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Unformatted text preview: AMSC/CMSC 667 Dr. Wolfe FINAL EXAM May 14, 2007 1. (30 points) Let g : R —> R be a C’1 function with g(a) = a. Suppose |g’(oz)l < 1. Prove that if x0 is chosen sufﬁciently close to a, the iterations xn+i = g(a:n), n = 0,1,... I (l)
converge to 04 I (b) Suppose" g E 02 with 9(a) = a,g’(a) = 0. Show that the convergence of (1) is (at
least) locally quadratic. . ' * (c) By consideration of the function g(x) = x —— x3,a = 0, show that the condition
[g’(a)| < 1 is not neccessary for the convergence of ‘2. (30 points) For solving 3/ = f (x, y) consider the numerical method h h2 ,, yn+1 = yn + 5% + 34:14.1] + ﬁlyn — 21144.1]-
Here y; = f(xn,yn) and .
aft? .31 8f a: ,y
yg = +f($nayn)_(—5§—n)" This formula is based on differentiating Y’ = f(a:, (a) Show that this isa fourth order method. Speciﬁcally, deﬁne the local truncation error
7;,(Y) and show that Tn(Y) = 0(h5). '
(b) Deﬁne the concept of absolute stability. That is, consider applying the method to the
~ case f (3:, y) = Ay with Real A < 0. Show that the region of absolute stability contains
the entire negative real axis of the complex hA plane. 3. (20 points) Consider the boundary value problem
-—u" +u = f(:c), u(0) = 0, u(l) = 0 (1) (a) Suppose that u E C'[0, 1] 0 02(0, 1) is a solution of (1)'with f 2 0. Show that u 2 0.
Hint: Argue by contradiction. The proof is two lines and involves only elementary calculus. Y
(b) Use the result of (a) to show that (1) has at most one solution. (c) Discretize the problem. Take a uniform partition of [0, 1]
wizih, i=0,l,2,...,n, h=1/n. (2) Use the three point difference formula for u” . Write the result in the form Ay: f. SCM -ww..~.i_~.—WMWWM
WWWm—~ What are A, y andﬂf: ? v
. (d) For the system (3), show that if fi 2 O for all 2' then 3412 2 0 for all 2'. (Compare with part (a).)
(e) Use the result of (d) to Show that (3) has a unique solution. 4. (20 points) We again consider the boundary value problem ( 1) introduced in problem
3. ‘ ' .
(a) Derive the variational formulation of (l): p '
Find u E H Such that B[u, v] ='F(’U) for all 1) E H. (4)
Deﬁne the bilinear form B [u, v], the linear functional F('u) and the space H.
(b) Prove (4) has a unique solution. .
(c) Deﬁne the piecewise linear ﬁnite element space used for ﬁnding approximate solutions
of this problem. (Use the partition Show that the ﬁnite element problem has a‘
unique solution uh. ‘ '
((1) Show that , ‘ . ' '
B[uh,uh] S B[u, I 0 (e) What is known about the size of — uh“ H1[0’t1] ? ll ...
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