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The problems for the applied analysis and numerical analysis qualifying exam held on august 13, 2011. The problems cover various topics such as watson's lemma, self-adjoint operators, green's functions, compact operators, and the heisenberg uncertainty principle. Students are required to solve any three problems and indicate which one they are skipping.
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August 13, 2011
Cover Sheet – Applied Analysis Part
Policy on misprints: The qualifying exam committee tries to proofread exams as carefully as possible. Nevertheless, the exam may contain a few misprints. If you are convinced a problem has been stated incorrectly, indicate your interpretation in writing your answer. In such cases, do not interpret the problem so that it becomes trivial.
Name
Combined Applied Analysis/Numerical Analysis Qualifier Applied Analysis Part August, 2011
Instructions: Do any 3 of the 4 problems in this part of the exam. Show all of your work clearly. Please indicate which of the 4 problems you are skipping.
Problem 1. Let I(λ) :=
0 e
−λt(1 + t)λdt. (a) State Watson’s lemma. (b) Find an asymptotic estimate for I(λ) as λ → ∞.
Problem 2. Let L[u] = − d
(^2) u dx^2 +^ u, 0^ ≤^ x^ ≤^ 1. Take D(L) := {u ∈ L^2 [0, 1] | u′′^ ∈ L^2 [0, 1], u′(0) = u(0), u′(1) = −u(1)}.
to be the domain of L.
(a) Show that L is self adjoint on D(L). (b) Find the Green’s function for the problem L[u] = f , u ∈ D(L). (c) Show that minimizing the functional
D[u] = u(0)^2 + u(1)^2 +
0
(u′^2 + u^2 )dx,
over all u subject to the constraint H[u] =
0 u
(^2) (x)dx = 1, leads to the eigenvalue problem L[u] = λu, u ∈ D(L). (Note: the minimization problem does not assume that u is in D.) (d) Suppose that the boundary condition at x = 0 is changed to u(0) = 0 instead of u(0) = u′(0). Is the lowest eigenvalue in the new problem larger or smaller than for the old one? Explain your reasoning.
Problem 3. Let H be a Hilbert space, with the inner product and norm being 〈·, ·〉 and ‖ · ‖. If K is a compact, self-adjoint operator having spectrum σ(K), then show that, for λ 6 ∈ σ(K), one has ‖(K − λI)−^1 ‖op = (dist(λ, σ(K))−^1. (Hint: use the spectral theorem for compact self-adjoint operators.)
Problem 4. In the following, use the Fourier transform conventions
fˆ (ω) := Ff =
−∞
f (x)eiωxdx
F−^1 fˆ =
2 π
−∞
f^ ˆ (ω)e−iωxdω.
State and prove the Heisenberg uncertainty principle, given that ∫ (^) ∞
−∞
x|f (x)|^2 dx = 0 and
−∞
ω| fˆ (ω)|^2 dω = 0.
Be sure to state any assumptions on the smoothness and decay of f. Is there an f that minimizes the “uncertainty product?” If so, what is it? (No need to justify your answer.)
Combined Applied Analysis/Numerical Analysis Qualifier Numerical analysis part August, 2011
In all questions below, Ω is a bounded polygonal domain with boundary ∂Ω and Th is a regular family of triangulations of Ω.
Problem 1. Let P 2 be the space of polynomials in two variables spanned by { 1 , x 1 , x 2 , x^21 , x 1 x 2 , x^22 }, let Tˆ be the reference unit triangle, ˆγ one side of Tˆ , and ˆπ the standard Lagrange interpolant in Tˆ with values in P 2. Recall that there exists a constant C only depending on the geometry of Tˆ such that
∀v ∈ H^3 ( Tˆ ) , inf p∈P 2
‖v + p‖H (^3) ( Tˆ ) ≤ C|v|H (^3) ( Tˆ ).
(a) State the trace theorem relating L^2 (ˆγ) and H^1 ( Tˆ ). (b) Prove that there exists a constant Cˆ only depending on the geometry of Tˆ and ˆγ such that
∀uˆ ∈ H^3 ( Tˆ ) , ‖uˆ − ˆπ(ˆu)‖L (^2) (ˆγ) ≤ Cˆ|uˆ|H (^3) ( Tˆ ).
(c) Let Xh = {vh ∈ C^0 (Ω) ; ∀T ∈ Th , vh|T ∈ P 2 }. Let T be a triangle of Th with diameter hT and diameter of inscribed disc ̺ T , and let γ be one side of T. Let FT be the affine mapping from Tˆ onto T and let π 2 ,h denote the standard Lagrange interpolant on Xh. Prove that there exists a constant C only depending on the geometry of Tˆ and ˆγ such that
∀u ∈ H^3 (T ) , ‖u − π 2 ,h(u)‖L (^2) (γ) ≤ CσT h2+1 T /^2 |u|H (^3) (T ), where σT = hT /̺ (^) T.
Problem 2. Let δ > 0 be a given constant parameter and u ∈ H 01 (Ω) a given function. Consider the problem: Find ϕδ^ ∈ H 01 (Ω) such that
−δ^2 ∆ ϕδ(x) + ϕδ(x) = u(x) a.e. in Ω , ϕδ(x) = 0 a.e. on ∂Ω.
(a) Define the bilinear form
aδ(u, v) = δ^2
Ω
∇ u(x) · ∇ v(x)dx +
Ω
u(x)v(x)dx.
Write the variational formulation of Problem (2.1) and prove that it has one and only one solution ϕδ^ ∈ H 01 (Ω). (b) Prove that
‖ϕδ‖L (^2) (Ω) ≤ ‖u‖L (^2) (Ω). (c) Prove that ‖∇ ϕδ‖L (^2) (Ω) ≤ ‖∇ u‖L (^2) (Ω). Hint: observe that ∆ ϕδ^ belongs to L^2 (Ω), take the scalar product of (2.1) with −∆ ϕδ^ and apply Green’s formula.
(d) Now let X 0 ,h = {vh ∈ C^0 (Ω) ; ∀T ∈ Th , vh|T ∈ P 1 , vh|∂Ω = 0}. Given uh in X 0 ,h, consider the discrete problem: Find ϕδh ∈ X 0 ,h satisfying
(2.2) ∀vh ∈ X 0 ,h , aδ(ϕδh, vh) =
Ω
uh(x)vh(x)dx.
(i) Prove that problem (2.2) has has one and only one solution ϕδh ∈ X 0 ,h. (ii) Prove that ‖ϕδh‖L (^2) (Ω) ≤ ‖uh‖L (^2) (Ω). (e) Assume that ϕδ^ belongs to H^2 (Ω). Let π 1 ,h denote the standard Lagrange interpolant on X 0 ,h. (i) Prove that
aδ(ϕδ^ − ϕδh, ϕδ^ − ϕδh) = aδ(ϕδ^ − ϕδh, ϕδ^ − π 1 ,h(ϕδ)) −
Ω
(u − uh)
ϕδh − ϕδ^ + ϕδ^ − π 1 ,h(ϕδ))dx.
(ii) Assuming that u is smooth enough, uh = π 1 ,h(u), and δ = h, derive an estimate for ‖ϕδ^ − ϕδh‖L (^2) (Ω).
Problem 3. Let T > 0 be a given final time, let ~b be a given vector valued function with components in L^2 (0, T ; H^1 (Ω)) ∩ C^0 (Ω × [0, T ]) and let u 0 be a given real valued function in C^0 (Ω). We suppose that div~b = 0 a.e. in Ω , ~b = ~0 on Γ.
Consider the time-dependent problem: Find u such that
∂u ∂t
(x, t) + ~b(x, t) · ∇ u(x, t) = 0 a.e. in Ω×]0, T [ , u(x, 0) = u 0 (x) a.e. in Ω ,
where ~b · ∇ u = b 1 ∂u ∂x 1
∂u ∂x 2
Accept as a fact that (3.1) has one and only one solution u that is sufficiently smooth. It is discretized as follows in space and time. Let
Xh = {vh ∈ C^0 (Ω) ; ∀T ∈ Th , vh|T ∈ P 1 }.
Choose an integer K ≥ 2, set k = T /K, tn = nk and u^0 h = π 1 ,h(u 0 ). For 1 ≤ n ≤ K, define unh ∈ Xh from un h −^1 recursively by
(3.2) ∀vh ∈ Xh ,
k
Ω
(unh − un h− 1 )(x)vh(x)dx +
Ω
(~b(x, tn) · ∇ unh(x))vh(x)dx = 0.
(a) Prove that
∀vh ∈ Xh ,
Ω
(~b(x, tn) · ∇ vh(x))vh(x)dx = 0.
(b) Show that, given un h −^1 ∈ Xh, (3.2) has one and only one solution unh in Xh. (c) Prove for 1 ≤ n ≤ K ‖unh‖L (^2) (Ω) ≤ ‖u^0 h‖L (^2) (Ω).
(d) Is the matrix of the system (3.2) symmetric? Justify your answer.
