Latest news
Welcome to the course! The schedule for the course can be found in TimeEdit.
Course representatives:
Oskar Liew, liewo@student.chalmers.se
Magnus
Fries, friesm@student.chalmers.se
Teachers
Course coordinator: Mihaly Kovacs
Teaching assistant: Maximilian Thaller
Course literature
Reference literature:
- S. Larsson and V. Thomée, Partial Differential Equations with Numerical Methods, Texts in Applied Mathematics 45, Springer, 2003. Corrected second printing 2005. Softcover, 2009.
Other material:
- A non-rigorous description of the variational formulation of certain PDEs in 2 and 3D and FEM in 2D.
- A note on the Projection Theorem.
- l_2 is a Hilbert space.
- Introduction to distributions (from previous years).
Program
Preliminary topics and where to find them in the book: functional analysis background and functions spaces (Appendix A.1), elliptic problems (Chapter 3), FEM in 2D (Chapter 5.2), parabolic problems (Chapters 8.1, 8.3, 8.4), hyperbolic problems (Chapters 11.1-11.3).
Lectures
| Day |
Sections | Contents |
|---|---|---|
| Oct 30 |
11.1, Appendix A.1 |
Characteristic polynomial/direction/surface, classification of
second order linear PDEs, inner product spaces |
| Oct 31 |
Appendix A.1 pp. 225-227 |
Linear spaces cont'd |
| Nov 6 |
Appendix A.1 pp. 227-232 |
Riesz representation theorem, coercivity, energy functional,
energy estimate, Lax-Milgram lemma, function spaces |
| Nov 7 |
Appendix A.1 pp. 232-234 |
Function spaces cont'd: L^p spaces, weak derivatives, Sobolev
spaces |
| Nov 10 |
Appendix A.1 pp. 235-237 | Sobolev spaces cont'd, trace theorem |
| Nov 13 |
Appendix A.1 p. 238; Chapter 3.5 pp. 32-33 | Poincare's inequality, the space H^1_0 and H^{-1}, weak formulation of BVP (Dirichlet homogeneous) |
| Nov 14 | Chapter 3.5 pp. 33-36 | Weak formulation of BVP cont'd (Dirichlet homogeneous, inhomogenous, Neumann homogeneous) |
| Nov 17 |
Chapter 3.5 pp. 36-37 | Weak formulation of BVP cont'd (Neumann homogeneous/inhomogeneous), regularity |
| Nov 20 | Chapter 5.2 pp. 57-58; Chapter 3.2 pp. 26-27 | FEM in 2D, maximum principle |
| Nov 21 | Chapter 3.2, pp. 27-28 | Maximum principle cont'd, Dirichlet problem for a disc: Poisson's integral formula |
| Nov 24 | Chapter 3.2 pp. 29-31 | Mean value property, strong maximum principle, fundamental solution |
| Nov 27 | Chapter 3.2 pp. 29-31 | Poisson's equation in R^d |
| Nov 28 | Chapter 3.2 p. 32; Chapter 8.1 pp. 109-111; Appendix A.3 | Laplace's equation, Green's function, the pure initial value problem for the heat equation: existence and continuity at t=0, properties of the Fourier transform |
| Dec 1 | Chapter 8.1 pp. 111-112; Chapter 8.4 pp 122-123 | The pure initial value problem for the heat equation: stability, a parabolic maximum principle |
| Dec 4 | Chapter 8.3 pp. 120-122 | Uniqueness for the pure initial value problem for the heat equation, variational formulation and energy estimates for the initial-boundary value problem for the heat equation |
| Dec 5 | Chapter 11.2 pp. 167-169 | The pure initial value problem for the wave equation: solution formulae, the method of descent (not in the book), energy estimates, finite propagation speed, uniqueness, energy estimate for the initial-boundary value problem |
| Dec 8 | Chapter 11.3 pp. 169-171 | First order scalar linear equations: the method of characteristics |
| Dec 11 | Introduction to the theory of distributions: definition, examples, derivatives | |
| Dec 12 |
Introduction to the theory of distributions cont'd: multiplication by smooth functions, convolution with test functions, weak convergence, application to divergent Fourier series |
Recommended exercises
- Solutions to selected problems in set 1.
- Solutions to selected problems in set 2.
- Solutions to selected problems in set 3.
- Solutions to selected problems in set 4.
- Solutions to set 5.
| Week |
Exercises |
|---|---|
| 1 |
Recommended problems set
1 |
| 2 |
Recommended problems set
2 |
| 3 |
Recommended problems set 3 |
| 4-5 |
Recommended problems set 4 |
| 6-7 |
Recommended problems set 5 |
Computer labs
Learning MATLAB, Tobin A. Driscoll ISBN: 978-0-898716-83-2 (The book is published by SIAM).
Course requirements
The learning goals of the course can be found in the course plan.
Projects
Project descriptions.
Examination
Examination procedures
In Chalmers Student Portal you can read about when exams are given and what rules apply on exams at Chalmers. In addition to that, there is a schedule when exams are given for courses at University of Gothenburg.
Before the exam, it is important that you sign up for the examination.
If you study at Chalmers, you will do this by the
Chalmers Student Portal, and if you study at University of
Gothenburg, you sign up via GU's
Student Portal.
At the exam, you should be able to show valid identification.
After the exam has been graded, you can see your results in Ladok by logging on to your Student portal.
At the annual (regular) examination:
When it is practical, a separate review is arranged. The date of the
review will be announced here on the course homepage. Anyone who can not
participate in the review may thereafter retrieve and review their exam
at the Mathematical
Sciences Student office. Check that you have the right grades and
score. Any complaints about the marking must be submitted in writing at
the office, where there is a form to fill out.
At re-examination:
Exams are reviewed and retrieved at the Mathematical
Sciences Student office. Check that you have the right grades and
score. Any complaints about the marking must be submitted in writing at
the office, where there is a form to fill out.
Old exams
Look at the homepage for this course for previous years.