Final Exam
Math. 610:600
(Last modified on 2015-12-14 15:40)
Hand in your exam by Wednesday, December 14, Noon. You can leave the
exam under my door (Blocker 505b) if it is not open.
These problems are to be done independently. You are allowed
to consult your class notes but no other references. You are not
allowed to discuss these problems with anyone! You may use results
from class without proof.
Problem 1. We consider the pure Neumann problem:
−∆u = f
∂u
=0
∂n
(1.1)
in Ω,
on ∂Ω.
Here n is the unit outward normal on ∂Ω, f ∈ L2 (Ω) and Ω = (0, 1)2 . A
variational formulation of (1.1) is given by: Find u ∈ H 1 (Ω) satisfying
Z
Z
f φ dx := F (φ), for all φ ∈ H 1 (Ω).
∇u · ∇φ dx =
(1.2) D(u, φ) :=
Ω
Ω
(a) Come up with a function f for which there does not exist a u ∈ H 1 (Ω)
satisfying (1.2). Explain your answer.
(b) Let
Z 1
V = {φ ∈ H 1 (Ω) such that G(φ) :=
φ(s, 0) ds = 0}.
0
You may assume that G is a bounded linear functional on H 1 (Ω) and
hence V is complete. Show that there is a positive constant c0 satisfying
c0 kvk2L2 (Ω) ≤ D(v, v),
for all smooth v in V . This inequality extends to all functions in V by
density.
(c) The above result guarantees a unique solution to the variational problem:
Find u ∈ V satisfying
(1.3)
D(u, φ) = F (φ), for all φ ∈ V.
Show that the unique solution u of (1.3) solves (1.2) when F (1) = 0,.
1
2
Problem 2. Consider the integral-differential equation
−u′′ + 2u + Gu = f,
in (0, 1),
u′ (0) = u′ (1) = 1,
where
(Gu)(y) =
Z
1
(y − x)u(x) dx.
0
(a) Derive a weak formulation for the above problem (make sure that you
identify the variational space V ).
(b) Show that the resulting bilinear form coerces the square of the H 1 norm
for functions in V .
(c) Consider the finite element approximation based on piecewise linear finite elements. Is the stiffness matrix symmetric? Is it sparse? Explain
your answers.
P
Problem 3. Consider the tensor product space Q2 = { 2i,j=0 ci,j xi y j } and
the the reference square τ̂ ≡ [0, 1]2 with nodes as illustrated below
v1
v5
v2
v8
v9
v6
v4
v7
v3 .
The space Q2 is unisolvent under these nodes, i.e., given values at the nodes,
there is a unique p ∈ Q2 interpolating those values (you need not show this).
(a) Let {φi }, i = 1, . . . , 9 be the nodal basis functions associated with the
above nodes. Compute φ4 explicitly.
(b) Consider the reduced space
8
4
X
X
′
p(vi ) = 0 .
p(vi ) − 2
Q2 = p ∈ Q2 : 4p(v9 ) +
i=1
i=5
Show that Q′2 contains P2 , the space of polynomials in x and y of total
degree at most 2.
(c) Let Ω be a rectangular domain and Ω = ∪τ̄i be an admissible partitioning
of Ω into smaller rectangular elements. Let Vh be the space of piecewise
(with respect to this partitioning) Q′2 functions with nodal values based
3
on the reference nodes {vi }, i = 1, . . . , 8 (this set of nodes forms a
unisolvent set for Q′2 and you need not prove it). Explain why Vh is
contained in H 1 (Ω)
Problem 4. Consider the time stepping method for the parabolic problem
with homogeneous right hand side: Given U0 ∈ Vh , set Uj ∈ Vh ≈ w(tj ) to
be the solution to
Uj+1 − Uj
, ψ + A(θUj+1 + (1 − θ)Uj , ψ) = 0,
for ψ ∈ Vh .
k
Here θ is a parameter in [0, 1] and we use the notation of the class lectures on
parabolic problems. Show that the above method is unconditionally stable
for θ ∈ [1/2, 1].
Problem 5. For t, k > 0, assume that g is in C 3 [t, t + k]. Show that
g(t + k) − g(t) g(t + k) + g(t)
−
+ O(k 2 ).
k
2
You need to give an expression for the error involving the derivatives of g
evaluated at various points.
g ′ (t + k/2) − g(t + k/2) =