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.

Information

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: https://chalmers.zoom.us/j/62768297711 pwd: 31415.
If Malin cannot give the exercises on campus, she will contact you via Canvas. Then, please use the zoom link exercise: https://gu-se.zoom.us/j/67019248484.

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/spring2022/tma372/home
Youtube introduction here: link

Link to our piazza page (password 31415): https://piazza.com/chalmers.se/spring2022/tma372/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!!

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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.

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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) and in blue of the compendium from 2018. Link to the book (via Chalmers library): https://onlinelibrary-wiley-com.proxy.lib.chalmers.se/doi/book/10.1002/9781119671688
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 5.3	Numerical integration, quadrature rule	Note6	Note6
Jan 31	5.1, 5.2 7.2-7.3	Finite element method (FEM), error estimates in energy norm	Note7	Note7
Feb 03	5.3 7.4	FEM for convection-diffusion-absorption BVPs	Note8	Note8
Feb 04	6.1-6.3 8.1-8.3	IVP: solution formula, stability, FD, Galerkin methods (change probably)	Note9	Note9
Feb 07	6.4 8.4	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 8.5-(8.6)	A priori error estimates for dG(0) (parabolic case) (change probably)	Note11	Note11
Feb 11	7.1 9.1	Heat equation	Note12	Note12
Feb 14	7.2 9.2	Wave equation	Note13	Note13
Feb 17	7.3 9.3	Convection-Diffusion problems	Note14	Note14
Feb 18	8.1-8.3 10.1-10.3	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 12.1-12.2	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

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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.

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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: malinni@chalmers.se.

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

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

Computational literature:

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.

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Written examination

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

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