MTH 655-9 (Numerical Analysis) Finite Element Methods Winter 2017
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General information
Assignments
Announcement

Solutions to HW have to be turned in on time. Please use correct vocabulary and describe what you are doing. (A bunch of graphs stapled together with code listing is NOT a solution.) Group work is OK, but (i) acknowledge your coworkers, (ii) you must write up your solution by yourself.

Below I use the symbols [EG] and [J] to denote the two textbooks by Ern/Guermond and Johnson.

All the LaTex sources are available in this folder, including mydef.sty.

Assignments and schedule
  1. 1/9: Class cancelled due to inclement weather
  2. 1/11: Introduction. Handout 1.

    3. Assignment 1 due Fri, 1/20. You will need fd1d_singular.m. For those needing more templates, see fd1d.m
    Please turn in this assignment on paper.

  3. 1/13: Functional spaces and variational formulation. O(h2). [J Chapter 1], [EG Appendix B]
  4. 1/16: No class (MLK holiday)
  5. 1/18: Variational formulation (1d, [J, 1.1]) Neumann and other: [J, 1.7]. Sobolev spaces Hk (lite; see [J, 1.5]). Handout 2 on weak derivatives.
  6. 1/20: First finite element examples.

    7. Assignment 2 due Wed, 2/1. You will need fem1d_2017.m. For the original template, see fem1d.m
    Please turn in this assignment on paper.

  7. 1/23/17: HW 1 discussion. Calculating distributional derivatives. More on shape functions. Calculations on reference element.
  8. 1/25/17: FE implementation using fem1d_2017.m
  9. 1/27/17: Inner product spaces. Seminorms. H1 seminorm is the H01 norm via Poincare-Friedrichs inequality. Interpolation estimates.
  10. 1/30/17: Galerkin ortogonality, Cauchy-Schwarz, and interpolation estimates ===> basic H01 estimate. Start bilinear forms.
  11. 2/1/17: Further examples and constants of coercivity and continuity for bilinear forms. Abstract variational formulation in Hilbert spaces.

    Assignment 3 lab part due Tue, 2/7 (first journal entry) and Mon 2/13 (complete assignment)
    Please turn in this assignment in CANVAS.
    Assignment 3 theory part due Mon 2/13.

  12. 2/3/17: Minimization principle.
  13. 2/6/17: When and why (V) is equivalent to (M).
    Beyond d=1. Domains and boundaries in Rd. Meshes over Ωh
  14. 2/6/17: Lab meeting, MLC computer lab Kidd 108 3:00-4:00 or 4:00-5:00.
  15. 2/8/17: Derivatives and norms in d>1. L2 error estimate (Aubin-Nitsche way, with regularity theory).
  16. 2/10/17: How to go back from (V) to (DE) and extract the essential boundary conditions from the definition of space and natural boundary conditions from the formulation.
    More, in the notes from Variational Methods for Hilbert spaces by Prof. R. Showalter, Sections 1-2.

    Assignment 4 due Wed, 2/22. You may need demoELEMENT.m
    Please turn in this assignment on paper or in PDF CANVAS.
    Work in groups is allowed if you solve the Extra parts of the assignment.

  17. 2/13/17: Details of FE computations in 2D/3D.
  18. 2/15/17: Discuss and present HW solutions.
    Local stiffness matrix computations. Assembly process in 2D.
  19. 2/17/17: Variational crime (when computing the rhs by quadrature). Dual functionals, and in particular, Dirac distribution and Green's function, as it relates to Pbm 4 on HW4.
  20. 2/20/17: Strang I and Strang II lemmas for the Petrov-Galerkin case. Inhomogeneous Dirichlet boundary conditions.

    21. Assignment 5 lab part due Monday, March 6, 2017.
    Theory part is due Monday, March 6.

  21. 2/20/17:
    22. Lab meeting, MLC computer lab Kidd 108 Monday 3:00-4:00 or 4:00-5:00.
  22. 2/22/17: Why you should avoid thin elements. Dependence of interpolation error on the angles in the grid.
  23. 2/24/17: ADR problems, and why you need velocity/flux. Stokes and mixed Darcy formulation for the flow problem
    24. 2/27, 3/1: no class meetings.
  24. 3/3: Mixed methods, cotinued. Minimization vs saddle points. Handout 9 on minimization and constrained minimization.
    Extra office hours Friday 3:00-4:00
  25. 3/6: Stable and unstable approximations in mixed FE settng.
  26. 3/8:
  27. 3/10: Time-dependent problems and nonlinear problems
  28. 3/13: Error estimation. Wrap-up
  29. 3/13: Lab wrap-up with a pi-surpise. Corrections to previous HW can be discussed and worked out. Discussion on final project.
    30. 3/15, 3/17: no class meetings.
    Final Project due Tuesday 3/21 at 2pm. Please send files by email so I can prepare some notes.
  30. Presentations Wednesday 3/22 noon-13:50. Prepare 3 slides/pages in PDF on your project: 1) overview, 2) theory, 3) numerics. Group work is allowed. Be ready to show on the computer, or on doc camera.