(Subject to revision according to progress in teaching plan).
Reference on PDEs
L.C. Evans: Partial Differential Equations, 1998. Amer. Math. Soc.. ISBN: 0-8218-0772-2. Graduate Studies in Mathematics, Volume 19.
Material covered in Evans' book : Linear second order elliptic equations, Lax Milgram theorem, Fredholm alternative, Eigenvalues, H^2 regularity. Second order parabolic equations. i.e.:
- Chapter 6 : §6.1, §6.2, §6.3, §6.5 (parts);
- Chapter 7 : §7.1 (except regularity);
- Some notions on weak convergence techniques;
- Chapter 5 and Appendices as needed by the above sections.
Reference on Finite Elements
D. Braess: Finite Elements (Theory, fast solvers, and applications in solid mechanics), 2001. Cambridge University Press. ISBN: 0-521-01195-7. 2. edition.
Material covered in Braess' book : Continuous piecewise polynomial finite elements for second order elliptic problems, Cea's lemma, Bramble-Hilbert Lemma, Inverse estimates, Aubin-Nitsche trick. i.e.:
- Chapter 2 : §1,§2 (already covered in Evans book)
- Chapter 2 : §4, §5, §6, §7
Additional References
W. Rudin: Functional Analysis, 1991. McGraw-Hill. ISBN: 0-07-054236-8. 2. edition.
Rudin's book will not be used explicitely in the course but can be useful to have at hand. Any other reference on functional analysis will do equally well (e.g. the books of Reed-Simon "Functional Analysis" (Academic Press) or Lax "Functional Analysis" (Wiley-Interscience)). Chapter 5 in Folland's book "Real analysis" (Wiley-Interscience) contains more than needed, in an elegant 30 page presentation.
Notes
Notes will sometimes be distributed as part of the curriculum. They contain exercises and background material.
Exam
Topics for the first part of the oral exam: Exam topics