General information

This is a first course on partial differential equations PDE 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 math and applied sciences and engineering. Students who are not following these programs are welcome to contact the instructor to get a scheme for an approproaite follow-up of the course.

Contents: Computation of approximate solutions to various PDE using the Finite Element Method. Interpolation, quadrature, and the solution of large, sparse linear systems. Stability and error estimates. Applications to problems of dynamical systems, heat conduction, wave propagation, convection-diffusion-reaction, etc.

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.

Schedule

Day Time Place Remarks
MON 8-10 Pascal Lecture
TUE 13-15 Euler Exercise
WED 15-17 Euler Lecture
THU 10-12 MVF31 Lecture/Exercise
  • Tuesday Feb 9 and Wednessday (afternoon) Feb 10 are CHARM DAYS. Therefore our excersie and Lecture on Feb 9 and Fen 10 are moved to:
    Mon Feb 8 Exercise 15-17 in Pascal,
    and
    Wed Feb 10 Lecture 10-12 in Euler.

    • Latest news
    Please, check the course diary for the latest news and info!

    Examiner and lecturer

    Mohammad Asadzadeh, mohammad@chalmers.se

    Questions about home assignments
    Fredik Lindgre :fredik.lindgren@chalmers.se

    Course literature

    1. M. Asadzadeh, (Lecture Notes, subject to minor changes)
    An Introduction to the Finite Element Method (FEM) for Differential Equations,     (pdf),
  • Please do not print out Part II yet. I might make some updatings.

    • 2. K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Computational Differential Equations, Studentlitteratur 1996.
  • Chapter and exercise numbers are referred to this book.


    • Reference literature
    1. M. Asadzadeh, Lecture Notes in Fourier Analysis, (pdf).
    2. S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Second edition, Springer 2002.
    3. C. Johnson, Numerical solutions of partial differential equations by the finite element method, reprinted by Dover, 2008
    4. M. Taylor, Partial Differential equations (basic theory), Springer 1996.
    5. W. Strauss, Partial Differential equations, An inroduction, 2008.

    Preliminary plan for lectures and classes

  • Chapter and excersice numers are referred to the book. We shall consider exercises from the Lecture Notes file as well.

    • Lectures

    Day
    Chapter
    Contents
    Mon 18/01 6 Classification of PDEs.
    Tue 19/01 6 Galerkin methods: global and piecewise polynomials.
    Wed 20/01 5 Polynomial and piecewise Interpolation.
    Mon 25/01 5 L_2 projection.
    Wed 27/01 7 Iterative methods.
    Thu 28/01 8 The finite element method.
    Mon 01/02 8 Error estimates in energy norm.
    Wed 03/02 9 Continuous and discontinuous Galerkin methods for scalar initial value problems.
    Thu 04/02 9 A posteriori error estimates for Galerkin methods for scalar initial value problems.
    Mon 08/02 9 A priori error estimates for Galerkin methods for scalar initial value problems.
    Wed 10/02 21 The abstract formulation.
    Thu 11/02 21 Poincare inequality, Lax-Milgram theorem.
    Mon 15/02 14 Variational formulation in R^2. Green's formula. Finite element basis in R^2.
    Wed 17/02 15 Stability and Finite elements for the Poisson's equation.
    Thu 18/02 15 Error estimates and adaptive error control in the energy norm for the Poisson's equation.
    Mon 22/02 16 Stability and piecewise linear Galerkin approximation for the heat equation.
    Wed 24/02 16 Error analysis of finite element methods for the heat equation.
    Thu 25/02 17 Conservation of energy. A finite element method for the wave equation.
    Mon 01/03 18 Finite element method for a convection-diffusion model problem.
    Wed 03/03 Exam 061218 Problems 1-4
    Thu 04/03 Old Exam Certain problems, answering questions, reserve

    Exercise coverage

  • I shall devote the exercise hours to a partial coverage of exercises below combined with some home-made, as well as old-exams, exercises. However, the uncovered material/exercises, that are listed in the plan, are equaly important. Work through them and talk to me if you have any questions/comments.

    • Recommended excercises
    Day
    		   Excersises
    Tu 26/01
    		 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.

    Tu 02/02 Chap 7: 7.3, 7.5, 7.24, 7.31(prove in addition that there is exactly one minimum), 7.54
    Tu 16/02 Chap 21: 21.1, 21.2, 21.3, 21.4, 21.5, 21.8, 21.13
    Thu 04/03 Chap 17: 17.8, 17.9, 17.10, 17.11, 17.13, 17.17, 17.33

    Demonstrated and/or recommended excercises

    Day
    		   Excersises
    Thu 21/01 Chapter 6: 6.1, 6.2, 6.3, 6.11
    Tu 26/01 Chapter 5: 5.12, 5.23, 5.27, 5.29
    Tu 02/02 Chapter 8: 8.1, 8.6, 8.7, 8.8, 8.11, 8.12,8.16, 8.18, 8.23
    Mon 08/02 Chapter 9: 9.9, 9.12, 9.13, 9.19, 9.43, 9.45, 9.46 (instead of Tu 09/02: CHARM DAY!)
    Tu 16/02 Chapter 14: 14.4, 14.7, 14.10, 14.21
    Tu 23/02 Chapter 15: 15.5, 15.13, 15.15, 15.20, 15.22, 15.27, 15.39, 15.44, 15.47
    Tu 02/03 Chapter 16: 15.5, 15.13, 15.15, 15.20, 15.22, 15.27, 15.39, 15.44, 15.47

    Compulsary Home Assignments, Computer labs and Matlab excercises

    Assignment 1: Do exercises 8.35, 18.6 a-c, and 9.12. Write a short yet detailed report, not exceeding ten pages, explaning your work and sumbit it by the end of study week 5 (Deadline: Friday February 19). Use MATLAB to do the coding parts. Hints: For 8.35 you need to read chapter 8, particularly, section 4, but not chapter 7. A good starting point for problem 18.6 might be the Matlab code, which solves -u''=f, u(0)=u(1)=0 using cG(1). In 9.12 consider only the case a=4.

    Assignment 2: Can be found here. Hand in a short report of your work before the final exam.

    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 each generating max 4 bonus points.

    • Written examination

  • Final exam is compulsory, written, and consists of 6 questions (5 problems + 1 theorm) with a maximum score of 42 (=6x7) points.
  • The theory question is choosen from the following list (see sample exam questions in the course diary.)
  • 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
    - <20 -
    3 20-29 G 20-35
    4 30-39 VG >35
    5 >39

    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

  • Old exams, with solutions are posted in the course diary.