Fachbereich Mathematik & Informatik
Freie Universität Berlin
Prof. Dr. Carsten Gräser, Tobias Kies
9th exercise for the lecture
Numerics IV
Winter Term 2016/2017
http://numerik.mi.fu-berlin.de/wiki/WS_2016/NumericsIV.php
Due: Exercise 1: Tuesday, Jan 17th, 2017
Exercise 2: Tuesday, Jan 10th, 2017
Exercise 1 (4 TP)
Let Ω ⊆ Rn be bounded, a : H 1 (Ω) × H 1 (Ω) → R a symmetric, continuous and coercive
bilinear form, ` : H 1 (Ω) → R linear and continuous, and ψ ∈ C 0 (Ω). We consider the
obstacle problem
min
u∈K
1
a(u, u) − `(u)
2
where
K := v ∈ H01 (Ω) | v ≥ ψ .
Suppose that Sh ⊆ H01 (Ω) and Kh ⊆ H 1 (Ω) are closed subspaces and that we approximate
the solution of the obstacle problem by the solution of
min
uh ∈Kh
1
a(uh , uh ) − `(uh ).
2
Furthermore, assume that there exists g ∈ L2 (Ω) such that for all v ∈ H 1 (Ω)
a(u, v) − `(v) = hg, viL2 (Ω) .
Show that both problems admit a unique solution, denoted by u and uh respectively, and
that the error estimate
2
2
cku − uh kH 1 (Ω) ≤
inf ku − vh kH 1 (Ω) + ku − vh kL2 (Ω) + inf kuh − vkL2 (Ω)
vh ∈Kh
v∈K
holds with the constant c depending on a and g.
Hint: You may want to state the variational inequalities corresponding to the given
minimization problems and to try a strategy that is similar to the proof of Céa’s lemma.
Please turn over...
Exercise 2 (4TP + 4 Bonus TP)
Let Ω = [−1, 1], and cf , cψ ∈ R, and define f (x) = cf , ψ(x) = cψ for all x ∈ Ω. Determine
for which pairs (cf , cψ ) the obstacle problem
Z
Z
2
f (x) u(x) dx
min k∇u(x)k dx −
u∈K
Ω
Ω
with
K := v ∈ H01 (Ω) | v ≥ ψ
admits a solution and compute an explicit expression for the solution (in case it exists).
When computing this expression you may assume without proof that there exists a (possibly
empty) interval I = [−a, a] such that u = ψ on I and u 6= ψ on Ω \ I.
Hint: Try to prove u0 (a) = 0. Using this, the boundary data and the knowledge what u
looks like on I and Ω \ I, you can derive a (small) finite-dimensional system of equations
that you can solve in order to obtain an analytical expression for u.
Have fun!