NUMERICAL ANALYSIS QUALIFIER
May, 2008
Do all of the problems below. Show all of your work.
Problem 1. Suppose that L1 and L2 are symmetric and positive definite n×n matrices.
We consider the alternating direction implicit (ADI) iteration
(1.1)
(I + kL1 )(I + kL2 )xi+1 = (I − kL1 )(I − kL2 )xi + 2kb
for solving the matrix equation (L1 + L2 )x = b. Here k > 0 is an iteration parameter.
(a) Show that this method is consistent and find the reduction matrix associated with
this iteration.
(b) Consider the norm k|xk| = k(I + kL2 )xk and let ek = x − xk . Here k · k denotes
the Euclidean norm on Rn . Show that
k|ei+1 k| ≤ γk|ei k|
for some γ < 1 depending on k and the spectrum of L1 and L2 . Do not assume
that L1 and L2 commute. (Hint: write k|ei+1 k| = kT (I + kL2 )ei k and bound kT k.)
(c) Suppose that the spectrum of L1 and L2 is contained in [µ1 , µ2 ] with µ1 > 0.
Choose a value of k which results in the optimal convergence rate. What is the
resulting rate?
Problem 2. Consider the initial value problem for y(t):
dy
= f (y, t), t > t0 , y(t0 ) = y0
dt
and its approximation by the Runge-Kutta method: for n = 0, 1, . . . ,
h
(2.1)
un+1 = un + (k1 + 3k2 ), u0 = y0 , tn+1 = tn + h,
4
where k1 = f (un , tn ) and k2 = f (un + 23 hk1 , tn + 32 h). Here un approximates y(tn ) and
h is the step-size.
(a) Use Taylor’s Theorem to show that the method is at least of order two.
(b) Define what it means for η = λh to be in the region of absolute stability of the
above scheme (here λ is an arbitrary complex number). Derive an inequality which
characterizes the region of absolute stability for (2.1).
(c) Determine which real values of η are in the region of absolute stability of (2.1).
Problem 3. Consider the boundary value problem
(3.1)
u(4) (x) + q(x)u = f (x),
0 < x < 1,
u(0) = 0, u(1) = 0,
u000 (0) = −γ, u0 (1) + u00 (1) = β,
where f (x) is a given function on (0, 1), β and γ are given constants and q(x) ≥ 0.
(a) Give a weak formulation of this problem in an appropriate space V , characterize
V , and prove that the corresponding linear form is coercive on V .
(b) Set up a finite dimensional space Vh ⊂ V of piece-wise cubic functions over a
uniform partition of (0, 1). Introduce the Galerkin finite element method for the
problem (3.1) for Vh . State an error estimate in V -norm assuming that u(x) ∈
H 4 (0, 1) (do NOT prove this).
1
2
(c) Assuming “full regularity” and using duality argument prove the following estimate
for the error of the Galerkin solution uh :
ku − uh kL2 ≤ Ch4 ku(4) kL2 .
(3.2)
Further prove the estimate ku0 − u0h kL2 ≤ Ch3 ku(4) kL2 .
Problem 4. Let τ and τb = {x : 0 < x
b1 < 1, 0 < x
b2 < 1−b
x2 } be a triangle and reference
triangle in R2 , correspondingly, and let Tτ be an affine transformation mapping τb to τ .
Denote by |τ | the area of τ . Let m1 , m2 , and m3 be the mid-points of the three edges
of τ .
(a) Prove that the following quadrature is exact for all polynomials in P2 :
Z
1
v(x)dx ≈ |τ |(v(m1 ) + v(m2 ) + v(m3 )).
3
τ
(b) In part (c) you will need an inequality of the following type: there is a constant
c(b
τ ) so that for all v ∈ H s (τ ) and s ≥ 0 an integer:
1
|b
v |H s (bτ ) ≤ c(b
τ )hsτ |τ |− 2 |v|H s (τ ) with vb(b
x) = v(Tτ (b
x)).
Prove this inequality for s = 1.
(c) Let hτ be the diameter of τ . Prove that there exists c > 0 (depending only on the
reference triangle τb) so that
Z
1
3
≤ ch3τ |τ | 21 |v|H 3 (τ ) .
∀v ∈ H (τ ),
|τ
|(v(m
)
+
v(m
)
+
v(m
))
v(x)dx
−
1
2
3
3
τ
Hint. Use Bramble-Hilbert Lemma.
Problem 5. Consider the boundary value problem: find u(x) such that
∂u
−∆u + b
+ u = f in Ω,
(5.1)
∂x1
u = 0 on ∂Ω.
Here Ω = (0, 1) × (0, 1), b is a given constant, and f is a given smooth function.
(a) Derive a up-wind finite difference approximation of this problem on a uniform
square mesh.
(b) Prove an a priori estimate for the solution in the maximum norm.
(c) Show that the local truncation error is O(h) and using the a priori estimate of part
(b) show O(h)-convergence rate in maximum norm.