MAT-51327 Numerical methods for partial differential equations (spring 2009)

Partial differential equations (PDE) are extremely important as they model a variety of phenomena in physics and technology.
Examples include heat transfer, electromagnetic radiation, water flow, and propagation of sound waves.
The three basic types of PDE are elliptic, parabolic and hyperbolic equations.

The goal of the course is that the students learn

  • basic principles of Finite Element Method and Finite Difference Method,
  • how variational formulation is used in both theoretical and practical solution of partial differential equations, and
  • to program solution methods from scratch using Matlab.

    Lectures

    Lecturer: professor Samuli Siltanen. Office hour: Tuesdays 15-16 in room TD321.

    Lectures on period 4:
    Feb 5, 2009, 9:15-12:00 in hall TC131. Introduction to weak derivatives and Sobolev spaces.


    Feb 12, 2009, 9:15-12:00 in hall TC131. Solution of elliptic equations using the variational principle. Finite element method.

    Note: There is no lecture on week 8 (16.2.-20.2.2009).

    After the two initial lectures by professor Siltanen, the students will explain parts of the textbook as a presentation (60 minutes long).
    The lectures will be graded, and that grade represents 40% of the final grade of the course.

    Here are some instructions for the presentations.
    The time for the presentation is 60 minutes including 5 minutes for questions. Try not to speak overtime as that will lower your grade. The talk should contain

  • Introduction to the type of PDE discussed (about 5 minutes),
  • Explanation of the theoretical solvability of the PDE (about 15 minutes),
  • Explanation of the numerical solution method (about 30 minutes),
  • Conclusion (about 5 minutes).
    The times above are suggestions only.
    The structure of the textbook is such that for each PDE type (elliptic, parabolic, hyperbolic) there is a theoretical section
    as well as a section for Finite Difference method and for Finite Element Method. It is advisable to follow these in your presentation.
    You can use the equipment available in the classroom (computer, projector, and whiteboard).


    Feb 26, 2009, 9:15-12:00 in hall TC131.
    Presentation by Bzdawka: cancelled.
    Presentation by Lamberg: Finite difference method for parabolic equations. Grade 4/5.


    March 5, 2009, 9:15-12:00 in hall TC131.
    Presentation by Peltola: Finite difference method for elliptic equations. Grade 4/5.
    Presentation by Basile: cancelled.


    March 12, 2009, 9:15-12:00 in hall TC131.
    Presentation by Pellikka: Finite element method for parabolic equations. Grade 4/5.
    Presentation by Müller: Finite element method for hyperbolic equations. Grade 5/5.

    Lectures on period 5:


    April 16, 2009, 9:15-12:00 in hall TC131.
    Presentation by Rautakorpi: Finite element method for elliptic equations.

    Exercises

    Here is Exercise 1 in pdf format.
    The exercise will be on Friday, February 13, at 12-14 in room SJ202.

    Here is Exercise 2.
    The exercise will be on Friday, February 20, at 12-14 in room SJ202.

    Project work

    Three project works have been completed with the following grades:
    Lamberg, Antti 4/5,
    MŸller, Juliane 5/5,
    Peltola, Juho 2/5.

    Period 5 is devoted to preparing a project work. If you have questions or problems during the preparation of the project work,
    please contact the course assistant Juho Linna by email and schedule a meeting with him if necessary.

    The idea of the project work is to implement and test one of the numerical solution methods discussed in the course.
    You can choose either finite difference or finite element method, and the equation type and dimension is one of the following:
    (a) Elliptic equation in dimension 2,
    (b) Parabolic equation in dimension 2+time,
    (c) Parabolic equation in dimension 1+time,
    (d) Hyperbolic equation in dimension 2+time,
    (e) Hyperbolic equation in dimension 1+time.

    Choose the simplest possible constant coefficient equation and the simplest possible domain (such as rectangle).
    Also, it is important to choose such a problem where analytic solution is also available for comparison with the numerical results.

    First deadline April 8, 2009: return written plan electronically by email to Juho Linna and Samuli Siltanen.
    No numerical work is needed at this point; you should just explain what will be computed (choice of equation, method, and domain).
    Please include the matrix form of the final computation.

    Final deadline May 8, 2009: Return the final report introduction, method, and numerical results.
    It is important to include a plot of the analytic solution and of the numerically computed solution.
    Also, a table is needed where it is shown how refining the computational discretization leads to smaller error in the computation.
    The report (in pdf format) and the Matlab codes are returned to the assistant for grading.

    Course material

    The course follows the book Partial Differential Equations with Numerical Methods by Stig Larsson and Vidar Thomée.
    Page updated May 28, 2009.