Problem 3.3: Weak formulation of Robin boundary value problem

IWR – Universit¨at Heidelberg Prof. Dr. Guido Kanschat Alec Gilbert

Due date:2.11.2018

Homework No. 3

Finite Elements, Winter 2018/19 Problem 3.1: Energy Norm

LetΩ⊂R^dbe a bounded domain, and letw∈L^∞(Ω)be such thatw(x)≥w0>0for allx∈Ω.

(a) Show that

a(u, v) :=

Z

Ω

w(x)∇u(x)· ∇v(x) dx, kuk_a := p a(u, u),

define a scalar product and a norm for the spaceH₀¹(Ω).

(b) Are they a scalar product and norm on the spaceH¹(Ω)?

Problem 3.2: H

Regularity

Consider the domainΩ ={x∈R²:|x|<1}. Determine whether the following functions belong toH¹(Ω).

(a)

u(x) = sin ln

1

|x|

(b)

u(x) = x

|x|

Problem 3.3: Weak formulation of Robin boundary value problem

LetΩ⊂R^dbe a bounded domain with a smooth boundary∂Ω, and letµ >0. Consider the following Robin-boundary problem

−∆u(x) =f(x), inΩ,

∂nu(x) +µu(x) =g(x), on∂Ω, (a) Formulate the problem weakly for functionsu∈H¹(Ω).

(b) EquipH¹(Ω)with an inner product and a norm.

(c) Prove the existence and uniqueness of a solution to your weak formulation. What conditions did you assume thatf and gsatisfy?

(d) Setµ= 0. Is there still a unique solution?

Problem 3.4: Bounded/Continuous Linear Operators

LetV, W be normed vector spaces, and letA : V → W be a linear operator. Prove that the following three statements are equivalent.

i. Ais continuous at0.

ii. Ais continuous for allv∈V.

iii. Ais bounded: there exists a constant0< C <∞such that

kAvk_W ≤ Ckvk_V for allv∈V .