Applied/Numerical Analysis Qualifying Exam

Transcription

1 Applied/Numerical Analysis Qualifying Exam 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 1

2 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) λ dt. (a) State Watson s lemma. (b) Find an asymptotic estimate for I(λ) as λ. Problem 2. Let L[u] = d2 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) (u 2 +u 2 )dx, over all u subject to the constraint H[u] = 1 0 u2 (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) Supposethattheboundaryconditionatx = 0ischangedtou(0) = 0insteadofu(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 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 ˆf(ω) := F[f](ω) = F 1 [ˆf](x) = 1 2π f(x)e iωx dx ˆf(ω)e iωx dω. State and prove the Heisenberg uncertainty principle, given that x f(x) 2 dx = 0 and ω ˆf(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.) 2

3 Applied/Numerical Analysis Qualifying Exam August 13, 2011 Cover Sheet Numerical 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 3

4 Combined Applied Analysis/Numerical Analysis Qualifier Numerical analysis part August, 2011 In all questions below, is a bounded polygonal domain with boundary and T h is a regular family of triangulations of. Problem 1. LetP 2 bethespaceofpolynomialsintwovariablesspannedby{1,x 1,x 2,x 2 1,x 1x 2,x 2 2 }, let ˆT be the reference unit triangle, ˆγ one side of ˆT, and ˆπ the standard Lagrange interpolant in ˆT with values in P 2. Recall that there exists a constant C only depending on the geometry of ˆT such that v H 3 (ˆT), inf p P 2 v +p H 3 (ˆT) C v H 3 (ˆT). (a) State the trace theorem relating L 2 (ˆγ) and H 1 (ˆT). (b) Prove that there exists a constant Ĉ only depending on the geometry of ˆT and ˆγ such that (c) Let û H 3 (ˆT), û ˆπ(û) L 2 (ˆγ) Ĉ û H 3 (ˆT). X h = {v h C 0 (); T T h, v h T P 2 }. Let T be a triangle of T h with diameter h T and diameter of inscribed disc T, and let γ be one side of T. Let F T be the affine mapping from ˆT onto T and let π 2,h denote the standard Lagrange interpolant on X h. Prove that there exists a constant C only depending on the geometry of ˆT and ˆγ such that where σ T = h T / T. u H 3 (T), u π 2,h (u) L 2 (γ) Cσ T h 2+1/2 T u H 3 (T), Problem 2. Let δ > 0 be a given constant parameter and u H0 1 () a given function. Consider the problem: Find ϕ δ H0 1 () such that (2.1) (a) Define the bilinear form δ 2 ϕ δ (x)+ϕ δ (x) = u(x) a.e. in, a δ (u,v) = δ 2 ϕ δ (x) = 0 a.e. on. u(x) v(x)dx+ u(x)v(x)dx. Write the variational formulation of Problem (2.1) and prove that it has one and only one solution ϕ δ H 1 0 (). (b) Prove that ϕ δ L 2 () u L 2 (). (c) Prove that ϕ δ L 2 () u L 2 (). Hint: observe that ϕ δ belongs to L 2 (), take the scalar product of (2.1) with ϕ δ and apply Green s formula. 4

5 (d) Now let X 0,h = {v h C 0 (); T T h, v h T P 1, v h = 0}. Given u h in X 0,h, consider the discrete problem: Find ϕ δ h X 0,h satisfying (2.2) v h X 0,h, a δ (ϕ δ h,v h) = u h (x)v h (x)dx. (i) Prove that problem (2.2) has has one and only one solution ϕ δ h X 0,h. (ii) Prove that ϕ δ h L 2 () u h L 2 (). (e) Assume that ϕ δ belongs to H 2 (). Let π 1,h denote the standard Lagrange interpolant on X 0,h. (i) Prove that a δ (ϕ δ ϕ δ h,ϕδ ϕ δ h ) = a δ(ϕ δ ϕ δ h,ϕδ π 1,h (ϕ δ )) (u u h ) ( ϕ δ h ϕδ +ϕ δ π 1,h (ϕ δ ))dx. (ii) Assuming that u is smooth enough, u h = π 1,h (u), and δ = h, derive an estimate for ϕ δ ϕ δ h L 2 (). Problem 3. LetT > 0beagivenfinaltime,let bbeagivenvectorvaluedfunctionwithcomponents in L 2 (0,T;H 1 ()) C 0 ( [0,T]) and let u 0 be a given real valued function in C 0 (). We suppose that div b = 0 a.e. in, 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 ]0,T[, (3.1) u(x,0) = u 0 (x) a.e. in, where u u b u = b1 +b 2. x 1 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 X h = {v h C 0 (); T T h, v h T P 1 }. Choose an integer K 2, set k = T/K, t n = nk and u 0 h = π 1,h(u 0 ). For 1 n K, define u n h X h from u n 1 h recursively by (3.2) v h X h, 1 (u n h k un 1 h )(x)v h (x)dx+ ( b(x,t n ) u n h (x))v h(x)dx = 0. (a) Prove that v h X h, ( b(x,t n ) v h (x))v h (x)dx = 0. (b) Show that, given u n 1 h X h, (3.2) has one and only one solution u n h in X h. (c) Prove for 1 n K u n h L 2 () u 0 h L 2 (). (d) Is the matrix of the system (3.2) symmetric? Justify your answer. 5