MTH 656/659: Numerical Analysis

THE FINITE ELEMENT METHOD FOR PARTIAL DIFFERENTIAL EQUATIONS
Spring 2015

Professor: Dr. Vrushali Bokil
Office: Kidder 292
Phone: 737-2609
Email: bokilv@math.oregonstate.edu
Office Hours:
  • Monday: 10:00-11:00 am
  • Wednesday: 4:00-5:00pm
  • Friday: 1-1:50pm

  • Time/Classroom: MWF 9:00 - 9:50 am WNGR 116

  • Course Description

    This course is an introduction to the finite element method (FEM) for the numerical discretizations of partial differential equations (PDEs). Along with finite difference and finite volume methods, FEMs are one of the standard tools for computing accurate solutions to PDEs in various disciplines such as electromagnetics, solid and structural mechanics, fluid dynamics, acoustics and thermal conduction.

    The FEM is based on the variational formulation of differential equations in Hilbert spaces which gives the method greater flexibility and matehmatical elegance. In addition, its ability to deal with complex geometries via unstructured meshes makes it one of the most popular techniques in many branches of engineering. The FEM is widely used in all areas of engineering and science in which numerical methods for PDEs are required; it forms the basis of commerical software such as COMSOL and is actively being used to develop web based numerical tools.

    In this course, students will obtain rigorous training in the basic mathematical and computational aspects of the FEM. Beginning with an introduction to PDEs, we will cover the basics of the finite element method with particular emphasis on its application to the Poisson equation in one and two dimensions. Analysis of the FEM including stability, dispersion and error analysis will be an integral part of the course as well as a discussion of adaptivity. Time permitting, topics such as mixed FEM, FEM for time dependent problems, other versions of the FEM such as hp-FEM, DG-FEM may be introduced Programs written in MATLAB will be used to demonstrate examples and students will receive training in writing their own programs.

    This course is geared towards graduate students in mathematics, science and engineering including physics, chemistry, oceanography and all engineering disciplines. Prerequisites are basic real analysis and differential equations. Some knowledge of numerical methods (e.g., MTH 351), computer programming and partial differential equations (e.g., MTH 453/553) will be helpful.

  • Advanced Subjects Sequence MTH 654/5/6
    MTH 656 is the third course in the year long sequence MTH 654/5/6 on advanced subjects in numerical analysis. MTH 656 can be taken independently of the others in this sequence.

  • Learning Outcomes
    After successfully completing this class students will:
    • be able to construct the variational or weak formuation for a simple ODE in 1 spatial dimension, in the appropriate Hilbert space, derive fully discrete schemes using finite element approximations and assess their accuracy.
    • be able to derive and analyze standard finite element methods for model elliptic problems in 2 spatial dimensions.
    • Be able to write simple codes in MATLAB to simulate finite element methods in 1 spatial dimension for the PDEs considered, and understand basic computational aspects related to demonstrating accuracy, stability and convergence of the discrete methods.

  • Textbook
    There is no required textbook for this class. Readings will be assigned from a number of different sources and available on Blackboard. There are a number of excellent textbooks that are available and that will be used as references for this course. These include
    • The Finite Element Method: Theory, Implementation, and Applications by Mats G. Larson, Fredrik Bengzon
      Texts in Computational Science and Engineering, vol 10, 2013. Springer.
    • Finite Elements : Theory, Fast Solvers, and Applications in Solid Mechanics by Dietrich Braess
      Cambridge University Press; 2007
    • Understanding and Implementing the Finite Element Method by Mark. S. Gockenbach, SIAM, 2006
    • The mathematical theory of finite element methods, Susanne C. Brenner, L. Ridgway Scott, Springer, 2002.
    • The Finite Element Method for Elliptic Problems, P. G. Ciarlet, SIAM, 2002
    • Finite Element Methods for Maxwell's Equations, Peter Monk, Oxford, 2003

  • MATLAB
    A scientific programming language, MATLAB is preferred due to the integration of computation and visualization. The following are options for accessing MATLAB at OSU:
    • The Mathematics Department computer lab is located in the Math Learning Center, Kidder 108. In order to use the computers there, you will need an ONID account If you have none please visit http://onid.orst.edu and sign up as soon as possible.
    • The computer lab in the Milne basement. Again, you will need an ONID account. Both labs may be scheduled for classes at certain times, so it is a good idea to find out in advance when they are open.
    • You may have access to MATLAB through a computer lab or network of your department.
    • If you would like to have MATLAB at home, consider purchasing the MATLAB Student Edition.

    • The following are online resources for learning MATLAB:

  • Grading
    There will be 4 written assignments for this class each worth 25% of the grade. Assignments will be a mix of theoretical and computational exercises. Each assignment will be posted on blackboard along with all the associated code and reading. Please check the Calendar for the due dates for the assignments.
    The last (fourth) assignment will be due during finals week.

  • Grade Scale (by percentage): Final grades for this class will be given based on the scale below. Each letter grade below corresponds to grades scored between the lower limit (including) and less than the upper limit (excluding).
    A 90 - 100%
    A- 87 - 90%
    B+ 84 - 87%
    B 80 - 84%
    B- 77 - 80%
    C+ 74 - 77%
    C 70 - 74%
    C- 67 - 70%
    D+ 64 - 67%
    D 60 - 64%
    D- 57 - 60%
    F below 57%

  • Contacting Dr. Bokil: The best way to contact me is via email. Best place/time to see me for questions is in my office during office hours. If you are unable to make it to office hours you may email your questions to me or setup an appointment by email. You can expect a response within 24 hours. Do not expect an immediate response .

  • Student Conduct Policies and Cheating Policy: I have a zero tolerance policy for cheating. All suspected cheating will be reported to the appropriate office, usually the Dean of your College. Provable cases of cheating will also result in a score of 0 on the assignment in question, and may lead to a grade of F in the class. This is highly detrimental to any long term career goals. It also will make it difficult for the student to obtain letters of recommendation. Please consult the OSU Student Conduct and Community Standards page.

  • Special arrangements: For students with disabilities please contact Professor Bokil and Disability Access Services. For make-up exams for official or emergency reasons and other special arrangements, please contact Professor Bokil. For all types of special arrangements appropriate documentation will be required.
  • Course Drop/Add Information: See Office of the Registrar and Academic Calendars