NUMERICAL ANALYSIS QUALIFIER
January 12, 2009
Problem 1. Let A and B be matrices in Rn×n , A be non-singular, and satisfy the inequality
kA−1 k2 kBk2 ≤ q with a constant q < 1. Here k · k2 is the matrix norm subordinate to the
Euclidean norm in Rn .
(a) Show that C = A + B is non-singular.
(b) Show that the iteration process Axj+1 = b − Bxj , j = 0, 1, . . . converges for any x0 to the
solution of the system Cx = b. Give an estimate for the Euclidean norm of error xj − x in
terms of q.
(c) Let A = 2I, where I is the identity matrix in Rn×n , and let B be the matrix with entries of
−1 on the two main co-diagonals and zeros elsewhere. Compute kA−1 k2 kBk2 in terms of n.
n
Problem
the space of polynomials of degree less than or equal to n. Let
R 1 2. 2Let P denote
2
kf kω = −1 |f (x)|(1 − x2 )−1/2 dx and define Πn f to be the best approximation to f in P n in the
norm k · kω , i.e.
kΠn f − f kω = min kf − φkω .
φ∈Pn
Finally, let Tk (x) denote the Chebyschev polynomial of order k. You may use the fact that Tk is
ω-orthogonal to P k−1 .
(a) Give a formula for Πn f in terms of the Chebyshev polynomials {Tk (x)}nk=0 .
(b) Let f ∈ P n+1 . Show that
kf − Πn f k∞ = inf kf − qk∞ .
q∈Pn
Problem 3. Consider the initial value problem for the ordinary differential equation
y ′ = f (y, t), y (t0 ) = y0
and a one-step numerical method of the form
un+1 = un + h M (un , tn ) ,
where u0 = y0 , h ∈ (0, 1) is the step size, un ≈ y(tn ), and tn+1 = tn + h.
(a) Rewrite the extrapolated method defined by
h
f (un , tn ),
2
h = u1n+1/2 + f u1n+1/2 , tn + h/2 ,
2
= un + h f (un , tn ) ,
u1n+1/2 = un +
u1n+1
u2n+1
un+1 = αu1n+1 + βu2n+1 .
as a one-step method and find all values of α and β for which the above scheme is consistent?
(b) Find the values of α and β which make the above method second order.
(c) Prove or disprove: The method of Part (b) is absolutely stable (A-stable).
Problem 4. Let Ω = (0, 1) and u be the solution of the second order hyperbolic problem:
utt − uxx = 0 for (x, t) ∈ Ω × (0, T ],
u(x, 0) = u0 (x),
u(0, t) = 0,
ut (x, 0) = u1 (x) for x ∈ Ω,
u(1, t) = 0 for t ∈ (0, T ).
1
2
(a) Describe the finite difference scheme for this initial value problem which results when the
spatial derivative is approximated by the three point stencil at time level n and the second
order partial derivative with respect to time is discretized by the central difference:
n − 2U n + U n
Ui−1
Uin+1 − 2Uin + Uin−1
i
i+1
and
u
(x
,
t
)
≈
.
tt
i
n
h2
k2
Here k and h = N 1+1 are the step sizes in time and space (respectively) and Uin approximates
the solution u(xi , tn ) where tn = nk and xi = ih, n = 0, 1, . . . , i = 0, 1, . . . , N . Make sure
that you are explicit about boundary conditions and the initial two steps (U 0 and U 1 ) for
this discretization.
(b) Estimate the local truncation error of the scheme.
(c) Derive a Courant condition for stability in time for the scheme of Part (a).
uxx (xi , tn ) ≈
Problem 5. Let Ω be a domain in R2 and (·, ·) denote the inner product in L2 (Ω). Let A(·, ·) be
a (nonsymmetric) bounded bilinear form on H01 (Ω) × H01 (Ω) satisfying the Gärding inequality:
(5.1)
kwk2H 1 (Ω) − c0 kwk2L2 (Ω) ≤ c1 A(w, w) for all w ∈ H01 (Ω)
(c0 , c1 are positive constants). For g ∈ L2 (Ω), assume that there is a unique v ∈ H01 (Ω) satisfying
A(φ, v) = (φ, g) for all φ ∈ H01 (Ω).
Assume further that the above problem is H 2 -regular, i.e., v ∈ H 2 (Ω) and satisfies
kvkH 2 (Ω) ≤ ckgkL2 (Ω) .
(a) Let Sh be a sequence of finite element spaces contained in H01 (Ω) with approximation parameter h ∈ (0, 1). Given w ∈ H01 (Ω), suppose that θ ∈ Sh solves
A(θ, χ) = A(w, χ) for all χ ∈ Sh .
Show that
(5.2)
kw − θkL2 (Ω) ≤ Chkw − θkH 1 (Ω) .
(b) Use (5.1) and (5.2) to show that there is an h0 > 0 such that if h ≤ h0 ,
kw − θk21 ≤ CA(w − θ, w − θ).
Taking w = 0 above shows that for h ≤ h0 , η = 0 is the only solution η ∈ Sh satisfying
A(η, χ) = 0 for all χ ∈ Sh .
(c) Use (b) above to show that for h ≤ h0 there exists a unique solution uh ∈ Sh to
A(uh , χ) = (f, χ) for all χ ∈ Sh ,
i.e., the finite element approximation exists and is unique.