MTH 654/659 (Numerical Analysis) Fall 2011
Finite Element Methods for Partial Differential Equations
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General information
Textbook and resources
Assignments
Assignments and schedule

Schedule. [Reading assignments]:
  1. 9/26/11: Introduction
  2. 9/28/11: Questionnaire. [Handout (Johnson's book)]. Introduce variational and FE formulation of 2-pt BVP, L^p spaces.
  3. 9/30/11: Weak derivatives, Sobolev spaces.
    [Read: Chapter I for general review/background. Chapter II.1]
    Assignment 1 due 10/17/11.
  4. 10/3/11: Sobolev spaces as inner product spaces. Poincare-Friedrichs inequality Basic error estimate for conforming FE formulation of the model problem: Galerkin orthogonality, Cea's lemma.
  5. 10/5/11: Matrix-vector form of the FE formulation of the model problem. Positive definite matrices. Bilinear forms and Lax-Milgram Theorem. [Read: Chapter II.1-2]
  6. 10/7/11: meet in MLC computer lab, KIdd 108C, 8:30-10:00.
    Assignment 2 (lab) due 10/17/11 (new date). Latex file for assignment 2 (lab1).
  7. 10/10/11: Continue discussion of FE formulation of the model problem, tools (e.g, numerical integration) needed to solve the computational problems. Variational and FE formulation of general BVP for elliptic PDEs in d=1,2,3 including nonsymmetric and those with Neumann and other boundary conditions. Minimization principle and Ritz method. [Read: Chapter II.2-4].
  8. 10/12/11 continued ...
  9. 10/14/11: Error estimate using a(.,.) for conforming FE (Galerkin orthogonality, Cea's lemma, interpolation). Triangulation of the domain; construction of V_h. [Read: Chapter II.2-5]
  10. 10/17/11: Piecewise polynomials in V_h: construction and how to make it conforming. [Read: Chapter II.5,6]
  11. 10/19/11: Spaces P_k and Q_k. Affine transformations. Reference element calculations. [Chapter II.6 and II.8]
  12. 10/21/11: Element stiffness matrices. [Chapter II.8]
  13. 10/24/11: Bramble-Hilbert lemma and interpolation estimates [II.6]
  14. 10/26/11: meet in MLC computer lab, Kidd 108C, 8:30-10:00.
    Assignment 3 (lab2) due 11/7/11. Latex file for assignment 3 (lab2). Code that you may find useful Extra : lookup other quadrature formulas on triangles or quadrilaterals (in the book by O. Zienkiewicz, Finite Elements in Engrg Science, 1971)
  15. 10/28: Duality theory (Aubin-Nitzsche trick). [Read Chapter II.7]
  16. 10/31: Superconvergence, Green's functions
  17. 11/2: Interpolation theory for quadrilateral elements. [Chapter II.6] FE computations for general elliptic BVP (general conductivity tensor).
  18. 11/4: FE computations for general elliptic BVP (low order terms, general boundary conditions).
  19. 11/7: continued.
    Assignment 4 due 11/28/11. Latex file for assignment 4 .
  20. 11/9: FE computations for problems with first order derivatives
  21. 11/11: structure of FE code; L^2 and H_0^1 projections.
  22. 11/14: meet in MLC computer lab
  23. 11/16: no class
  24. 11/18: L^2 projections and estimates. [Chapter II.7]. Clement quasi-interpolant. [Chapter II.6.8]
  25. 11/21: A-posteriori estimation; general ideas and Richardson extrapolation.
  26. 11/23: HW hints ... residual a-posterior estimator. [Read Chapter III.8]
  27. 11/28: Bank-Weiser/Oden estimator. Mesh adaptivity.
  28. 11/30: Mixed methods for Poisson's equation. [Read overview from Chapter III.4-5]
    Final Assignment due 12/9/11 10:00am.
  29. 12/2: Recap.
    Finite Element Holiday ornament made from students' final project results.