Applied/Numerical Analysis Qualifying ExamJanuary 8, 2012Cover Sheet – Applied Analysis PartPolicy on misprints: The qualifying exam committee tries to proofread exams as carefully aspossible. Nevertheless, the exam may contain a few misprints. If you are c onvinced a problem hasbeen st ate d incorrectly, indicate your interpretation in writing your answer. In such cases, do notinterpret the problem so that it becomes trivial.Name1Combined Applied Analysis /Num er ic al Analysis QualifierApplied Analysis PartJanuary 8, 2012Instructions: 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 D be the set of compactly supported functions defined on R and let D′be thecorresponding set of distributions.(a) Define convergence in D and D′.(b) Consider a function f ∈ C(1)(R) such that both f and f′are in L1(R), andRRf(x)dx = 1.Define the sequence of functions {Tn(x) := n2f′(nx): n = 1, 2, . . .}. Show that, in the senseof dis tr i bu ti ons — i.e., in D′—, Tnconverges to δ′.Problem 2. Let M : C[0, 1] → C[0, 1] be defined by M[u](t) :=R10(2 + st + u(s)2)−1ds, 0 ≤ t ≤ 1.Let k · k := k · kC[0,1]. Let Br:= {u ∈ C[0, 1] | kuk ≤ r}.(a) Show that M : B1→ B1/2⊂ B1.(b) Show that M is Lipschitz continuous on B1, with Lipschitz constant 0 < α < 1 – i.e.,kM[u] − M[v]k ≤ αku − vk.(c) Show that M has a fixed point in B1. State the theorem you are usi ng to show that thefixed point exists.Problem 3. Let Lu = −d2udx2, −π ≤ x ≤ π, with the domain of L given byDL:= {u ∈ L2[−π, π] : u′′∈ L2[π, π], u(−π) = −u(π), u′(−π) = −u′(π)}.(a) Show that L is self adjoint on D(L).(b) Find the Gre e n’ s function G(x, y) for the problem Lu = f, u ∈ DL.(c) Show that Ku :=Rπ−πG(·, y)u(y)dy is a compact self-adjoint operator.(d) Without actually finding them, show that the eigenfunctions of L form a complete, or-thogonal set for L2[−π, π]. (Hint: Relate the eigenfunctions of L to those of K. Usecompactness.)Problem 4. Let T be a (possibly unbounded) linear operator on a Hilbert space H, defined onthe domain DT.(a) Define these: the resolvent set of T , ρ(T ); the discrete spectrum, σd(T ); the continuousspectrum, σc(T ); and th e resi du al spectrum, σr(T ).(b) Assume T is bounded. Show that the set {λ ∈ C : |λ| > kT k} ⊆ ρ(T ). (Hint: Use aNeumann series expansion.)(c) Let H = ℓ2, wi th the usual inner product. Define T to be th e shift operatorT (x1, x2, . . .) = (0, x1, x2, . . .).Show that every |λ| > 1 is in ρ(T ), that λ = 1 is in σc(T ), and th at λ = 0 is in σr(T ).2Applied/Numerical Analysis Qualifying ExamJanuary 8, 2012Cover Sheet – Numerical Analysis PartPolicy on misprints: The qualifying exam commit te e tries to proofread exams as carefully aspossible. Nevertheless, the exam may contain a few misprints. If you are c onvinced a problem hasbeen st ate d incorrectly, indicate your interpretation in writing your answer. In such cases, do notinterpret the problem so that it becomes trivial.Name3Combined Applied Analysis/Numerical Analysis QualifierNumerical analysis par tJanuary 8, 2012Problem 1:Let Ω = (0, 1) × (0, 1), f ∈ C0(Ω) and q ∈ R with q ≥ 0. Consider the boundary value problem−∆u + qu = f in Ω; u = 0 on ∂Ω.We are interested in approximating the quantity α :=R∂Ωn·∇u where n is the outward unit normalof Ω.1. The boundary problem has a weak formulation: Find u ∈ V such that∀v ∈ V : a(u, v) = L(v).Identify V, a(u, v) and L (v). Show that there exists a unique soluti on u ∈ V sati s fy i ng theabove weak formulation.2. Let {Th}0<h<1be a seque nc e of conforming shape-regular subdivisions of Ω such that diam(T ) ≤ h,for al l T ∈ Thand de fin eVh:=v ∈ C0(Ω) ∩ V | ∀T ∈ Th, v|Tis l i n ear.Write the weak formulation satisfied by the finite element approximation uh∈ Vhof u. Provethat the function uhexists and is uni q u e.3. Assume from now that u ∈ H2(Ω). Derive the error estimateku − uhkH1(Ω)≤ c1hkukH2(Ω),where c1is a constant independent of h and u.Hint: you can use without proof the fact that there exists a constant C i nd ependent of h suchthat for any v ∈ H2(Ω)infvh∈Vhkv − vhkV≤ Chkv kH2(Ω).4. Show that that for the constant function w(x) = 1 we haveα = a(u, w) − L(w).Now let αh:= a(uh, w) − L(w). Using the previous parts, show that when q > 0 there holds|α − αh| ≤ c2h2kukH2(Ω),where c2is a constant independent of h and u . What can you say about |α − αh| when q = 0?Problem 2:Let K be a polyhedron in Rd, d ≥ 1. Let h = diam(K) and defineˆK = {ˆx = x/diam(K), x ∈ K} .Show that there exi st s a constant c solely depending onˆK such that for any v ∈ H1(K),kvkL2(∂K)≤ ch−1/2kvkL2(K)+ h1/2k∇vkL2(K).4Problem 3:Let u0: (0, 1) → R be a given smooth initial condition and T > 0 be a given final time. Letu : [0, T ] × Ω → R be a smooth function satisfying u(t, 0) = u(t, 1) = 0 f or any t ∈ [0, T ] and∀v ∈ C∞c([0, T ) × (0, 1)) :−ZT0Z10u(t, x) vt(t, x) dx dt −Z10u0(x)v(0, x) dx+ZT0Z10ux(t, x) vx(t, x) dx dt +ZT0Z10u(t, x) v(t, x) dx dt = 0.(4.1)Here C∞c([0, T ) × (0, 1)) is the space of functions belonging to C∞([0, T ] × [0, 1]) and compactlysupported in [0, T ) × (0, 1).1. Derive the corresponding strong formulation.2. Let N > 0 be an integer, h = 1/N and xn= n h, n = 0, ... , N. Derive the semi-discreteapproximation of (4.1) using continuous piecewise linear finite elements.3. In addition, let M > 0 be an integer, τ = T /M and tm= mτ for m = 0, .., M. Write the fullydiscrete schemes corresponding to backward Euler and forward Euler methods, respectively.4. Prove that the backward (implicit) Euler scheme is uncondi t i onal l y stable while the forward(explicit) E ul er method is stable provided τ ≤ ch2, where c is a constant independent of h