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