TMA373 / MMG801 Partial differential equations, first course
Content of this page: lectures, exercises, computer assignments , files and summaries, as well as previous exams.
Student representatives are listed at the end of the page.
Information
CourseInfo.pdf (05.01.24). Please read this before we start.
ExamInfo.pdf (05.01.24). Information from Chalmers on examination can be found here
Lectures and exercises are planned to be given on campus.
If David cannot give the lecture on campus, I'll contact you via Canvas. Then, please use the zoom link for the lecture: https://chalmers.zoom.us/j/69859657554 pwd: 31415.
If Michael cannot give the exercises on campus, he will contact you via Canvas. Then, please use the zoom link for the exercises: https://chalmers.zoom.us/j/66384775046 pwd 689581.
TA: Michael Roop. Office hours: Wednesdays 10-16.
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: https://piazza.com/chalmers.se/spring2024/tma373
Youtube introduction here: link
Link to our piazza page (password 31415): https://piazza.com/chalmers.se/spring2024/tma373
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 (15.01.24)
Chapter2.pdf (19.01.24)
Chapter3.pdf (25.01.24)
Chapter4.pdf (30.01.24)
Chapter5.pdf (02.02.24)
Chapter6.pdf (08.02.24)
Chapter7.pdf (11.02.24)
Chapter8.pdf (15.02.24)
Chapter9.pdf (19.02.24)
Chapter10.pdf (22.02.24)
Chapter11.pdf (23.02.24)
Chapter12.pdf (28.02.24)
Chapter13.pdf (29.02.24)
intro.pdf (05.01.24)
Photos and infos on some mathematicians (05.01.24)
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).
Link to the book (via Chalmers library): https://onlinelibrary-wiley-com.proxy.lib.chalmers.se/doi/book/10.1002/9781119671688
GU students can get access to the (physical and online) Chalmers library, please visit the library to get more information on how to.
Observe that the notes are from 2021 and 2022. They may (will) differ from this year's lecture.
This table is listed as an indication and may be subject to change.
| Day | Sections | Content (2023) | Content (2023) | Content (2021-2022) | Notes 2021 | Notes 2022 |
|---|---|---|---|---|---|---|
| Jan 15 | 1.1-1.2, 1.5 |
Classification of PDEs, derivation of heat and wave equations | Classification of PDEs, derivation of heat and wave equations | Classification of PDEs, derivation of heat and wave equations | Note1 | Note1 |
| Jan 17 | 2.1,2.2 |
Vector spaces, Spaces differentiable-and integrable functions, Sobolev spaces | Vector spaces, Spaces differentiable-and integrable functions, Sobolev spaces | Vector spaces, n differentiable-and integrable functions, Sobolev spaces | Note2 | Note2 |
| Jan 18 | 2.3, 2.5, 2.6, 2.7 | Basic inequalities, Riesz and Lax-Milgram theorems |
Basic inequalities, Riesz and Lax-Milgram theorems |
Basic inequalities, power of abstraction, Riesz and Lax-Milgram theorems | Note3 | Note3 |
| Jan 19 | 3.2, 3.3 |
Polynomial approximation, Lagrange interpolation | Polynomial approximation, Lagrange interpolation | Polynomial approximation, Forward Euler for IVP, Galerkin for BVP, Finite difference | Note4 | Note4 |
|
Jan 22 |
3.5 |
Lagrange interpolation |
Numerical integration, quadrature rule |
Preliminaries, Lagrange interpolation | Note5 | Note5 |
| Jan 25 | 3.7 |
PW linear interpolation. Numerical integration/quadrature rules. |
IVP and forward Euler for IVP |
Numerical integration, quadrature rule | Note6 | Note6 |
| Jan 29 | 5.1, 5.2 |
IVP and forward Euler for IVP. |
Galerkin for BVP. Finite element method (FEM), error estimates in energy norm |
Finite element method (FEM), error estimates in energy norm | Note7 | Note7 |
| Feb 01 | 5.3 |
FEM for elastic deformation of a bar | FEM for convection-diffusion-absorption BVPs | FEM for convection-diffusion-absorption BVPs | Note8 | Note8 |
| Feb 02 | 6.1-6.3 |
A priori-error estimates in energy norm A posteriori-error estimates in energy norm |
A posteriori error estimates for cG(1), adaptivity | IVP: solution formula, stability, FD, Galerkin methods (change probably) | Note9 | Note9 |
| Feb 05 | 6.4 |
Adaptivity FEM for further BVP |
A priori error estimates | A posteriori error estimates error estimates for cG(1) and dG(0), adaptivity for dG(0). (change probably) | Note10 | Note10 |
| Feb 08 | 6.5-6.6 |
Heat equation in 1d and discretisation with FEM in space and backward Euler in time |
Heat equation |
A priori error estimates for dG(0) (parabolic case) |
Note11 | Note11 |
| Feb 09 | 7.1 |
Wave eq. in 1d, conservation of energy. Discretisation with FEM in space and CN in time | Wave equation | Heat equation | Note12 | Note12 |
| Feb 12 | 7.2 |
Green's formula in Rd. Variational formulation of Poisson eq. in 2d. Triangulation Space of linear polynomials. |
Approximation in several variables, construction of finite element spaces | Wave equation | Note13 | Note13 |
| Feb 15 | 7.3 |
Space of cont. pw linear functions. L2 projection Notes on implementation of FEM in 2d |
Interpolation, Poisson equation, fundamental solution, stability, cG(1) error estimates | Convection-Diffusion problems | Note14 | Note14 |
| Feb 16 | 8.1-8.3 |
Notes on implementation of FEM in 2d | PDE in higher dimensions, heat equation, stability | Approximation in several variables, construction of finite element spaces | Note15 | Note15 |
| Feb 19 | 8.4, 9.1 |
A priori error estimates, in the energy norm, for FEM for Poisson eq. in 2d. Heat eq. in higher dimension |
FEM for heat and wave equations in higher dimensions | Interpolation, Poisson equation, fundamental solution, stability, cG(1) error estimates | Note16 | Note16 |
| Feb 22 | 10.1 |
Variational formulation and FEM for heat in higher dimension. Semi-discrete error estimates. Conservation of energy for wave eq. | Finite difference | PDE in higher dimensions, heat equation, stability | Note17 | Note17 |
| Feb 23 | 10.1-10.2 |
Variational formulation and FEM for wave eq. in higher dimension. Error estimates for wave eq. The concept of finite element. | TBA | FEM for heat and wave equations in higher dimensions | Note18 | Note18 |
| Feb 26 | TBA | The concept of finite element. Higher order FE. Variational crimes. | TBA | TBA | Note19 | Note19 |
| Feb 29 | TBA | Stability of time integrators. Dahlquist test equation. Finite difference | TBA | TBA | Note20 | Note20 |
| Mar 01 | Repetition | Summaries (if time) and exam from 13.03.23 | Repetition | Repetition | Note21 | Note21 |
Demonstrated exercises (with Michael)
Below are indications of exercises that will be discussed during the sessions. This may be subject to slight changes in the ordering.
The below displayed exercises are from the book An Introduction to the Finite Element Method for Differential Equations (2020).
| Day | Exercises | Solution 2022 | Solution 2024 |
|---|---|---|---|
| Jan 24 | Problem File: New_Problems.pdf Problems 53-60. Book: 2.13, 2.15 |
Solutions_w1_2022.pdf | week1_2024.pdf |
| Jan 26 | Problem File: New_Problems.pdf . Problems 6-10. Book: 3.24, 3.25 |
Solutions_w2_2022.pdf | week2_2024.pdf |
| Jan 31 | Problem File: New_Problems.pdf Problems 1-3, 5. Book: 3.5-3.7, 6.11 |
Solutions_w3_2022.pdf | week3_2024.pdf |
| Feb 07 | Problem File: New_Problems.pdf Problems 4, 11-20. Book: 5.3-5.8, 5.10, 5.16-5.19 |
Solutions_w4_2022.pdf | week4_2024.pdf |
| Feb 14 | Problem File: New_Problems.pdf Problems 22-23, 26-27. Book: 6.14, 7.5-7.8, 8.10, 8.11 |
|
|
| Feb 21 | Problem File: New_Problems.pdf Problems 34-40. Book: 9.10, 9.12 |
Solutions_w6_2022.pdf | |
| Feb 28 | Problem File: New_Problems.pdf Problems 43-52. Book: 10.4, 10.9, 10.16, 10.17 |
Solutions_w7_2022.pdf | week7_2024.pdf |
Recommended exercises (self-study):
The below displayed exercises are from the book An Introduction to the Finite Element Method for Differential Equations (2020).
| 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. Book: 1.1-1.5, 1.11, 1.12, 1.21, 1.22 |
| SW3 | Read through iterative methods of chapter 4 (self study not included in the exam). Book: 2.3, 3.1-3.4, 3.17-3.19 |
| SW4,5 | Book: 5.1, 5.3, 5.9, 6.3-6.6, 7.3, 7.6, 7.9 Problem File: 21 = 6.8 (book) (look at the solution in week_5_corr.pdf above and try to improve it) |
| SW6 | Book: Problems in Chapters 8-10. |
Computer labs:
You may work in a group of 2-3 persons but hand in only one report for the group (don't forget to include all names and relevant information).
Submit your report via Canvas.
Assignment (09.01.24, update 20.02.24). Hint.pdf for the derivation of cG(2) FEM from Michael Roop (09.01.24).
File with element stiffness and mass matrices in 2D: Element matrices 2D.pdf
Templates: template_lab4.m (09.01.24), template_lab5.m (20.02.24), Chicken.txt,
template-PlotSolutionHelmoltz.m, template-elementmassmatrix.m, template-plotmygrid.m, template-FEHelmoltz2D.m (09.01.24).
Literature and reference on matlab:
- 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. - https://maths.dundee.ac.uk/dfg/MatlabNotes.pdf
- http://ubcmatlabguide.github.io/
Previous exams
2024: 1st session: Exam and proposition for solutions. 2nd session: Exam and proposition for solutions. 3nd session: Exam and proposition for solutions.
2023: 1st session: Exam and solutions. 2nd session: Exam and solutions. 3nd session: Exam and solution.
2022: 1st session: Exam and solutions. 2nd session: Exam and solutions. 3nd session: Exam and solution.
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:
Petter Dittmer (MPENM)
Andreas Führ (MPCAS)
Mathilda Gustafsson (MPCAS)
Anxhela Hysa (MPENM)
Lech Kazimierz Kula (TKELT)
For more infos, please see https://student.portal.chalmers.se/sv/chalmersstudier/minkursinformation/kursvardering/Sidor/Att-vara-studentrepresentant.aspx
and https://www.chalmers.se/utbildning/dina-studier/planera-och-genomfora-studier/kursvardering/ and https://intranet.chalmers.se/verktyg-stod/utbildning/kurs/kursvardering/