Diploma thesis as pdf file - Johannes Kepler University, Linz

3.4 Numerical experiments 35
3.4.2 Example 2
We consider another radially symmetric problem when: Ω is a unit circle, constant
force is acting on the membrane f = −10, and the obstacle is described as
{ √
R
ψ(x,y) =
2 − x 2 − y 2 − R − 1, if x 2 + y 2 ≤ R 2 ,
−5, if x 2 + y 2 > R 2
where R = 0.7
The exact solution to this problem is :
{ √
R
u =
2 − x 2 − y 2 − R − 1, if x 2 + y 2 < r
5(x 2 + y 2 )/2 − alnr − 5/2, if x 2 + y 2 ≥ r
where r = 0.3976, a = r 2 (5 + 1/ √ R 2 − r 2 ).
Figure 3.5 depicts the final iterate of the approximate solution.
Figure 3.5: Approximate solution for the obstacle with the spherical surface
The table 3.4.2 reports the results when the stopping criterion for the Newton
iterations was changed to δu
u−ψ
≤ 0.1. It illustrates that stricter stopping criterion
doesn’t improve the accuracy of the solution for the original constrained problem,
but only increases the number of iterations. This confirms the fact that we
don’t need to solve each corrector problem with high accuracy but it’s sufficient