**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 *L ^{2}*-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} * |

** 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 }
= k_{n} *

** 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'' * || * < C _{i} * ||

and the

||

where

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

Stig Larsson Last modified: Wed Dec 10 13:23:08 MET 1997