TMA372 / MMG800 Partial differential equations, first course

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 (12.01.21). Please read this before we start, contains also zoom links to the lecture.
ExamInfo.pdf (18.02.21). Information from Chalmers on examination under covid time can be found here

Zoom link lecture: https://chalmers.zoom.us/j/62768297711 pwd: 31415
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: piazza.com/chalmers.se/spring2021/tma372
Youtube introduction here: link

Link to our piazza page (password 31415):

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 (18.01.21)
Chapter2.pdf (25.01.21)
Chapter3.pdf (25.01.21)
Chapter4.pdf (29.01.21)
Chapter5.pdf (03.02.21)
Chapter6.pdf (08.02.21)
Chapter7.pdf (12.02.21)
Chapter8.pdf (15.02.21)
Chapter9.pdf (18.02.21)
Chapter10.pdf (22.02.21)
Chapter11.pdf (25.02.21)
Chapter12.pdf (01.03.21)
Chapter13.pdf (01.03.21)
Chapter14.pdf (04.03.21)

intro.pdf (18.01.21)

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. This is listed as an indication. Link to the book (via Chalmers library): https://onlinelibrary-wiley-com.proxy.lib.chalmers.se/doi/book/10.1002/9781119671688

Day Sections Content Notes
Jan 18 1.1-1.2, 1.5
1, (2.2)
 
Classification of PDEs, derivation of heat and wave equations Note1
Jan 20 2.1,2.2 
3.1, 3.3, 3.5
Vector spaces, n differentiable-and integrable functions, Sobolev spaces Note2
Jan 21 2.3, 2.5, 2.6, 2.7 3.6-3.8 Basic inequalities, power of abstraction, Riesz and Lax-Milgram theorems Note3
Jan 22 3.2, 3.3
4.1, (4.2), 4.3
Polynomial approximation, Forward Euler for IVP, Galerkin for BVP, Finite difference Note4
Jan 25 3.5
5.1, 5.2
Preliminaries, Lagrange interpolation Note5
Jan 28 3.7 
5.3
Numerical integration, quadrature rule Note6
Feb 01 5.1, 5.2 
7.2-7.3
Finite element method (FEM), error estimates in energy norm Note7
Feb 04 5.3 
7.4
FEM for convection-diffusion-absorption BVPs Note8
Feb 05 6.1-6.3 
8.1-8.3
IVP: solution formula, stability, FD, Galerkin methods (continuous/discontinuous) Note9
Feb 08 6.4 
8.4
A posteriori error estimates error estimates for cG(1) and dG(0), adaptivity for dG(0).  Note10
Feb 11 6.5-6.6 
8.5-(8.6)
A priori error estimates for dG(0) (parabolic case) Note11
Feb 12 7.1 
9.1
Heat equation Note12
Feb 15 7.2 
9.2
Wave equation Note13
Feb 18 7.3 
9.3
Convection-Diffusion problems Note14
Feb 19 8.1-8.3
10.1-10.3
Approximation in several variables, construction of finite element spaces Note15
Feb 22 8.4, 9.1 
10.4, 11.1-11.3
Interpolation, Poisson equation, fundamental solution, stability, cG(1) error estimates Note16
Feb 25 10.1 
12.1-12.2
PDE in higher dimensions, heat equation, stability Note17
Feb 26   10.1-10.2 
12.3, 12.5
FEM for heat and wave equations in higher dimensions Note18
March 01 TBA TBA Note19
March 04 TBA TBA Note20
March 05 Repetition Repetition Note21

 

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Demonstrated exercises (with Malin) 

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

Day Exercises
Jan 27 Problem File: New_Problems.pdf Problems 53-60. Books: 2.13, 2.15/3.13, 3.15 Solutions_w1.pdf  
Jan 29 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.pdf  
Feb 03 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.pdf  
Feb 10 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.pdf  
Feb 17 Problem File: New_Problems.pdf Problems 21-23, 26-27. Books: 8.10,8.11/10.10, 10.11 Solutions_w5.pdf 
Feb 24 Problem File: New_Problems.pdf Problems 34-40. Books: 9.10, 9.12/11.9, 11.11 Solutions_w6.pdf  
March 03 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.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.
 
(26.01.21): I was informed today that only assignment 2 is mandatory for MVE455 students. 
However I strongly recommend to do (part) of assignment 1 as it may help to understand the lecture 
better (we could perhaps even give one bonus point for the exam).

Assignment 1: Assignment1.pdf (26.01.21, typo corrected). Deadline: Monday February 15.

Assignment 2: Assignment2.pdf (15.02.21). Deadline: Monday March 8. 

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.

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Written examination

Information on the written exam will be communicated in due time (still not clear if it will be online or not).

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Previous exams 

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.

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Student representatives: 

MPCAS andab@student.chalmers.se Andreas Andersson
MPALG ekgrim@student.chalmers.se Christoffer Ekgrim
TKTEM foraxel@student.chalmers.se Axel Forsman
TKTEM preiman@student.chalmers.se Jessica Preiman
MPENM roosemi@student.chalmers.se Emilia Roos

For more infos, please see https://student.portal.chalmers.se/sv/chalmersstudier/minkursinformation/kursvardering/Sidor/Att-vara-studentrepresentant.aspx