NUMERICAL ANALYSIS QUALIFIER
May, 2007
Do all of the problems below. Show the whole work.
Problem 1. Consider the iterative method
xk+1 = Mxk + b,
where xk and b are vectors in Rn , M is an n × n real matrix, and x0 is a given
initial iterate. Assume that kMk < 1 where kMk is the induced matrix norm
from a vector norm kxk.
(a) Show that the matrix I − M is nonsingular and therefore the linear system
x = Mx + b has unique solution x ∈ Rn .
(b) Show also that
kxk − xk ≤ kMkk kx0 k +
kMkk
kbk
1 − kMk
so that the above iteration converge to the solution of x = Mx + b for any
initial iterate x0 .
(c) Prove that
kxk − xk ≤ k(I − M)−1 k kxk+1 − xk k.
Problem 2. Consider the initial value problem
dy
= f (y, t), t > t0 , y(t0) = y0
dt
and its approximation by the Runge-Kutta method: For n = 0, 1, . . . ,
h
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.
(c) Derive an inequality which can be used to determine when η = λh is in the
region of absolute stability for this scheme.
(d) Determine which real values of η are in the region of absolute stability.
Problem 3. Let w be a positive integrable function on the interval [a, b] such
that w(x) ≥ w0 > 0 for all x ∈ [a, b]. Let Pn be the set of polynomials on
[a, b] of degree at most n. Let A0 , . . . An , x0 , . . . , xn be real numbers such that
xi ∈ [a, b], i = 0, . . . , n and
Z b
n
X
Ai f (xi ),
∀f ∈ P2n+1 .
f (t)w(t)dt =
a
i=0
1
2
(a) Show that the polynomial πn (x) := Πni=0 (x − xi ) is orthogonal to Pn with
Rb
respect to the inner product (p, q)w := a p(t)q(t)w(t)dt.
(b) Prove that the coeffcients Ai are all positive.
(c) Assume f ∈ C(2n+2) [a, b]. Derive a representation for
Z b
n
X
E=
f (t)w(t)dt −
Ai f (xi )
a
in terms of f
(2n+2)
i=0
.
Problem 4. Consider the boundary value problem
(4.1)
u(4) (x) = f (x),
0 < x < 1,
u(0) = 0,
u′′ (0) = 0,
′
′′
u (1) + u (1) = β, −u′′′ (1) = γ,
where f (x) is a given function on (0, 1) and β and γ are given constants.
(a) Give the weak formulation of this problem in an appropriate space V and
characterize V .
(b) Show that the corresponding linear form is coercive and continuous in V and
the linear form is continuous in V .
(c) Set up a finite dimensional space Vh ⊂ V of piece-wise polynomial functions
over a uniform partition of (0, 1). Define the “nodal” basis in terms of the
degrees of freedom.
(d) Introduce the Galerkin method for the problem (4.1) for Vh ; state the error
estimate in the V -norm assuming smooth solution u(x).
Problem 5. Consider the boundary value problem: find u(x) such that
∂u
∂u
−∆u + α
+ βx1
= f (x), x := (x1 , x2 ) ∈ Ω
(5.1)
∂x1
∂x2
u(x) = 0,
x ∈ ∂Ω.
Here Ω is a bounded convex polygonal domain in R2 , α and β are given constants,
and f (x) is a given smooth function. These guarantee full regularity of the
solution for any α and β, i.e. u ∈ H 2 (Ω) and kukH 2 ≤ Ckf kL2 .
(a) Derive a weak form of this problem in an appropriate space V (describe the
space !).
(b) Show that the corresponding form is coercive in the norm of the space V .
(c) Set up a finite element approximation on triangulation of Ω; describe the
space Vh ⊂ V of piece-wise linear finite elements.
(d) Write down the a priori error estimate in V -norm; using Aubin-Nitsche (duality) argument derive an error estimate in the L2 -norm.