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 (20.01.23, update without the link to an old .pdf file). Please read this before we start.
ExamInfo.pdf (05.02.23). Information from Chalmers on examination can be found here

Lectures and exercises are planned to be given on campus.

If I (=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/62768297711 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/68802328665 pwd 946530.

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/spring2023/tma373/home
Youtube introduction here: link

Link to our piazza page (password 31415): https://piazza.com/chalmers.se/spring2023/tma373/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!!

Back to the top

Summaries and various files

Chapter1.pdf (16.01.23)
Chapter2.pdf (21.01.23)
Chapter3.pdf (26.01.23)
Chapter4.pdf (30.01.23)
Chapter5.pdf (06.02.23)
Chapter6.pdf (09.02.23)
Chapter7.pdf (10.02.23)
Chapter8.pdf (16.02.23)
Chapter9.pdf (20.02.23)
Chapter10.pdf (23.02.23)
Chapter11.pdf (24.02.23)
Chapter12.pdf (27.02.23)
Chapter13.pdf (02.03.23)

intro.pdf (12.01.23)
Photos and infos on some mathematicians (20.02.23)

Quiz1, Quiz2, Quiz3, Quiz4, Quiz5, Quiz6, Quiz7 (07.03.23)

Back to the top

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 from 2021 and 2022 may (will) differ from this year's lecture.
This is listed as an indication and may be subject to change.

Day	Sections	Content (2023)	Content (2021-2022)	Notes 2021	Notes 2022
Jan 16	1.1-1.2, 1.5	Classification of PDEs, derivation of heat and wave equations	Classification of PDEs, derivation of heat and wave equations	Note1	Note1
Jan 18	2.1,2.2	Vector spaces, Spaces differentiable-and integrable functions, Sobolev spaces	Vector spaces, n differentiable-and integrable functions, Sobolev spaces	Note2	Note2
Jan 19	2.3, 2.5, 2.6, 2.7	Basic inequalities, Riesz and Lax-Milgram theorems	Basic inequalities, power of abstraction, Riesz and Lax-Milgram theorems	Note3	Note3
Jan 20	3.2, 3.3	Polynomial approximation, Lagrange interpolation	Polynomial approximation, Forward Euler for IVP, Galerkin for BVP, Finite difference	Note4	Note4
Jan 23	3.5	Numerical integration, quadrature rule	Preliminaries, Lagrange interpolation	Note5	Note5
Jan 26	3.7	IVP and forward Euler for IVP	Numerical integration, quadrature rule	Note6	Note6
Jan 30	5.1, 5.2	Galerkin for BVP. Finite element method (FEM), error estimates in energy norm	Finite element method (FEM), error estimates in energy norm	Note7	Note7
Feb 02	5.3	FEM for convection-diffusion-absorption BVPs	FEM for convection-diffusion-absorption BVPs	Note8	Note8
Feb 03	6.1-6.3	A posteriori error estimates for cG(1), adaptivity	IVP: solution formula, stability, FD, Galerkin methods (change probably)	Note9	Note9
Feb 06	6.4	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 09	6.5-6.6	Heat equation	A priori error estimates for dG(0) (parabolic case) (change probably)	Note11	Note11
Feb 10	7.1	Wave equation	Heat equation	Note12	Note12
Feb 13	7.2	Approximation in several variables, construction of finite element spaces	Wave equation	Note13	Note13
Feb 16	7.3	Interpolation, Poisson equation, fundamental solution, stability, cG(1) error estimates	Convection-Diffusion problems	Note14	Note14
Feb 17	8.1-8.3	PDE in higher dimensions, heat equation, stability	Approximation in several variables, construction of finite element spaces	Note15	Note15
Feb 20	8.4, 9.1	FEM for heat and wave equations in higher dimensions	Interpolation, Poisson equation, fundamental solution, stability, cG(1) error estimates	Note16	Note16
Feb 23	10.1	Finite difference	PDE in higher dimensions, heat equation, stability	Note17	Note17
Feb 24	10.1-10.2	TBA	FEM for heat and wave equations in higher dimensions	Note18	Note18
Feb 27	TBA	TBA	TBA	Note19	Note19
Mar 02	TBA	TBA	TBA	Note20	Note20
Mar 03	Repetition	Repetition	Repetition	Note21	Note21

Back to the top

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.

Day	Exercises	Solution 22
Jan 25	Problem File: New_Problems.pdf Problems 53-60. Book: 2.13, 2.15	Solutions_w1_2022.pdf
Jan 27	Problem File: New_Problems.pdf Problems 1-5. Book: 3.5-3.7, 3.24, 3.25	Solutions_w2_2022.pdf
Feb 01	Problem File: New_Problems.pdf Problems 6-12. Book: 5.3-5.8, 5.10, 5.16-5.19	Solutions_w3_2022.pdf
Feb 08	Problem File: New_Problems.pdf Problems 13-20. Book: 6.8, 6.11, 6.14, 7.5-7.8	Solutions_w4_2022.pdf
Feb 15	Problem File: New_Problems.pdf Problems 21-23, 26-27. Book: 8.10, 8.11	Solutions_w5_2022.pdf week_5_corr.pdf
Feb 22	Problem File: New_Problems.pdf Problems 34-40. Book: 9.10, 9.12	Solutions_w6_2022.pdf
Mar 01	Problem File: New_Problems.pdf Problems 43-52. Book: 10.4, 10.9, 10.16, 10.17	Solutions_w7_2022.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. 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
SW6	Book: Problems in Chapters 8-10.

Back to the top

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 by sending an email to Michael.

Assignment (12.01.23)
Templates: template_lab4.m (09.02.23), template_lab5.m (if needed), Chicken.txt (12.01.23).
template-PlotSolutionHelmoltz.m, template-elementmassmatrix.m, template-plotmygrid.m (28.02.23). template-FEHelmoltz2D.m (03.03.23)

Literature 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.

Back to the top

Written examination

Information on the written exam will be communicated in due time. See top of the page.

Back to the top