TMA372/MMG800, Partial differential equations, first course, 2017/18

General information

Welcome to the course Partial Differential Equations I . The schedule for the course can be found in TimeEdit.

Except the general introduction below all material should be consided as preliminary and subject to minor alternations.

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 and applied sciences and engineering are also welcome to take this course.

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

The course code is:
for students registered at Chalmers: TMA372,
for students registered in GU: MMG800.
For Kf3 students: MVE455.

Contents: The main topic of this course is: Application of finite element methods to problems of, e.g., dynamical systems, Poisson's (Laplace; stationary heat) equation, heat conduction, wave propagation, convection-diffusion-reaction equations. The goal is to provide the participants with some
i) Theoretical backgroud: wellposedness (existence, uniqueness and stability), regularity, convergence analysis and conservation properties (when applicable). The highlights in this part are Reisz representation and Lax-Milgram theorems.
ii) Modern approximation skills: Approximation procedures to various PDEs (ODEs) using various Finite Element Methods, Polynomial Interpolation, Quadrature rules, and the solution techniques of the resulting large, sparse linear system of equations. Error estimates in both a priori and a posteriori settings.
Compulsary home assignments,(in group) contain both analytic approaches as well as coding aspects, ranging from iterative algorithms to problems involving complex multiphysics programing. This can provide, and also challenge, the students implementation skills.

Participating and following the course, actively, you should gain some analytical intutions (e.g., not all PDEs: as is, can be assigned a closed form solution), and learn some approximation skill 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 consisting of both theoretical aspects as stability and convergence of approximate solutions, as well as numerical analysis and implementations.

The course consists of 36 lecture hours, 20 exercise hours and gives 7.5 points.

course description/PM.

Latest news

For all current and most recent information please check the
course diary.


The schedule for the course can be found via the link to webTimeEdit top of the page.
Below is the concise schedule:
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:10-12, L2032-2; assignments
THU 10-12 Pascal Lecture
FRI 8-10 Pascal Lecture/Exercise

Teachers

Course coordinator:

Mohammad Asadzadeh, mohammad@chalmers.se

Teaching assistants:

Maximilian Thaller, @maxtha@chalmers.se

Lab supervisor:

Maimilian Thaller, maxtha@chalmers.se

Course literature




M. Asadzadeh,
An Introduction to the Finite Element Method (FEM) for Differential Equations. Available in Cremona

M. Asadzadeh,
Lecture Notes: An Introduction to the Finite Element Method (FEM) for Differential Equations: Part II Problems in ${\mathbb R}^d,\,\, d>1$. ( Chapters 10-12; will appear by Feb 15),




Reference literature:

  • K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Computational Differential Equations, Studentlitteratur 1996.
  • M. Asadzadeh, Lecture Notes in Fourier Analysis, (pdf).
  • 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)

  • Program



    Lectures (preliminary plan)
    Week Book chapters
    Contents
    3, Study w1
    1, 3.1-3.3, 3.7-3.8
    Classification of PDEs, Math Tools, Power of abstraction, Reisz and Lax-Milgram theorems
    4, study W2
    4, 5
    Polynomial approximation, Polynomial Interpolation, L_2 projection, Gauss quarature rule
    5, study W3
    7
    Two-points boundary value problems, Finite element approximations, "a priori" and "a posteriori" error estimates in energy norm.
    6, study W4
    8
    Continuous and discontinuous Galerkin and a priori/a posteriori error estimates for Galerkin methods for scalar initial value problems.
    7, study W5
    9
    Initial-boundary value problems: one- space dimension. Stability for the heat conduction, finite element approximation for the heat equation, error analysis. The wave equation, conservation of energy, finite elements for the wave equation.
    8, study W6
    Lecture Notes: Poisson/Heat in ${\mathbb R}^d, \, d>1$
    Error estimates and adaptive error control in the energy norm for the Poisson's equation. Stability and piecewise linear Galerkin approximation for the heat equation. Error analysis of finite element methods for the heat equation.
    9, study W7
    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, 2012-2017.





    Recommended exercises
    Week Excersises
    4, study w2
    1: Give a varitional formulation of -u''+u=f in (0,1), with u(0)=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
    5, study w3
    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
    6,7 study w 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,
    8, study w6
    Chap 10-12: Lecture Notes: Problems in Chapters 10-12.



    Demonstrated/recommended Exercises
    Week Excersises:Problem file (will be completed by exercises from the Lecture notes)
    4, study w2
    Problem file: Problems 53-60 Book: 3.5-3.7,
    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, 1.10, 7.16-7.19
    7, study w5
    Problem file: Problems 13-20 Book: 8.10-8.14, 8.19, 9.6-9.9
    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



    Computer labs





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

    Reference literature:

    Learning MATLAB, Tobin A. Driscoll ISBN: 978-0-898716-83-2 (The book is published by SIAM).

    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 9) . 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 February 26).

    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.
  • The two compulsory home assignments should be handed in before the final exam 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) Grades GU Points
    - <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 will do this by 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.

    Old exams



    Student Representativres:


    Please come to course coordinator's office: MVL:2035; Monday 19/2; 12-12:30.
    Name Programe E-mail
    ANDERAS DAHLBERG TKTEM andrdahl@student.chalmers.se
    SIMON LARSSON TKTEM simla@student.chalmers.se
    JIE CHIEN LIM Utbyte a.limjiechien@gmail.com
    THEMIS MOULIAKOS EJREG themis@student.chalmers.se
    ELINA OLSSON TKITE elinao@student.chalmers.se