Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
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: Interpret, Eigenvalues, Courant Minimax Principle, Weierstrass Approximation, Theorem, Set of Functions, Complete Orthonormal, Right Shift Operator, Operator and Compute, Compact Operator
Typology: Exams
1 / 5
This page cannot be seen from the preview
Don't miss anything!
i=
Combined Applied Analysis/Numerical Analysis Qualifier Numerical analysis part January, 2012
In all questions below, you may use standard estimates for finite element interpolation operators without proving them.
Problem 1. (a) You may assume the inequality
‖u‖^2 H (^1) (ˆτ ) ≤ C
ˆτ
|∇u|^2 dˆx + ¯u^2
, for all u ∈ H^1 (ˆτ ).
Here ˆτ is the reference triangle in R^2 , ¯u denotes the mean value of u on ˆτ and Pk^ denotes the polynomials of (x, y) of degree at most k. Let τ denote a general triangle in R^2. Show that
‖u‖^2 H (^1) (τ ) ≤ Cθ
τ
|∇u|^2 dx + h^2 ¯u^2
, for all u ∈ P^1.
Here θ denotes the minimum angle of τ and h its diameter. Now ¯u denotes the mean value of u on τ. (You may assume, without proof, standard properties involving the dependence on θ of the affine map of ˆτ onto τ .) (b) Let Vh be the space of continuous piecewise linear functions with respect to a quasi-uniform mesh Ω = ∪Ni=1τi. Consider the one point quadrature approximation
Qτi (g) := |τi| g(bi) ≈
τi
g,
where |τi| is the area of τi and bi is its barycenter. Consider the finite element problem: Find uh ∈ Vh satisfying Ah(uh, φ) = Fh(φ), for all φ ∈ Vh. Here for uh, vh ∈ Vh,, Ah and Fh are given by
A(uh, vh) :=
i=
(Qτi (∇uh · ∇vh) + Qτi (uhvh)) and Fh(vh) :=
i=
Qτi (f vh).
respectively. Show that
Qτi (|∇u|^2 ) =
τi
|∇u|^2 and Qτi (|u|^2 ) = |τi| ¯u^2 , for all u ∈ P^1.
(c) Use Parts (b) and (c) above to show that the form Ah(·, ·) is Vh-elliptic, i.e., Ah(vh, vh) ≥ c‖vh‖^2 H (^1) (Ω), for all vh ∈ Vh,
holds with c independent of h.
Problem 2. Let Ω be a convex polygonal domain of R^2. Given f ∈ L^2 (Ω), we denote by u ∈ H 01 (Ω) the solution of the Poisson problem:
−∆u = f in Ω, u = 0 on ∂Ω.
We note that u satisfies full elliptic regularity, i.e., u ∈ H^2 (Ω). We consider a non conforming finite element method to approximate u. Let {Th} 0 <h< 1 be a sequence of conforming shape regular subdivisions of Ω such that diam(T ) ≤ h. Denote by Xh the spaces of continuous, piecewise linear polynomials subordinate to the subdivisions Th, 0 < h < 1.
2
The numerical method consists of finding uh ∈ Xh such that for all vh ∈ Xh:
ah(uh, vh) :=
Ω
∇uh · ∇vh −
∂Ω
∂ν uh vh +
α h
∂Ω
uh vh =
Ω
f vh.
Here ν denotes the outward pointing unit normal (defined almost everywhere), ∂ν u := ∇u · ν and α > 0 is a constant yet to be determined. Note that Xh 6 ⊂ H 01 (Ω) but Xh ⊂ H^1 (Ω).
(a) Explain why ah(u, vh) makes sense for any vh ∈ Xh and show Galerkin orthogonality, i.e.,
ah(u − uh, vh) = 0, for all vh ∈ Xh.
(b) For any vh ∈ Xh, defined the mesh dependent norm
‖vh‖h :=
‖∇vh‖^2 L 2 (Ω) +
α h
‖vh‖^2 L 2 (∂Ω)
Show that there exists a constant c 0 independent of h such that for all vh ∈ Xh ∫
∂Ω
|∇vh|^2 ≤
c 0 h
Ω
|∇vh|^2.
Using this fact, deduce that for all vh ∈ Xh,
ah(vh, vh) ≥
‖vh‖^2 h,
provided α ≥ c 0. (c) Let Ih denote the Lagrange finite element interpolation operator associated with Xh. You may use the following estimate without proof: For i = 1, 2, ∥ ∥∥ ∥
∂(u − Ihu) ∂xi
L^2 (e)
≤ Ch^1 /^2 ‖u‖H (^2) (τ ).
Take α = c 0 and derive an optimal error estimate for ‖u − uh‖h.
Problem 3. Given the boundary value problem: find u(x, t) such that
∂u ∂t
= κ
∂^2 u ∂x^2
− b(x)
∂u ∂x
u(0, t) = 0, u(1, t) = 0, 0 < t ≤ T u(x, 0) = v(x), 0 ≤ x ≤ 1 ,
where κ = const > 0, b(x) ∈ C^0 [0, 1], v(x), and f (x) are given smooth functions. Let xi = ih with h = 1/N and tn = nτ , with n = 0, 1 ,... , J and (time step size) τ = T /J.
(1) Write down a forward (explicit) Euler fully discrete scheme for the above problem based on a finite difference discretization in space which upwinds the b(x) term. (2) Find a Courant (CFL) condition and show that if this condition is satisfied,
‖U n+1‖∞ ≤ ‖U n‖∞ + τ ‖f (tn)‖∞. Here U n^ is the approximation at tn of part (a). (3) Define the fully discrete method but with backward (implicit) Euler time stepping and show that this scheme is unconditionally stable, i.e., prove that for any positive τ , ‖U n+1‖∞ ≤ ‖U n‖∞ + τ ‖f (tn+1)‖∞ holds.
