Dr. Vrushali Bokil
Office: Kidder 048
Email: bokilv at math dot oregonstate dot edu
Office Hours: M: 3:00pm-3:50pm, W: 10am-10:50am, or by Appt.
- Time/Classroom: MWF 9:00 - 9:50 am
- MTH 655: 33398/159/3
- MTH 659: 33782/155/3
- Course Description
This course is an introduction to the
finite element method (FEM) for the numerical discretization of partial differential
equations (PDEs). Along with finite difference and finite volume
methods, FEMs are one of the standard tools for computing numerical solutions to PDE based models that arise in a wide variety of scientific and engineering applications including 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 mathematical 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 forms the basis of commerical software such as COMSOL and is actively being used to develop web based
In this course, students will obtain rigorous training in the basic mathematical
and computational aspects of the standard continuous Galerkin FEM. We will cover the basics of the
finite element method with particular emphasis on its application to linear elliptic equations and time harmonic Maxwell equations in one and two spatial dimensions. For the time harmonic Maxwell system, we will consider the construction of the famous Nedelec edge finite element methods in two dimensions.
Error analysis of the FEM will be an integral part of the course as well as a discussion of adaptivity. Time permitting, we will also explore mixed FEM and other versions of the FEM such as hp-FEM, Discontinuous Galerkin (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. A discussion of available numerical software will also be included.
This course is geared towards graduate students in mathematics, science and engineering
including physics, chemistry, oceanography and all
engineering disciplines. Prerequisites include familiarity with numerical methods and graduate standing, or instructor's consent.
In addition, familiarity with basic real analysis, differential equations, linear algebra, prior computer programming and partial
differential equations (e.g., MTH 453/553) is recommended.
- Advanced Subjects Sequence MTH 654/5/6
MTH 655 is the second course in the year long sequence MTH 654/5/6 on
advanced subjects in numerical analysis. MTH 655 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 linear 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 linear elliptic and time harmonic 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.
There is no required textbook for this class.
There are a number of excellent textbooks that are available in the Valley library. 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
- Computing with hp-ADAPTIVE Finite Elements Method, volume I: One and Two Dimensional Elliptic and Maxwell Problems, Leszek Demcowicz.
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
- If you would like to have MATLAB at home, consider purchasing the MATLAB Student Edition.
- The following are online resources for learning MATLAB:
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 Canvas along with all the associated code and reading.
Please check the Canvas for assignments and their due dates. 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
| 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.
Special Arrangements and Course Drop/Add Information
- Disability Access:
Accommodations for students with disabilities are determined and approved by Disability Access Services (DAS). If you, as a student, believe you are eligible for accommodations but have not obtained approval please contact DAS immediately at 541-737-4098 or at Disability Access Services.
DAS notifies students and faculty members of approved academic accommodations and coordinates implementation of those accommodations. While not required, students and faculty members are encouraged to discuss details of the implementation of individual accommodations.
- Makeup Policy: For make-up exams for official or emergency
reasons and other special arrangements, please read the University's official policy
Student Petitions to Change the Time of an Exam Please contact Professor Bokil to make arrangements. For
all types of special arrangements appropriate documentation will be
- Course Drop/Add Information: See Office of the Registrar