## 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!!*

### 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.

### 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) |
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 |

### 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. |

### 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.*

**Written examination**

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

### 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.

### Student representatives:

Gustav Birath Blom (TKTEM)

Safwan Dieb (MPENM)

Alessandro Dordoni (MPENM)

Karl WennerstrÃ¶m (MPDSC)

Leo Ã…nestrand (TKTEM)

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