Qualifying Exam, Fall 2013

Numerical Analysis
DO NOT FORGET TO WRITE YOUR SID NO. ON YOUR EXAM.
There are 8 problems. Problems 1-4 are worth 5 points and problems 5-8 are worth
10 points. All problems will be graded and counted towards the final score.
You have to demonstrate a sufficient amount of work on both groups of problems[1-4] and [5-8] to obtain a passing score.

[1] (5 Pts.) Consider a piecewise linear interpolant L(x) to sin(kx), k ∈ Z, x ∈ [0, 2π], based uponN + 1 equispaced points xj , where xj = N1 .
(a) Give a derivation of an estimate of the smallest integer value N, N∗ , depending on k, such thatmax | sin(kx) − L(x) | < .01x∈[0, 2π](b) Does the number of points per wavelength required to insure such a bound depend upon thevalue of k?

[2] (5 Pts.) Consider the (m1 + m2 ) × (m1 + m2 ) block matrix
A1,1 A1,2
A =A2,1 A2,2
where A1,1 is m1 × m1 , A1,2 is m1 × m2 , A2,1 is m2 × m1 , and A2,2 is m2 × m2 .
(a) Derive an expression for a block lower triangular matrix L and a block upper triangular matrix
U in terms of the block components of A, such that LA = U(b) Consider the system of equations




A1,1 A1,2
x1
f1=A2,1 A2,2x2
f2
Derive an expression for the m2 × m2 matrix S and the vector f˜ in terms of the block componentsof A and components of f such that
Sx2 = f˜is the set of equations that determines the x2 component of the solution of the original system. Theequations you derive should not include x1 .

Qualifying Exam, Fall 2013

Numerical Analysis
R1f (x)dx using a Gaussian quadrature formula[3](5Pts.)
Find
theapproximation
oftheintegral−1
Pncf(x)
withn=2.
Givethedegreeof
precisionofthe approximation.
ii=1 i[4] (5 Pts.) Let f (0), f (h) and f (2h) be the values of a real valued function at x = 0, x = h andx = 2h.
(a) Derive the coefficients c0 , c1 and c2 so that
Dfh (x) = c0 f (0) + c1 f (h) + c2 f (2h)is as accurate an approximation to f 0 (0) as possible.
(b) Derive the leading term of a truncation error estimate for the formula you derived in (a).
dy[5] (10 Pts.) Consider the following ODE method for creating approximate solutions of= F(y)dtwith timestep k,
yn =1
2
4 n−1
y− y n−2 + k F(y n )333(a) Derive an expression for the local truncation error.
(b) Show that this method satisfies the root-condition.
(c) Is this a convergent method, and, if it is, what is the global order of accuracy of this method?
(d) Derive the conditions that determine the region of absolute stability for this method.
[6] (10 Pts.) Consider the equation
ut + (u2 + 1)ux = uxx
to be solved for0 ≤ x ≤ 1, t ≥ 0for  > 0, with u(x, 0) = u0 (x) smooth.
(a) Set up a well posed problem by imposing boundary conditions at x = 0 and x = 1.
(b) Write a finite difference equation that will remain stable and convergent for all  > 0 and t > 0.
(b) Justify your answers.

Qualifying Exam, Fall 2013

Numerical Analysis[7](10 Pts.) Consider the equation
utt = uxx + uyy − 2cuxy
to be solved for 0 ≤ x, y ≤ 1, t > 0, with smooth initial datau(x, y, 0) = ϕ(x, y)ut (x, y, 0) = ψ(x, y)with periodic boundary conditions and parameter c.
(a) Write a convergent finite difference scheme for this and choose the appropriate range of values
for c.
(b) Justify your answer.

[8] (10 Pts.) Consider the problem in two dimensions,−4u + uuu∂u∂n= f (x, y), (x, y) ∈ T,
= g1 (x), (x, y) ∈ T1 ,= g2 (y), (x, y) ∈ T2 ,= h(x, y), (x, y) ∈ T3 ,whereT
T1T2T3===={(x, y)| x > 0, y > 0, x + y < 1}{(x, y)| y = 0, 0 < x < 1}{(x, y)| x = 0, 0 < y < 1}{(x, y)| x > 0, y > 0, x + y = 1}.

(a) Find the weak variational formulation of the problem and verify the assumptions of the LaxMilgram Lemma by analyzing the appropriate bilinear and linear forms (impose the weakest necessary assumptions on the functions f , g1 , g2 and h).
(b) Develop and describe the piecewise linear Galerkin finite element approximation of the problem
and a set of basis functions such that the corresponding linear system is sparse. Show that thislinear system has a unique solution.