TMA371 Partial Differential Equations 1997

Course diary

This page will contain more detailed information about the course than the course programme. For example, information will be given about differences between the programme and what was actually covered. Also additional information about exercises and projects can be found here.

Lecture 1 (Tue 28/10)
The first lecture covers Chapter 6 in "Computational Differential Equations". The exercise part covers some problems from the same chapter, but also something from Chapter 4. It is important to be familiar with the basic concepts from linear algebra: scalar products, projections, norms, linear mappings etc. In the lectures I will assume that the material covered in Chapter 4 is known.

Lecture 2 (Thu 30/10)
I will finish Chapter 6, and then do relevant parts of Chapters 5 and 7. This deals with interpolation of functions by polynomials (Ch 5), and with the methods for solving linar systems of equations (Ch 7).

Exercise class, week 1
The following problems have been solved: 4.18, 4.22 (only L2 - norm), 5.2, 6.1, 6.3, 6.11. Also some theory concerning linear algebra was discussed.

Lecture 3 (Tue 4/11)
The topic for this lecture is the first half of Chapter 8 in the book.

Lecture 4 (Thu 6/11)
The second part of Chapter 8 will be studied. This deals with error estimation of the solutions, and it is one of the central themes of the course.

Lecture 5 (Tue 11/11) and Lecture 6 (Thu 13/11)
The first part of the lecture deals with chapter 14, i.e., with piecewise polynomials in more than one dimension, and with interpolation estimates. The remaining part of the week deals mostly with Poisson's equation in two dimensions: The variational formulation, the Galerkin method with piecewise linear functions, error estimates, and adaptivity. It is a very good idea to look at chapter 13 and to recall some calculus in several variables from the earlieer courses.

Lecture 7 (Tue 18/11)
The lecture covers the remaining part of Chapter 15: a posteriori estimates in two dimensions, regularity of solutions, and optimal error estimates in the L2-norm. At the end, there will be a brief recapitulation of the material covered so far.

Lecture 8 (Thu 20/11)
From now on the lectures will ge given by Stig Larsson. This lecture started with an introduction to the three "archetypes": elliptic, parabolic, and hyperbolic problems. We then covered the beginning of Chapter 9, sections 9.1-9.2.

Lecture 9 (Tue 25/11)
Sections 9.3, 9.4, 9.5 (Theorem 9.5 without proof. There is a misprint here: in Theorem 9.5 and Lemma 9.6 we must assume that k n | a | I n < 1/2. This is not needed in the parabolic case.). You may skip Section 9.6.

Lecture 10 (Thu 27/11)
I emphasized the follosing parts of Chapter 10: Section 10.5 (10.5.2, 10.5.3), Section 10.6, Section 10.7 (Theorem 10.2, Lemmas 10.3, 10.4, 10.5). The error estimates in the case of a system of ODEs are essentially the same as for the case of a scalar equation (Chapter 9). The main difference is that it is slightly more complicated to compute the stability factors. I described how this is done for problems of parabolic and hyperbolic types, using both the spectral method and the energy method.

Lecture 11 (Tue 2/12)
In Chapter 16 we study Sections 16.2 and 16.3 in detail (note that 16.3 is an introduction to problem 1a of assignment 2b, and that 16.2 gives the background to problem 1b). Sections 16.4 and 16.5 are studied without details; just note that we simply (?) combine the numerical method for spatial discretization from Chapter 15 with the time stepping methods from Chapter 9. Sections 16.1 and 16.6 may be skipped.

Lecture 12 (Thu 4/12)
In Chapter 17 we emphasize Sections 17.1, 17.2.2, 17.2.3, 17.2.4, 17.3.2, 17.3.3, 17.4 without details, in 17.5 only Theorem 17.2 (without proof).

Hint for Assignment 2b: observe that w j = kn -1 S I n Uj (t)dt (= the average of Uj ) at the top of page 443.

Lecture 13 (Tue 9/12)
Sections 18.1, 18.2, 18.3, 18.4.1. If there is time we will cover the beginning of Chapter 21. Chapters 19 and 20 may be skipped.

Note for GU students: you will have a separate exam, covering also Chapter 20.

Lecture 14 (Thu 11/12)
Chapter 21. We emphasize Sections 21.1, 21.2, 21.3, 21.4.1--21.4.3, 21.5.

Please note that there was a small misprint in Assignment 1b. In the first equation, in the left hand side, we wrote c(x), but this should be c(x) u(x) .

A note for the assignment
For the beam problem it is necessary to solve problems 8.38, 8.39 and 8.40, or at least to find the a priori and a posteriori error bounds. The a priori bound is
|| e'' || < Ci || h2 u (4) ||,
and the a posteriori error bound is
|| e'' || < Ci || h2 R(U) || ,
where R(U) = f - U (4) on each sub-interval.

For those of you who use Windows, there is now a version of adfem with short filenames. A link to this can be found on the main page for this course.

Suggestions for exercises
(Some of these exercises will be demonstrated)

Chapter 5: 5.11, 5.12, 5.17, 5.23, 5.27, 5.56

Chapter 6: 1: Give a variational formulation of -u'' + u = f in (0,1) , with u(0) = u(1) = 0 .
2: Write a FEM-formulation with piecewise linear, continuous functions, and a uniform stepsize h = 1/4.
3: The same as above, but with piecewise quadratic functions.

Chapter 7: 7.3, 7.5, 7.24, 7.31 (prove in addition that there is exactly one minimum), 7.54

Chapter 8: 8.1, 8.3, 8.6, 8.7, 8.8, 8.11, 8.12, 8.15, 8.16 8.18, 8.21, 8.22, 8.23, 8.32, 8.38, 8.41

Chapter 14: 14.7, 14.10,

Chapter 15: 15.5, 15.9, 15.11, 15.15, 15.20, 15.22, 15.24, 15.35, 15.39, 15.44, 15.45, 15.47

Chapter 9: 9.4, 9.5, 9.7, 9.9, 9.10, 9.12, 9.13, 9.14, 9.19, 9.22, 9.23, 9.24, 9.26, 9.27, 9.28, 9.33, 9.43, 9.45, 9.46

Chapter 16: 16.4, 16.7, 16.11, 16.14, 16.15, 16.18, 16.20

Chapter 17: 17.4, 17.8, 17.9, 17.10, 17.11, 17.13, 17.17, 17.18, 17.19, 17.20, 17.33

Chapter 18: 18.1, 18.3, 18.4, 18.5, 18.6, 18.9

Chapter 21: 21.1, 21.2, 21.3, 21.4, 21.5, 21.8, 21.9, 21.13