Numerical Methods for PDEs
MATH 610 - 600
Alan Demlow
Homework 1
Spring 2015
Homework 1
Exercise 1
30%
Recall that for sufficiently smooth functions u : R → R, the following relation holds
Z x
u(x) = u(y) +
u0 (s)ds.
y
1
Let C ([0, 1]) be the set of function u : [0, 1] → R that are continuous and have continuous first
derivative on [0, 1]. Consider the following subsets of C 1 ([0, 1]) :
A = {u ∈ C 1 ([0, 1]) : u(0) = u(1) = 0}, B = {u ∈ C 1 ([0, 1]) : u(0) = 0}, C = C 1 ([0, 1]).
Derive any five of the following six inequalities (6 points each) :
Z
Z 1
1 1 02
u dx, ∀u ∈ B,
u2 dx 6
2 0
0
Z
Z 1
1 1 02
u dx, ∀u ∈ A.
u2 dx 6
4 0
0
Show that for u ∈ C the following inequalities are valid :
Z 1
Z
Z 1
2
1 1 02
2
u dx 6
u dx +
udx ;
6 0
0
0
Z 1
max |u(x)|2 6 2u2 (1) + 2
u02 dx;
x∈[0,1]
Z
0
1
u2 (x)dx 6 2u2 (1) + 2
0
1
u02 (x)dx;
0
2
Z
max |u(x)| 6 2
x∈[0,1]
Exercise 2
Z
1
(u2 + u02 )dx.
0
30%
We say that u ∈ L2 (a, b), a, b ∈ R, has a weak derivative in L2 (a, b) if there exists v ∈ L2 (a, b)
satisfying
Z b
Z b
0
u(x)φ (x)dx = −
v(x)φ(x)dx, ∀φ ∈ C0∞ ([a, b]).
a
a
The function v is called the weak derivative of u and the space of functions in L2 (a, b) with weak
derivative in L2 (a, b) is denoted by H 1 (a, b).
2
1. Consider the function u(x) = |x| 3 − 1 defined on the interval (−1, 1). Determine whether
u ∈ H 1 (−1, 1) and if so, give an explicit formula for its weak derivative.
2. Let u1 ∈ H 1 (−1, 0) and u2 ∈ H 1 (0, 1). Show that if u1 (0) = u2 (0) then the function
u1
−1 < x < 0;
u(x) =
u2
0 < x < 1;
belongs to H 1 (−1, 1). Explain why if u1 (0) 6= u2 (0), then u 6∈ H 1 (−1, 1).
Exercise 3
20%
Let V be the closure of the set {v ∈ C 1 [0, 1] : v(0) = 0} in the usual H 1 norm kvkH 1 (0,1) =
kukL2 (0,1) + ku0 kL2 (0,1) . Consider the variational formulation : Find u ∈ V such that
Z
1
0 0
Z
(k(x)u v + uv)dx − u(1)v(1) =
0
1
f vdx ∀v ∈ V.
0
Assuming that 2 ≤ k(x) ≤ 4, show that the corresponding bilinear form is coercive and continuous
in V (equipped with the usual H 1 norm). Derive the strong form of this problem.
Exercise 4
20%
Consider the following second order boundary value problem : Seek u : (0, 1) → R such that
(−k(x)u0 )0 + b(x)u0 + q(x)u = f,
x ∈ (0, 1)
supplemented with the boundary conditions u(0) = u(1) = 0. Here f ∈ L2 (0, 1) and k, q, b, b0 ∈
L∞ (0, 1) satisfy k(x) > 0 and q(x) − 12 b0 (x) > 0 for almost every x ∈ (0, 1).
1. Determine a weak formulation of the above strong problem.
2. Show that there exists a unique weak solution u ∈ H01 (0, 1).