MTH 654-9 FALL 2023

FINITE ELEMENTS

MALGORZATA PESZYNSKA

SCHEDULE

-------------- Week 0

1. 9/27/23: Introduction. Why study Finite Elements. 

2. 9/29/23:  FD view of the approximation error: truncation error+ data error and stability. Variational (weak) formulation and FE approximation for a model problem. 

DUE:  First Day Survey

-------------- Week 1

3. 10/2/23: Functional spaces: norms, inner product. C^k, L^p spaces. FSpaces-worksheet.jpeg

4. 10/4/23: Distributional (weak) derivatives. Sobolev spaces. Dirac delta. Functionals J(v): integral, Dirac, (strain) energy.

5. 10/6/23: Continue (M): the minimization of energy functional is equivalent to (V): variational formulation of (D). FE formulation.

-------------- Week 2

6. 10/9/23: Details of FE formulation in 1d, with piecewise linear Galerkin FE. Notation on inner product spaces, and start error estimate.  

DUE:  HW1.  

7. 10/11/23: Finish error estimate. Outline interpolation estimates for norm(u-I_h u). 

8. 10/13/23: No class today (SIAM PNW conference). 

-------------- Week 3

9. 10/16/23: Poincare-Friedrichs inequality allows to define H_0^1 norm which is easier to work with. Forms: bilinear, symmetric, continuous, coercive allow us to formulate (V), (FE), (M) efficiently.  

10. 10/18/23: Reference element calculations of stiffness and mass matrix; local and global numbering of nodes and elements. Assembly process. 

11. 10/20/23: Bilinear forms: worksheet with examples. 

DUE on 10/22/23: HW2

-------------- Week 4

12. 10/23/23: Blinear forms with integrals: checking on continuity and coercivity constants for selected self-adjoint PDEs depending on norm choice and boundary conditions.

13. 10/25/23: Bilinear forms for nonsymmetric elliptic PDEs  and those with singular behavior. Use in (V) formulation: well posedness. 

14. 10/27/23: Lax-Milgram theorem and FE error in abstract variational formulation with bilinear forms. HW2 discussion. 

-------------- Week 5

15. 10/30/23: No lecture. Please focus on HW2 redo's: error calculations in L^2 and H^1 norms. 

16. 11/1/23: Online lecture on software for HW3.   notes are here MTH654_notes.pdf

17. 11/3/23: Group work (no lecture) on software as outlined in HW3. 

-------------- Week 6

18. 11/6/23:  HW remarks: loops, MATLAB constructs and so on. Abstract error estimate with bilinear forms. 

DUE: HW 3. 

19. 11/8/23: Boundary conditions other than essential (homogeneous) Dirichlet. Neumann (natural). Conneciton to well-posedness of BVP.  

DUE: redos of HW2. 

(*). 11/10/23: VETERAN's Day

-------------- Week 7

20. 11/13/23: Why the use of grid norms (MATLAB norms) is a bad idea in 1d, with k=1. Proof of L^2 error estimates via Aubin-Nitsche duality method.

DUE: HW4.

21. 11/15/23: Strang second lemma proof, and how to use it for numerical integration. (Worksheet summarizing dependence of error estimates on polynomial degree k and regularity degree r).  

22. 11/17/23: Formal set-up of FE spaces in D>1. Grid: conforming and nonconforming (worksheet on grids). Affine and not. The triple (K,P,\Siggma).  

-------------- Week 8

23. 11/20/23: Examples of eleemnts: confirming and not. (Crouzeix-Raviart). Barycentric coordinates.

24. 11/22/23: Element computations; use numerical integration on triangles. What is required for V_h \subset V (lemma from Ciarlet).   

(*). 11/24/23: THANKSGIVING HOLIDAY

-------------- Week 9

25. 11/27/23: Projections (general context). How to work with quadrature on  triangles. Where do Gaussian points come from. Q_k elements as tensor products. 

26. 11/29/23: HW5 pre-discussion. How to set-up your 2D solutions with gridding->mesh2acf->ACF. Raviart-Thomas elements. Inhomogeneous boundary conditions.

DUE: HW5.

27. 12/1/23 Projections on FE spaces. 

-------------- Week 10

28. 12/4/23 Residual a-posterior estimate. Time dependent problems: which FE to use/which not to use.

29. 12/6/23 Parabolic problems: error estimate, algoorithms.  Stokes problem leads to mixed formulation.

30. 12/8/23: Other problems solved with FE: elasticity. REVIEW.

-------------- Week 11 (FINALS)

HW6 due December 13. 

No FINAL EXAM.