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!!
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)
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://onlinelibrarywileycom.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 (20212022)  Notes 2021  Notes 2022 

Jan 16  1.11.2, 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 differentiableand integrable functions, Sobolev spaces  Vector spaces, n differentiableand integrable functions, Sobolev spaces  Note2  Note2 
Jan 19  2.3, 2.5, 2.6, 2.7  Basic inequalities, Riesz and LaxMilgram theorems 
Basic inequalities, power of abstraction, Riesz and LaxMilgram 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 convectiondiffusionabsorption BVPs  FEM for convectiondiffusionabsorption BVPs  Note8  Note8 
Feb 03  6.16.3 
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  6.56.6 
Heat equation 
A priori error estimates for dG(0) (parabolic case) 
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  ConvectionDiffusion problems  Note14  Note14 
Feb 17  8.18.3 
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  10.110.2 
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 
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 5360. Book: 2.13, 2.15 
Solutions_w1_2022.pdf 
Jan 27  Problem File: New_Problems.pdf Problems 15. Book: 3.53.7, 3.24, 3.25 
Solutions_w2_2022.pdf 
Feb 01  Problem File: New_Problems.pdf Problems 612. Book: 5.35.8, 5.10, 5.165.19 
Solutions_w3_2022.pdf 
Feb 08  Problem File: New_Problems.pdf Problems 1320. Book: 6.8, 6.11, 6.14, 7.57.8 
Solutions_w4_2022.pdf 
Feb 15  Problem File: New_Problems.pdf Problems 2123, 2627. Book: 8.10, 8.11 

Feb 22  Problem File: New_Problems.pdf Problems 3440. Book: 9.10, 9.12 
Solutions_w6_2022.pdf 
Mar 01  Problem File: New_Problems.pdf Problems 4352. Book: 10.4, 10.9, 10.16, 10.17 
Solutions_w7_2022.pdf 
Recommended exercises (selfstudy):
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 FEMformulation with piecewise linear, continuous functions, and a uniform stepsize h=1/4. 3: The same as above, but with piecewise quadratic functions. Book: 1.11.5, 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, 3.13.4, 3.173.19 
SW4,5  Book: 5.1, 5.3, 5.9, 6.36.6, 7.3, 7.6, 7.9 
SW6  Book: Problems in Chapters 810. 
Computer labs:
You may work in a group of 23 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).
templatePlotSolutionHelmoltz.m, templateelementmassmatrix.m, templateplotmygrid.m (28.02.23). templateFEHelmoltz2D.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 ebook 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. See top of the page.
Previous exams
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_20170315(pdf),
2016: Ordinary Exam and solutions: tenta+sol_20160316(pdf),
2015: Ordinary Exam and solutions: tenta_20150318(pdf).
Previous exams for MVE455, please see here.
Student representatives:
Martin Bergström
Elliot Källander
Simon Rödén
Xinxin Tan
For more infos, please see https://student.portal.chalmers.se/sv/chalmersstudier/minkursinformation/kursvardering/Sidor/Attvarastudentrepresentant.aspx