General information

Welcome to the course tma372/MMG800.

Except this introduction all material should be consided as preliminary and subject to some minor alternation.

This is a first course on partial differential equations PDEs intended for students following math and computation oriented studies in master programs at Chalmers and the University of Gothenburg, the International Mathematics Master Program, students in "Teknisk Matematik"(=TM), E3 students at Chalmers, as well as PhD students in computational/applied math and applied sciences and engineering. Students from other disciplines who are not following these programs are welcome to register for the course and encouraged to contact the instructor to get an approproaite follow-up scheme, if necessary.

Contents: The course covers topics as: Approximating solutions to various PDEs (ODEs) using the Finite Element Method, Polynomial Interpolation, Quadrature rules, and the solution of large, sparse linear system of equations. Stability and convergence analysis, error estimates in a priori and a posteriori settings. Reisz representation and Lax-Milgram theorems, Application of finite element methods to problems of dynamical systems, heat conduction, wave propagation, convection-diffusion-reaction, etc.

Compulsary home assignments contain both analytic approaches as well as coding aspects, ranging from iterative algorithms to problems involving complex multiphysics programing.

Following the course, actively, you should gain some modeling skills relevant to the PDE of your own field of interest, knowledge on weak/variational formulation, and a great deal of 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. The course code is for engineering schools (students registered at Chalmers): tma372, and for students registered in GU: MMG800. See also course description:

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
TU 8-10 Pascal Lecture Mohammad:13-15, L2035; questions
THU 10-12 Pascal Exercise Kristin:13-15, L2105; assigmnents
FRI 8-10 Pascal Lecture/Exercise

Teachers
Course coordinator:
Mohammad Asadzadeh, mohammad@chalmers.se

Teaching assistant:
Kristin Kirchner, kristin.kirchner@chalmers.se

lab Supervisor:
Kristin Kirchner, kristin.kirchner@chalmers.se

Course litterature

M. Asadzadeh,
An Introduction to the Finite Element Method (FEM) for Differential Equations. (Lecture Notes) (pdf),
  • The first 210 pages are updated as posted in course diary (version5). I might make some updatings of the rest. Otherwise, current file would serve as our lecture notes.





  • 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.



  • Programme


    Lectures (preliminary plan)
    Week Chapter
    Contents
    4, Study w1
    0-3
    Classification of PDEs, Polynomial approx, Galerkin Method
    5, study W2
    3-5
    Piecewise Polynomial Interpolation, L_2 projection, The finite element method
    6, study W3
    6,7
    Error estimates in energy norm, Continuous and discontinuous Galerkin methods for scalar initial value problems, A posteriori error estimates for Galerkin methods for scalar initial value problems
    7, study W4
    7,8
    A priori error estimates for Galerkin methods for scalar initial value problems,Variational formulation in R^2. Green's formula. Finite element basis in R^2
    8, study W5
    9-10
    Abstract formulation,Poincare inequality, Lax-Milgram theorem, Stability and Finite elements for the Poisson's equation.
    9, study W6
    10-12
    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.
    13, study W7
    12-13
    Conservation of energy. A finite element method for the wave equation. Finite element method for a convection-diffusion model problem. Selected problems for exams, 2010-2013





    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.
    Lecture Notes: 2.6, 2.8, 2.9
    6, study w3
    Chapter 4: Read through iterative methods of chapter 4(self study not included in the exam). Lecture Notes: 5.7, 5.9, 5.12, 5.16
    7, study w4
    Chapter 9: Problems 9.3-9.7
    8, study w6
    Chap 11-12: Problems in Chapters 11-12.



    Demonstrated/recommended Exercises
    Week Excersises:Problem file (will be completed by exercises from the Lecture notes)
    4, study w1
    Problem file: Problems 1-5 Lecture Notes: 2.5, 2.7.
    5, study w2
    Problem file: Problems 6-12, Lecture Notes: 3.10, 3.15, 3.16
    6, study w3
    Problem file: Problems 13-20 Lecture Notes: 5.1, 5.4, 5.6, 5.8, 5.10, 5.18
    7, study w4
    Problem file: Problems 21-23, 26-27 Lecture Notes: 6.5, 6.5, 6.9, 6.10, 6.19
    8, study w5
    Problem file: Problems 34-40 Lecture Notes: 7.4, 7.6, 7.19-7.22, 9.5, 9.7
    9, study w6
    Problem file: chosen problems from the list: 43-52
    10, study w7
    Problem file: chosen problems from the list: 53-60, demonstrate problems from an old exam





    Computer labs


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

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

    A good knowledge of linear algbra, calculus of several variables 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 14) . Use MATLAB to do the coding parts. Hints: For problem 1 you need to read chapter 5, but not chapter 4. In 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. You may use, e.g. COMSOL Multiphysics (or other computational code/environment) instead of PDE TOLBOX. Hand in report of your work beginning of study week 7 (Deadline: Monday March 3).

    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 5 (2+3) bonus points.
  • Written examination

  • Final exam is compulsory, written, and consists of 5 questions (4 problems + 1 theorm) with a maximum score of 30 (=6x5) 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 on web-site, pp 229-231.
  • 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 Grades GU Points
    - <15 -
    3 15-20 G 15-27
    4 21-27 VG >28
    5 >28

    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.
    At the link Scedule you can find when exams are given for courses at University of Gothenburg.
    At the exam, you should be able to show valid identification.
    Before the exam, it is important that you report that you want to take the examination. If you study at Chalmers, you will do this by the Chalmers Student Portal, and if you study at University of Gothenburg, so sign up via GU's Student Portal.

    You can see your results in Ladok by logging on to the Student portal.

    At the annual examination:
    When it is practical a separate review is arranged. The date of the review will be announced here on the course website. Anyone who can not participate in the review may thereafter retrieve and review their exam on Mathematical sciences study expedition, Monday through Friday, from 9:00 to 13:00. 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 picked up at the Mathematical sciences study expedition, Monday through Friday, from 9:00 to 13:00. Any complaints about the marking must be submitted in writing at the office, where there is a form to fill out.
    Old exams
    Old exams are posted in the course diary.