# TMA372/MVE455/MMG800, Partial differential equations, first course, 2018/19

## General information

Welcome to the course Partial Differential Equations I . The course schedule is posted in TimeEdit.

Except this general information, all material should be consided as preliminary and subject to minor alternation.

This is a first course on partial differential equations (PDEs) intended for upper undergraduate and master students following engineering programs at Chalmers and math and physics programs at the University of Gothenburg. Due to recommendation from their advisers, PhD students in computational/applied math, applied sciences and engineering disciplines are also welcome to take the course.

After the first 2 weeks, the third year students of Physical Chemistry (Kf3) programe are joining the course.

Course cods and credit points:
For students registered at Chalmers: TMA372, (7.5 points).
For students registered in GU: MMG800, (7.5 points).
For Kf3 students: MVE455, (4.5 poits).
For PhD students as in TMA372/MMG800 or the supervisor's suggestion.

Contents: The main topic of this course is: Application of finite element methods to PDEs. More specifically, we consider problems of, e.g., dynamical systems, Poisson's (Laplace; stationary heat) equation, heat conduction, wave propagation, convection-diffusion-reaction equations. The goal is two-fold:

i) To provide the students with some theoretical backgroud: wellposedness (existence, uniqueness and stability), regularity, convergence analysis and conservation properties (when applicable) for classical PDEs. The highlights in this part are Poincare inequaly, Riesz representation and Lax-Milgram theorems.

ii) To introduce the participants some modern approximation skills: Approximation procedures for PDEs (and ODEs) using Finite Element Methods (FEMs) based on Polynomial Approximation/Interpolation. This results to large, sparse linear system of equations (that are solved using techniques from numerical linear algebra).
Stability and error analysis in both a priori and a posteriori regi, in order to verify efficiency and reliability of the considered FEMs.
There are two compulsary, bonus generating, home assignments that are recommended as group work. They contain both analytic approaches as well as coding aspects ranging from iterative algorithms to problems involving multiphysics programing.

Participating and following the course actively, you should gain some analytical intutions (e.g., "in which mathematical environment a solution to a particular problem can make sense", or "not all PDEs: as is, can be assigned a closed form solution"), and learn some approximation techniques to solve the PDEs (of e.g., your own field of interest), consisting of knowledge on correct weak/variational formulation, and a great deal about finite element analysis containing both theoretical aspects as stability and convergence of approximate solutions, as well as numerical analysis and implementations.

The course consists of 36 lecture hours (Lh) and 20 exercise hours (Eh):
3x2 hours/week: 5h lectures+1 h Excercises (Totally 35Lh+7Eh) and
1x2 hours exercise/week (Totally 14Eh).

course description/PM.

## Latest news

For all current and most recent information please visit!
course diary.

Below is the concise schedule:

Day Time Place Remarks Office Hours
MON 8-10 Pascal Lecture Mohammad:10-12, L2035; questions
WED 8-10 Pascal Exercise (Lecture W1) Maximilian:13-15, L2032-2; assignments
THU 10-12 Pascal Lecture
FRI 8-10 Pascal Lecture/Exercise

## Teachers

Course coordinator:

Teaching assistants:

Maximilian Thaller, maxtha@chalmers.se

Lab supervisor:

Maimilian Thaller, maxtha@chalmers.se

## Course literature

An Introduction to the Finite Element Method (FEM) for Differential Equations. Available in Cremona
OBS! The 2018 version.

Lecture Notes: An Introduction to the Finite Element Method (FEM) for Differential Equations: Part II Problems in ${\mathbb R}^d,\,\, d>1$.

Reference literature:

• K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Computational Differential Equations, Studentlitteratur 1996.
• M. Asadzadeh, Lecture Notes in Fourier Analysis.
• S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Second edition, Springer 2002.
• C. Johnson, Numerical solutions of partial differential equations by the finite element method, reprinted by Dover, 2008
• M. Taylor, Partial Differential equations (basic theory), Springer 1996.
• W. Strauss, Partial Differential equations, An inroduction, 2008.
• Tobin A. Driscoll, Learning MATLAB, ISBN: 978-0-898716-83-2 (The book is published by SIAM)
• English-Swedish mathematical dictionary

• ## Program

Lectures (preliminary plan)
Week Book chapters
Contents
4, Study week 1
1, 3.1-3.3, 3.7-3.8, 4
Classification of PDEs, Math Tools, Power of abstraction, Reisz and Lax-Milgram theorems: Summary of Chapter 1. Summary of Chapter 3.
(proofs: later!) Polynomial approximation. Galerkin spaces, Stiffness matrix.
5, study Week 2
5, 7
Polynomial Interpolation, Gauss quarature rule, Boundary value problem (BVP), Variational formulation (VP), minimization problem (MP)
"a priori" and "a posteriori" error estimates in energy norm.
6, study Week 3
7, 8
Finite element approximations: Examples of Convection-diffusion-advection/absorption problems .
Scalar initial value problem (IVP): Exact solution, theoretical stability.
Finite Differences for (IVP) .

7, study Week 4
8
Continuous and discontinuous Galerkin and a priori/a posteriori error estimates for Galerkin methods for scalar initial value problems. . Initial-boundary value problems: one- space dimension. Stability for the heat conduction, finite element approximation for the heat equation, error analysis..
8, study Week 5
9, Lecture Notes: Poisson in ${\mathbb R}^d, \, d>1$
Initial-boundary value problems: one- space dimension. The wave equation, conservation of energy, finite elements for the wave equation. Error estimates and adaptive error control in the energy norm for the Poisson's equation. A canonical example.
9, study Week 6
Lecture Notes: Heat in ${\mathbb R}^d, \, d>1$
Stability and piecewise linear Galerkin approximation for the heat equation. Error analysis of finite element methods for the heat equation.
10, study Week 7
Lecture Notes: Heat/Wave in ${\mathbb R}^d, \, d>1$
Conservation of energy. Wave equation as a system of IVP. Finite elements for the wave equation. Finite element method for a convection-diffusion model problem. Selected problems from exams, 2015-2018.
11, (study) Week 8
Reserved material
The week is reserved for going through uncovered material , answering questions, solving some old-exams and other interesting problems.

#### Recommended exercises

Week Excersises
5, study week w2
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: 2.1-2.5, 2.11, 2.12, 2.21, 2.22
6, study week 3
Chapters 3-5: Read through iterative methods of chapter 5(self study not included in the exam). Book: 3.3, 4.1-4.4, 5.8-5.10
7,8 study week 4,5
Chapters 7-9: Book: Problems in Chapters 7.1, 7.3, 7.9, 8.3-8.6, 9.3, 9.6, 9.9,
9, study week 6
Chap 10-12: Lecture Notes: Problems in Chapters 10-12.

Demonstrated/recommended Exercises
Week Excersises From:Problem file, Book and Lecture note
4, study w2
Problem file: Problems 53-60
Book: 3.13-3.15,
5, study w3
Problem file: Problems 1-5,
Book: 4.5-4.7, 5.15, 5.16
6, study w4
Problem file: Problems 6-12
Book: 7.3-7.8, 7.10, 7.16-7.19
7, study w5
Problem file: Problems 13-20
Book: 8.8-8.11, 8.16, 9.5-9.8
8, study w6
Problem file: chosen problems from the list: 21-23, 26-27
Lecture Notes: 10.10, 10.11
8, study w6
Problem file: chosen problems from the list: 34-40 43-52
Lecture Notes: 11.9, 11.11
9, study w7
Problem file: chosen problems from the list: 43-52,
Lecture Notes:12.4, 12.9, 12.13, 12.14
10, study w8
Reserve

## Computer labs

Compulsary Home Assignments,
Computer labs and Matlab excercises are included in the assignments below

## Course requirements

The learning goals of the course can be found in the course plan.

Here are some crucial concepts:
A good knowledge of linear algbra, calculus of several variables, ordinary differential equations and Fourier analysis are fundamental to follow the course and gain an optimal result from it.

## Assignments

You may work in a group of 2 persons but hand in only one report for the group.

Assignment 1: See the file. For this assignment write a short yet detailed report, not exceeding ten pages, explaning your work and sumbit it by the end of study week 4 (Deadline: Friday February 15) . Use MATLAB to do the coding parts. Hints: For problem 1 you need to read chapter 7. problem 2 consider only the case a=4. A good starting point for problem 3 might be the Matlab code, which solves -u''=f, u(0)=u(1)=0 using cG(1).
If you don't have access to FEM-LAB, then you may skip FEM-LAB comparisons.

Assignment 2: Can be found here. Hand in report of your work beginning of study week 7 (Deadline: Monday March 4).

Kf3 students will hand in only assignment 2

## Examination

• To pass this course you should pass the written exam and the assignments 1 and 2.
• To pass each assignment you need to get at least 1 point in that assigment.
• Assignments 1 and 2 have total of 3, respectively, 5 poits. Hence maximum bonus poits for them are 2 and 4, respectively.
• The two compulsory home assignments should be handed in before the due times above. They are generating max 6 (2+4) bonus points.
• For full points in assigment 2 you need to use a posteriori estimates and perform adaptive mesh-generation.
• Written examination

• Final exam is compulsory, written, and consists of 6 questions (5 problems + 1 theorm) with a maximum score of 30 (=6x5) points. This means that the proportion between the points in home assignments and the exam is 6/30=1/5=20%.
• Same proportion for Kf3 is then 4/P=1/5, i.e. Kf3 exam will have maximum score of P=20 points for (4 problems + 1 theorm each having maximum score of 4 points).
• The theory question is choosen from a list that will appear in (see sample exam questions in the course diary).
• As for the proof of Lax-Milgram theotrm, you may use the proof in lecture notes I.
• No aids are allowed.
• You should be able to state and explain all definitions and theorems given in the course and also apply them in problem solving.
• Grades are set according to the table below.

Grades Chalmers Points (Kf3)
- <15 (<10)
-
3 15-21 (10-15)
G 15-26
4 22-28 (16-20)
VG 27-
5 29- (>20)

The exam takes place at ..
Bring ID and receipt for your student union fee

Solutions to the exam will be published in the course diary.
You will be notified the result of your exam by email from LADOK (This is done automatically as soon as the exams have been marked an the results are registered.)
The exams will then be kept at the students' office in the Mathematical Sciences building.
Check that the number of points and your grade given on the exam and registered in LADOK coincide.
Complaints of the marking should be written and handed in at the office. There is a form you can use, ask the person in the office.).

The following link will tell you all about the examination room rules at Chalmers: Examination room instructions

• ## Examination procedures

In Chalmers Student Portal you can read about when exams are given and what rules apply on exams at Chalmers. In addition to that, there is a schedule when exams are given for courses at University of Gothenburg.

Before the exam, it is important that you sign up for the examination. If you study at Chalmers, you can do this from the Chalmers Student Portal, and if you study at University of Gothenburg, you sign up via GU's Student Portal.

At the exam, you should be able to show valid identification.

After the exam has been graded, you can see your results in Ladok by logging on to your Student portal.

At the annual (regular) examination:
When it is practical, a separate review is arranged. The date of the review will be announced here on the course homepage. Anyone who can not participate in the review may thereafter retrieve and review their exam at the Mathematical Sciences Student office. Check that you have the right grades and score. Any complaints about the marking must be submitted in writing at the office, where there is a form to fill out.

At re-examination:
Exams are reviewed and retrieved at the Mathematical Sciences Student office. Check that you have the right grades and score. Any complaints about the marking must be submitted in writing at the office, where there is a form to fill out.

## Student Representativres:

Please come to course coordinator's office: MVL:2035; Friday 2/15; 11:50-12:30.
+
Name Programe E-mail
Kajsa Ahlgren KF3 kajsaah@student.chalmers.se
Lova Wilske KF3 lova@student.chalmers.se
Hanna Ek TMTEM ekhanna@student.chalmers.se
Henrik Nordell TMTEM hennord@student.chalmers.se
Casper Opperud TMTEM opperud@student.chalmers.se