: Weak formulation of Robin boundary value problem
Given is the following Robin-boundary problem
−∆u(x) =f(x), inΩ,
∂nu(x) +µu(x) =g(x), on∂Ω,
with a bounded domainΩ⊂Rn, which has a smooth boundary∂Ωandµ >0.
(a) Formulate the problem weakly for functionsu∈H1(Ω).
(b) EquipH1(Ω)with an inner product and a norm, such that you can prove existence and uniqueness of a solution to your weak formulation by Riesz representation theorem. Bonus points for showing that the inner product is indeed one.
(c) Setµ= 0. Is there still a unique solution?
Jede Aufgabe 4 Punkte.
