Problem 6.2: Meshes (4 points)

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

Due date:6.12.2013

Homework No. 6

Numerical Methods for PDE, Winter 2013/14 Problem 6.1: Corner singularity (10 points)

Let the domainΩ⊂R²be the sector

ω

(0,0)

R

with radiusR= 1and interior angleω. In polar coordinates, this domain is described byr∈(0,1)andϑ∈(0, ω).

(a) Verify: The Laplace equation in polar coordinates is

−∂²

∂r²u(r, ϑ)−1 r

∂

∂ru(r, ϑ)− 1 r²

∂²

∂ϑ²u(r, ϑ) =f(r, ϑ).

(b) Verify that the function

u(r, ϑ) =r^πωsin ^π_ωϑ

solves the Laplace equation with zero boundary values on the legs of the angle and smooth boundary valuessin ^π_ωϑ on the circumference.

(c) Show thatu6∈W^2,2(Ω)ifω > π.Hint: It is sufficient to consider the derivative∂rru.

(d) Show that on a triangle of sizehcontaining the origin, this function cannot be approximated by linear functions better than

|u−uh|1.h^πω.

Here, the operator “.” means: there is a positive constantcindependent ofh(but in this case depending onu) such that

|u−uh|1≤ch^πω

Problem 6.2: Meshes (4 points)