MTH 655
and
MTH 659 (Numerical Analysis)
Numerical Functional Analysis with Applications
- Winter 2011
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Schedule |
- 1/3-7: 7.1. 7.2-3 Sobolev spaces. (read 7.3, 7.4)
Weak derivatives, motivation.
- 1.10-14: Variational (weak) solutions to elliptic BVPs (first
pass). 8.1, 8.2, 8.4.
9.1, 10.1 Galerkin method, Galerkin FE solution to BVP.
- Error estimates for FE solution, Cea's lemma.
- Construction of FE algorithms, show the difference from FD. How
to verify order of convergence.
- Superconvergence of FE in 1D as Riesz representation result.
- Error estimates in non-energy norms; stability of FE solutions.
- 1/19: Dual solution, examples. Aubin-Nitzsche lemma;
- 1/21: Examples of dual spaces. Riesz representation in Hilbert
spaces. E/U result for symmatric elliptic BVP (to be extended to
Lax-Milgram Thm).
- 1/24-28:
3.1, Abstract interpolation; Interpolation using Lagrange polynomials.
Newton divided differences. Error estimate (Prop.3.2.4).
Piecewise polynomials. Error estimates. quasi-interpolators.
- 1/31: 3.3. Approximation: abstract setting in normed spaces.
- 2/2: 3.4 Approximation in inner product spaces. Projections.
- 2.2-2/4: [other sources] Orthogonal polynomials as solutions of
differential equations; other properties of orthogonal polynomials:
recursion formula, location of roots.
- 2/7-11: [classical reults on numerical integration including
Stroud and Sechrest, Abramowitz/Stegun, Hoffmann/Hammerlin,
Atkinson'78, Isaacson/Keller] Numerical integration: Newton-Cotes
formulas and lack of convergence with polynomial order. Composite
quadratures. Gauss quadrature construction, proof of convergence, from
Hermite interpolation + zeros of ortogonal polynomials.
- 2/11-14: [book by Bangerth/Rannacher]: representation formulas for
error in different quantities of interest using DWR (dual weighted residuals)
approach. A-posteriori estimates for the error in the energy/L2 norms,
pointwise norm.
- 2/16 Eigenvalue problems, error estimates.
- 2/16-21: [read 8.3-8.4]. Weak solutions and FE formulations of
general elliptic BVP in d>1. Lax-Milgram Theorem. Nonhomogeneous
boundary conditions, variable coefficients, non-selfadjoint problems.
- 2/23-25: [Chapter 12, see also background on integral equations
in Chapter 2]. Numerical solution of integral equations using
projections methods: Galerkin and collocation methods.
- 2/28-3/2: Saddle point problems. solving problems with
constraints (optimization with inequality and equality constraints) in
Banach spaces. [Ito/Kunisch]. Lagrange multipliers. Examples in
R^n. Existence and uniqueness of solutions to saddle point problems:
[Ekeland/Temam] setting for convex/concave functionals. Inf-sup
condition for saddle point problems with linear constraints. [Chapter
8.6]. Examples in Sobolev spaces: PDE with constraint (obstacle
problem), parameter identification problem. Variational inequalities.
Energy (primal) and complementary energy (dual) functionals. [Handouts
on examples and inf-sup condition].
- 3/4-: [Quarteroni-Valli]. Variational saddle point
problems. Discrete solution, LBB condition. Solving elliptic BVP in
primal and dual formulation. Inhomogeneous boundary condition as a
constraint. Hybrid methods (primal and dual). Stokes system, domain
decomposition.
- Finite element spaces in d>1 [Chapter 10]. Mixed finite element
spaces [Braess]. Convergence of FE methods. Construction of
algorithms, solving the resulting linear saddle point problems.
- REVIEW
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