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.


CourseInfo.pdf (25.01.22). Please read this before we start.
ExamInfo.pdf (14.02.22). Information from Chalmers on examination 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 lecture: pwd: 31415.
If Malin cannot give the exercises on campus, she will contact you via Canvas. Then, please use the zoom link exercise:

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:
Youtube introduction here: link

Link to our piazza page (password 31415):

If you have any problems or feedback for the developers, email
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 (17.01.22)
Chapter2.pdf (20.01.22)
Chapter3.pdf (24.01.22)
Chapter4.pdf (24.01.22)
Chapter5.pdf (31.01.22)
Chapter6.pdf (04.02.22)
Chapter7.pdf (05.02.22)
Chapter8.pdf (05.02.22)
Chapter9.pdf (08.02.22)
Chapter10.pdf (13.02.22)
Chapter11.pdf (15.02.22)
Chapter12.pdf (15.02.22)
Chapter13.pdf (24.02.22)
Chapter14.pdf (01.03.22)

intro.pdf (16.01.22)
Proof a posteriori (04.02.22)
Photos and infos on some mathematicians (15.02.22)

Solutions from exercise sessions are found below.

Back to the top


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. Link to the book (via Chalmers library):
Observe that the notes from 2021 may (will) differ from this year's lecture.
This is listed as an indication and may be subject to change.

Day Sections Content Notes 2021 Notes 2022
Jan 17 1.1-1.2, 1.5
1, (2.2)
Classification of PDEs, derivation of heat and wave equations Note1 Note1
Jan 19 2.1,2.2 
3.1, 3.3, 3.5
Vector spaces, n differentiable-and integrable functions, Sobolev spaces Note2 Note2
Jan 20 2.3, 2.5, 2.6, 2.7 3.6-3.8 Basic inequalities, power of abstraction, Riesz and Lax-Milgram theorems Note3 Note3
Jan 21 3.2, 3.3
4.1, (4.2), 4.3
Polynomial approximation, Forward Euler for IVP, Galerkin for BVP, Finite difference Note4 Note4
Jan 24 3.5
5.1, 5.2
Preliminaries, Lagrange interpolation Note5 Note5
Jan 28 3.7 
Numerical integration, quadrature rule Note6 Note6
Jan 31 5.1, 5.2 
Finite element method (FEM), error estimates in energy norm Note7 Note7
Feb 03 5.3 
FEM for convection-diffusion-absorption BVPs Note8 Note8
Feb 04 6.1-6.3 
IVP: solution formula, stability, FD, Galerkin methods (change probably) Note9 Note9
Feb 07 6.4 
A posteriori error estimates error estimates for cG(1) and dG(0), adaptivity for dG(0). (change probably) Note10 Note10
Feb 10 6.5-6.6 

A priori error estimates for dG(0) (parabolic case)
(change probably)

Note11 Note11
Feb 11 7.1 
Heat equation Note12 Note12
Feb 14 7.2 
Wave equation Note13 Note13
Feb 17 7.3 
Convection-Diffusion problems Note14 Note14
Feb 18 8.1-8.3
Approximation in several variables, construction of finite element spaces Note15 Note15
Feb 21 8.4, 9.1 
10.4, 11.1-11.3
Interpolation, Poisson equation, fundamental solution, stability, cG(1) error estimates Note16 Note16
Feb 24 10.1 
PDE in higher dimensions, heat equation, stability Note17 Note17
Feb 25  10.1-10.2 
12.3, 12.5
FEM for heat and wave equations in higher dimensions Note18 Note18
Feb 28 TBA TBA Note19 Note19
Mar 03 TBA TBA Note20 Note20
Mar 04 Repetition Repetition Note21 Note21


Back to the top

Demonstrated exercises (with Malin) 

Below are indications of exercises that will be discussed during the sessions. 

Day Exercises Solution 22 (tba)
Jan 26 Problem File: New_Problems.pdf  Problems 53-60. Books: 2.13, 2.15/3.13, 3.15 Solutions_w1_2022.pdf 
Jan 27 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_2022.pdf   
Feb 02 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_2022.pdf 
Feb 09 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_2022.pdf   
Feb 16 Problem File: New_Problems.pdf Problems 21-23, 26-27. Books: 8.10,8.11/10.10, 10.11 Solutions_w5_2022.pdf 
Feb 23 Problem File: New_Problems.pdf Problems 34-40. Books: 9.10, 9.12/11.9, 11.11 Solutions_w6_2022.pdf 
Mar 02 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_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.
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.

Back to the top

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:
Assignment 1: Assignment1.pdf (13.01.22). Deadline: Monday February 14.

Assignment 2: Assignment2.pdf (13.01.22). Deadline: Monday March 07. 

Computational literature:

  1. 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.
  2. 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

Previous exams 

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.

Back to the top

Student representatives: 

Gustav Birath Blom (TKTEM)
Safwan Dieb (MPENM)
Alessandro Dordoni (MPENM)
Karl Wennerström (MPDSC)
Leo Ånestrand (TKTEM)

For more infos, please see