TMA372 / MMG800 Partial differential equations, first course
Content of this page: lectures, exercises, assignments , files and summaries, as well as previous exams.
Student representatives are listed at the end of the page.
Information
CourseInfo.pdf (12.01.21). Please read this before we start, contains also zoom links to the lecture.
ExamInfo.pdf (18.02.21). Information from Chalmers on examination under covid time can be found here
Zoom link lecture: https://chalmers.zoom.us/j/62768297711 pwd: 31415
Zoom link exercise: https://gu-se.zoom.us/j/67019248484
Piazza Discussion Forum:
We will be using Piazza for class discussion.
The system is highly catered to getting you help fast and efficiently from classmates, the TA, and myself. Rather than emailing questions to the teaching staff, I encourage you to post your questions on Piazza.
In order to encourage your active participation, a student will earn 1 bonus point at the exam if she/he, at least, posts one question and answers two questions. The questions/answers must be relevant to the course.
Students can also ask questions anonymously (such questions cannot be counted for a bonus point).
Find our class signup link at: piazza.com/chalmers.se/spring2021/tma372
Youtube introduction here: link
Link to our piazza page (password 31415): https://piazza.com/chalmers.se/spring2021/tma372/home
If you have any problems or feedback for the developers, email team@piazza.com.
If one of you wants to initiate and organise a discord discussion for a group of students, please fee free to do so!!
Summaries and various files
Chapter1.pdf (18.01.21)
Chapter2.pdf (25.01.21)
Chapter3.pdf (25.01.21)
Chapter4.pdf (29.01.21)
Chapter5.pdf (03.02.21)
Chapter6.pdf (08.02.21)
Chapter7.pdf (12.02.21)
Chapter8.pdf (15.02.21)
Chapter9.pdf (18.02.21)
Chapter10.pdf (22.02.21)
Chapter11.pdf (25.02.21)
Chapter12.pdf (01.03.21)
Chapter13.pdf (01.03.21)
Chapter14.pdf (04.03.21)
intro.pdf (18.01.21)
Solutions from exercise sessions are found below.
Lectures
The schedule of the course is in TimeEdit. The below displayed sections are from the book An Introduction to the Finite Element Method for Differential Equations (2020) and in blue of the compendium from 2018. This is listed as an indication. Link to the book (via Chalmers library): https://onlinelibrary-wiley-com.proxy.lib.chalmers.se/doi/book/10.1002/9781119671688
| Day | Sections | Content | Notes |
|---|---|---|---|
| Jan 18 | 1.1-1.2, 1.5 1, (2.2) |
Classification of PDEs, derivation of heat and wave equations | Note1 |
| Jan 20 | 2.1,2.2 3.1, 3.3, 3.5 |
Vector spaces, n differentiable-and integrable functions, Sobolev spaces | Note2 |
| Jan 21 | 2.3, 2.5, 2.6, 2.7 3.6-3.8 | Basic inequalities, power of abstraction, Riesz and Lax-Milgram theorems | Note3 |
| Jan 22 | 3.2, 3.3 4.1, (4.2), 4.3 |
Polynomial approximation, Forward Euler for IVP, Galerkin for BVP, Finite difference | Note4 |
| Jan 25 | 3.5 5.1, 5.2 |
Preliminaries, Lagrange interpolation | Note5 |
| Jan 28 | 3.7 5.3 |
Numerical integration, quadrature rule | Note6 |
| Feb 01 | 5.1, 5.2 7.2-7.3 |
Finite element method (FEM), error estimates in energy norm | Note7 |
| Feb 04 | 5.3 7.4 |
FEM for convection-diffusion-absorption BVPs | Note8 |
| Feb 05 | 6.1-6.3 8.1-8.3 |
IVP: solution formula, stability, FD, Galerkin methods (continuous/discontinuous) | Note9 |
| Feb 08 | 6.4 8.4 |
A posteriori error estimates error estimates for cG(1) and dG(0), adaptivity for dG(0). | Note10 |
| Feb 11 | 6.5-6.6 8.5-(8.6) |
A priori error estimates for dG(0) (parabolic case) | Note11 |
| Feb 12 | 7.1 9.1 |
Heat equation | Note12 |
| Feb 15 | 7.2 9.2 |
Wave equation | Note13 |
| Feb 18 | 7.3 9.3 |
Convection-Diffusion problems | Note14 |
| Feb 19 | 8.1-8.3 10.1-10.3 |
Approximation in several variables, construction of finite element spaces | Note15 |
| Feb 22 | 8.4, 9.1 10.4, 11.1-11.3 |
Interpolation, Poisson equation, fundamental solution, stability, cG(1) error estimates | Note16 |
| Feb 25 | 10.1 12.1-12.2 |
PDE in higher dimensions, heat equation, stability | Note17 |
| Feb 26 | 10.1-10.2 12.3, 12.5 |
FEM for heat and wave equations in higher dimensions | Note18 |
| March 01 | TBA | TBA | Note19 |
| March 04 | TBA | TBA | Note20 |
| March 05 | Repetition | Repetition | Note21 |
Demonstrated exercises (with Malin)
Below are indications of exercises that will be discussed during the sessions.
| Day | Exercises | |
|---|---|---|
| Jan 27 | Problem File: New_Problems.pdf Problems 53-60. Books: 2.13, 2.15/3.13, 3.15 | Solutions_w1.pdf |
| Jan 29 | Problem File: New_Problems.pdf Problems 1-5. Books: 3.5-3.7, 3.24, 3.25/4.5-4.7, 5.15, 5.16 | Solutions_w2.pdf |
| Feb 03 | Problem File: New_Problems.pdf Problems 6-12. Books: 5.3-5.8, 5.10, 5.16-5.19/7.3-7.8, 7.10, 7.16-7.19 | Solutions_w3.pdf |
| Feb 10 | Problem File: New_Problems.pdf Problems 13-20. Books: 6.8, 6.11, 6.14, 7.5-7.8/8.8, 8.11, 8.16, 9.5-9.8 | Solutions_w4.pdf |
| Feb 17 | Problem File: New_Problems.pdf Problems 21-23, 26-27. Books: 8.10,8.11/10.10, 10.11 | Solutions_w5.pdf |
| Feb 24 | Problem File: New_Problems.pdf Problems 34-40. Books: 9.10, 9.12/11.9, 11.11 | Solutions_w6.pdf |
| March 03 | Problem File: New_Problems.pdf Problems 43-52. Books: 10.4, 10.9, 10.16, 10.17/12.4, 12.9, 12.13, 12.14 | Solutions_w7.pdf |
Recommended exercises (self-study):
| Study Week (SW) | Exercises |
|---|---|
| SW2 | 1: Give a varitional formulation of -u''+u' +u=f in (0,1), with u'(0) =1 and 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. Books: 1.1-1.5, 1.11, 1.12, 1.21, 1.22/2.1-2.5, 2.11, 2.12, 2.21, 2.22 |
| SW3 | Read through iterative methods of chapter 4 (self study not included in the exam). Books: 2.3, 3.1-3.4, 3.17-3.19/3.3, 4.1-4.4, 5.8-5.10 |
| SW4,5 | Books: 5.1, 5.3, 5.9, 6.3-6.6, 7.3, 7.6, 7.9/7.1, 7.3, 7.9, 8.3-8.6, 9.3, 9.6, 9.9 |
| SW6 | Book: Problems in Chapters 8-10. |
Computer labs:
You may work in a group of 2 persons but hand in only one report for the group (don't forget to include all names and relevant information).
Submit your report by sending an email to Malin: malinni@chalmers.se.
(26.01.21): I was informed today that only assignment 2 is mandatory for MVE455 students.
However I strongly recommend to do (part) of assignment 1 as it may help to understand the lecture
better (we could perhaps even give one bonus point for the exam).
Assignment 1: Assignment1.pdf (26.01.21, typo corrected). Deadline: Monday February 15.
Assignment 2: Assignment2.pdf (15.02.21). Deadline: Monday March 8.
Computational literature:
- Learning MATLAB, Tobin A. Driscoll. Provides a brief introduction to Matlab to the one who already knows computer programming. Available as e-book from Chalmers library.
- Physical Modeling in MATLAB 3/E, Allen B. Downey
The book can be downloaded from the web. The book gives an introduction for those who have not programmed before. It covers basic MATLAB programming with a focus on modelling and simulation of physical systems.
Written examination
Information on the written exam will be communicated in due time (still not clear if it will be online or not).
Previous exams
2021: 1st session: Exam, Exam MVE455 and solutions. 2nd session: Exam, Exam MVE455, and solutions. 3nd session: Exam, Exam MVE455, and solution.
2020: tenta_200316.pdf, tenta_200609.pdf, tenta_KF_200316.pdf, tenta_KF_200609.pdf
2019: Ordinary Exam and solutions: tenta+sol_20190320(pdf),
2018: Ordinary Exam and solutions: tenta+sol_20180314A(pdf),
2017: Ordinary Exam and solutions: tenta+sol_2017-03-15(pdf),
2016: Ordinary Exam and solutions: tenta+sol_2016-03-16(pdf),
2015: Ordinary Exam and solutions: tenta_2015-03-18(pdf).
Previous exams for MVE455, please see here.
Student representatives:
MPCAS andab@student.chalmers.se Andreas Andersson
MPALG ekgrim@student.chalmers.se Christoffer Ekgrim
TKTEM foraxel@student.chalmers.se Axel Forsman
TKTEM preiman@student.chalmers.se Jessica Preiman
MPENM roosemi@student.chalmers.se Emilia Roos
For more infos, please see https://student.portal.chalmers.se/sv/chalmersstudier/minkursinformation/kursvardering/Sidor/Att-vara-studentrepresentant.aspx