Literature: S. C. Brenner and L. R. Scott, The mathematical theory of finite element methods, 3rd ed., Springer, 2008.
Course description:
The core of the course is the theory of finite elements in Chapters 3 and 4. In addition, we study some preparations from Chapters 0, 1, 2 and selected topics from Chapters 5-14.
In this course we first study how to construct finite element function spaces based on triangular or rectangular element domains and piecewise polynomials. Then we develop the associated approximation theory based on averaged Taylor polynomials and Riesz potentials. This leads to interpolation error estimates in Sobolev norms. As a by-product we also obtain a proof of Sobolev's inequality.
Depending on the available time and the interests of the participants we may also study some additional topics, such as, multigrid methods, maximum norm estimates, mixed methods.
Required background:
Some experience with partial differential equations, finite element methods, functional analysis, and Sobolev spaces, corresponding to, for example, Chapter 5 and Appendix of S. Larsson and V. Thomée, Partial Differential Equations with Numerical Methods, Texts in Applied Mathematics 45, Springer, 2003.
Teacher: Stig Larsson
First meeting: Monday January 16, 12.00-12.30, room MV L14 (to determine the schedule).
Schedule:
Lecture Monday 10.00-11.45 room MV L14 (beginning January 17).
Lecture Tuesday 10.00-11.45 room MV L14.
Lecture and Exercises Friday 10.00-11.45 room MV L14.
Lectures:
X = Stig will travel abroad. Self study, week 4-5: read Chapter 3 of "The FEniCS Book" and prepare a presentation of two finite elements which are not discussed in Brenner-Scott. The book can be downloaded from FEniCS project web site.
1. Tue Jan 17.
Introduction. Chapt 0. (notes)
2. Fri Jan 20. Notes. Weighted norm
estimates (notes).
US=UI (notes).
3. Mon Jan 23. Chapt 3.1-3.2 (notes)
4. Tue Jan 24. Chapt 3.3-3.4 (notes)
5. Fri Jan 27. Chapt 3.4-3.7 (notes)
6. Mon Jan 30. Chapt 1, Chapt 4.1 (notes)
7. Tue Jan 31. Chapt 4.1-4.2 (notes)
8. Fri Feb 3. Chapt 4.3 (notes)
Mon Feb 6. X Self study: read Chapter 3 of "The
FEniCS Book" and prepare a presentation of two finite elements which
are not discussed in Brenner-Scott. The book can be downloaded from
FEniCS project web site.
Tue Feb 7. X Self study.
Mon Feb 13. X Self study.
Tue Feb 14. X Self study.
9. Mon Feb 20. Chapt 4.4 (notes)
10. Tue Feb 21. Chapt 4.4 cont'd (notes)
11. Fri Feb 24. Chapt 4.5 (notes)
12. Mon Feb 27. Chapt 4.6 (notes)
13. Tue Feb 28. Student presentations. Chapt 4.7 (notes)
14. Fri March 2. Chapt 4.8 (notes)
Exercises:
1. Jan 20. Hand in: Problems 0.x.6, 0.x.10, and prove (0.7.1)
on page 13. (0.x.6)
2. Jan 27. Hand in: Problems 3.x.3, 3.x.9, 3.x.10
3. Feb 3. Hand in: Problems 4.x.18, 4.x.20, 4.x.21
4. Feb 24. Hand in: Bramble-Hilbert.
7. March 2. Hand in: 4.x.8, 4.x.10
/stig