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These are the notes of Exam of Applied Analysis which includes Interpretation, Product and Norm Being, Asymptotic Estimate, Self Adjoint, Function, Minimizing The Functional etc. Key important points are: Nevertheless, Linearly Independent, Compact Operator, Fredholm Alternative, Compact Operators, Eigenvalue, Diffierentiable, Piecewise Cubic, Fourier Transform, Ifinitely Diffierentiable
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August 9, 2012
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 9, 2012
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 ψj and φj , j = 1,... , n, be in L^2 [0, 1]. Assume the sets {ψj }nj=1 and {φj }nj=
are linearly independent. Consider the kernel κ(x, y) =
∑n j=1 ψj^ (x)φj^ (y). (a) Define the term compact operator. (b) Show that the operator Ku =
0 κ(·, y)u(y)dy^ is compact on^ L
(c) State and sketch a proof for the Fredholm alternative for compact operators on a Hilbert space. (d) With K as in part (b), show that the equation (I − λK)u = f has an L^2 -solution for all f ∈ L^2 [0, 1] if and only if 1/¯λ is not an eigenvalue of the matrix A, where Ajk = 〈φj , ψk〉.
Problem 2. Find the first term of the asymptotic series for F (x) :=
x t
xe−tdt, x → +∞.
Problem 3. Let n > 2 be an integer and let xj = j/n, j = 0,... , n. Consider the func-
tional J[y] = (^12)
0 (y
′′) (^2) dx. The admissible functions are in C (^1) [0, 1]. On each closed interval
[xj , xj+1], they are in C^4 [xj , xj+1], for j = 0,... , n − 1. Finally, for each j, y(xj ) = yj is fixed.
(a) Assume that the functional is Fr´echet differentiable. Show that for η ∈ C^2 [0, 1], η(xj ) = 0, j = 0,... , n, one has
∆J[y, η] =
0
y(iv)ηdx +
n∑− 1
j=
y′′η′
x− j+
x+ j
(b) If the minimizer y of J exists, use the result above to show that y is a piecewise cubic spline that is in C^2 [0, 1].
Problem 4. Let f ∈ L^2 (R). Use the following formulas for the Fourier transform and its inverse:
f̂ (ω) =
−∞
f (t)e−iωtdt and f (x) =
2 π
−∞
f^ ˆ (ω)eiωtdω.
(a) Define the term band-limited function. (b) Show that if f is band-limited, then it is infinitely differentiable on R. (Actually, it’s analytic.) (c) State and prove the the Shannon Sampling Theorem.
Combined Applied Analysis/Numerical Analysis Qualifier Numerical Analysis Part August 9, 2012
Problem 1. Consider the variational problem: find u ∈ H^1 (Ω), such that a(u, v) = L(v) for all v ∈ H^1 (Ω), where Ω = (0, 1) × (0, 1), Γ is its boundary, and
(1.1) a(u, v) =
Ω
∇u · ∇v dx dy +
0
u(s, 0)v(s, 0) ds and L(v) =
Γ
gv ds.
Let Vh ⊂ H^1 (Ω) be a finite dimensional space of conforming piece-wise linear finite elements (Courant triangles) over regular partition of Ω into triangles. For continuous v, w defined on
Γ^ ˜ ⊆ Γ, let the bilinear form QΓe(v, w) come from the quadrature
(1.2) QeΓ(v, w) =
e⊆eΓ
|e| 2
(v(P 1 e )w(P 1 e ) + v(P 2 e )w(P 2 e )) ≈
eΓ
vw ds.
Here e is an edge of the triangulation of length |e| with end points P 1 e and P 2 e. Consider the FEM: find uh ∈ Vh such that
(1.3) ah(uh, v) = Lh(v), ∀v ∈ Vh,
where ah(uh, v) and Lh(v) are defined from a(uh, v) and L(v) with the boundary integrals approximated using quadrature (1.2).
Complete the following tasks:
(a) Derive the strong form to the problem (1.1). (b) Prove that the bilinear form a(u, v) is coercive on H^1. (c) Prove that for Γ =˜ {(x, 0), 0 < x < 1 }, there are constants c 1 and c 2 , independent of h, such that
c 1 QeΓ(v, v) ≤
0
v(x, 0)^2 dx ≤ c 2 QeΓ(v, v), ∀v ∈ Vh.
Note that this inequality and part (b) immediately imply ah(v, v) ≥ α‖v‖^2 H (^1) (Ω), ∀v ∈ Vh for some α > 0 independent of h. (d) Apply Strang’s First Lemma to estimate the error in H^1 -norm for the FEM (1.3). You may assume that g is as regular (smooth) as needed by your analysis and you can use (without proof) standard approximation properties for the finite element space Vh.
Problem 2. Consider the following initial boundary value problem: find u(x, t) such that
∂t
(u − ∆u) − μ∆u = f, x ∈ Ω, T ≥ t > 0 , u(x, t) = 0, x ∈ ∂Ω, T ≥ t > 0 , u(x, 0) = u 0 (x), x ∈ Ω,
where Ω is a polygonal domain in R^2 , μ > 0 is a given constant, and f (x, t) and u 0 (x) are given right hand side and initial data functions.
(a) Derive a weak formulation of this problem and derive an a priori estimate for the solution in the norm
(2.2) ‖u(t)‖H (^1) (Ω) =
‖u(t)‖^2 L (^2) (Ω) + ‖∇u(t)‖^2 L (^2) (Ω)
in terms of the right-hand side and the initial data. (b) Write down the fully discrete scheme based on implicit (backward) Euler approx- imation in time and the finite element method in space with continuous piece-wise linear functions. Prove unconditional stability in the H^1 -norm for the resulting approximation. (c) Consider now the forward Euler approximation for the derivative in t. Find the Courant condition for stability of the resulting method in a norm of your choice.
Problem 3. Let Th be a partition of (0, 1) into finite elements of equal size h = 1/N , N > 1 an integer, and xi = ih, i = 0, 1 ,... , N. Consider the finite dimensional space Vh of continuous piece-wise quadratic functions on Th. The degrees of freedom on finite element (xi− 1 , xi) are
(3.1)
v(xi− 1 ), v(xi),
h
∫ (^) xi
xi− 1
v dx
(1) Explicitly find the nodal basis of Vh over the finite element (xi− 1 , xi), correspond- ing to these degrees of freedom. (2) Prove that
sup φ∈H^1 (0,1)
0
(u − Πhu)φ dx
‖φ‖H^1 (0,1)
≤ Ch ‖u − Πhu‖L^2 (0,1), ∀u ∈ H^1 (0, 1).
Here Πhu is the finite element interpolant of u with respect to the nodal basis of Vh defined by (3.1).
