Welcome to the course TMA373/MMG801!

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.

General information

CourseInfo.pdf (20.01.25): General information on the course, the examination, the literature and bonus points. Please read this before we start.
ExamInfo.pdf (20.01.25). Information from Chalmers on examination can be found here

The prerequisites and the syllabus can be found here: Chalmers and GU.

We are all here to study and learn something new!
We thus do not allow the use of chatbots in the course, especially for the computational assignments and the written exams.
Further information on AI systems at Chalmers 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 (password 31415): https://piazza.com/chalmers.se/spring2025/tma373
Youtube introduction here: link

Link to our piazza page (password 31415): https://piazza.com/chalmers.se/spring2025/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 (22.01.25)
Chapter2.pdf (02.02.25)
Chapter3.pdf (05.02.25)
Chapter4.pdf (07.02.25)
Chapter5.pdf (07.02.25)
Chapter6.pdf (13.02.25)
Chapter7.pdf (13.02.25)
Chapter8.pdf (20.02.25)
Chapter9.pdf (23.02.25)
Chapter10.pdf (26.02.25)
Chapter11.pdf (28.02.25)
Chapter12.pdf (03.03.25)
Chapter13.pdf (06.03.25)

Intro.pdf (20.01.25)
Photos and infos on some mathematicians (20.01.25)

FDnotes and FDclean (06.03.25)

Back to the top

Lectures and lecture notes

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.

A really really preliminary version of lecture notes can be found here (update from 05.03.25).

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 (from the book, see above)	Content (2025)	Content (2023)	Content (2021-2022)	Notes 2021	Notes 2022
Jan 20	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 22 (obs. lecture)	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 23	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 24	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 27	3.5	Lagrange interpolation	Numerical integration, quadrature rule	Preliminaries, Lagrange interpolation	Note5	Note5
Jan 30	3.7	PW linear interpolation. Numerical integration/quadrature rules.	IVP and forward Euler for IVP	Numerical integration, quadrature rule	Note6	Note6
Feb 03	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 06	5.3	FEM for elastic deformation of a bar	FEM for convection-diffusion-absorption BVPs	FEM for convection-diffusion-absorption BVPs	Note8	Note8
Feb 07	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 10	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 13	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) (change probably)	Note11	Note11
Feb 14	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 17	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 20	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 21	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 24	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 27	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 28	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
Mar 03	TBA	The concept of finite element. Higher order FE. Variational crimes.	TBA	TBA	Note19	Note19
Mar 06	TBA	Stability of time integrators. Dahlquist test equation. Finite difference	TBA	TBA	Note20	Note20
Mar 07	Repetition	Summaries (if time) and exam from 13.03.23	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.
The below displayed exercises are from the book An Introduction to the Finite Element Method for Differential Equations (2020).

Day	Content 2025	Exercises	Solution 2022	Solution 2024
Jan 29	Sobolev spaces, trace theorem, Poincaré-Friedrich inequality, Lax-Milgram theorem	Problem File: New_Problems.pdf Problems 5, 53-60. Book: 2.13, 2.15	Solutions_w1_2022.pdf	week1_2024.pdf
Jan 31	Interpolation (linear, p.w. constant), quadrature rules, error estimates	Problem File: New_Problems.pdf . Problems 6-10. Book: 3.24, 3.25	Solutions_w2_2022.pdf	week2_2024.pdf
Feb 5	IVP for ODEs: global Galerkin method, cG(1) for IVP, dG(0) for IVP	Problem File: New_Problems.pdf Problems 1-3, 22, 23. Book: 3.5-3.7, 6.11	Solutions_w3_2022.pdf	week3_2024.pdf
Feb 12	cG(1) and cG(2) for BVP in 1D	Problem File: New_Problems.pdf Problems 4, 11-16, cG(2) for BVP. Book: 5.3-5.8, 5.10	Solutions_w4_2022.pdf	week4_2024.pdf
Feb 19	A priori and a posteriori error estimates	Problem File: New_Problems.pdf Problems 18-20. Book: 5.16-5.19	Solutions_w5_2022.pdf week_5_corr.pdf
Feb 26	Poisson equation in 2D: fundamental solution, discretization	Problem File: New_Problems.pdf Problems 26-27, 34-40. Book: 8.10, 8.11, 9.10, 9.12	Solutions_w6_2022.pdf
Mar 5	Heat equation in 1D and 2D, fundamental solution; wave equation in 1D (d'Alembert's formula)	Problem File: New_Problems.pdf Problems 43-52. Book: 7.5-7.8, 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.

Back to the top

Computer labs

You may work alone or 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 via Canvas (under assignments).

Assignment (20.01.25). 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 (20.01.25), template_lab5.m (20.01.25), Chicken.txt,
template-PlotSolutionHelmoltz.m, template-elementmassmatrix.m, template-plotmygrid.m, template-FEHelmoltz2D.m (20.01.25).

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.
http://ubcmatlabguide.github.io/

Back to the top

Previous exams

2025: 1st session: Exam and proposition for solutions. 2nd session: Exam and proposition for solutions. 3nd session: Exam and proposition for solutions.
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.

Back to the top

Student representatives

Oscar Ekström (MATGRUND)
Karl Hallström (MPENM)
Huining Liu (MPENM)
Sebastian Wåhlström (TIELL)

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/