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, elastodynamics, thermal conduction among others.
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 science and engineering. 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 in one and two spatial dimensions. Error analysis of the FEM will be an integral part of the course. Programs written in MATLAB or Python will be used to demonstrate examples and students will receive training in writing their own programs. A discussion of available FEM software will also be included, and students will be asked to read papers that describe the implementation of some software packages as well as implement model problems using some of these packages. Advanced topics including adaptivity, high order hp and p FEM, time dependent problems and nonlinear problems will be included as time permits. This course is geared towards graduate students in mathematics, science and engineering disciplines. Familiarity with basic real analysis, differential equations, linear algebra, prior computer programming and partial differential equations (e.g., MTH 453/553) is recommended for this topic.
Advanced Subjects Sequence MTH 654/5/6: MTH 656 is the third course in the yearlong sequence MTH 654/5/6 on advanced subjects in numerical analysis. MTH 656 can be taken independently of the other courses in this sequence. This sequence includes advanced topics in numerical analysis, such as finite volume methods and finite element methods for partial differential equations, numerical methods for inverse problems, and image processing. All courses used to satisfy MTH prerequisites must be completed with a C grade or better. This course is repeatable for 12 credits.MTH 656/659 is an advanced topics course in numerical analysis.
Textbooks:
There is no required textbook for this class. There are a number of excellent textbooks which I will refer to over the course and which students will be asked to read chapters of. All of the following books are available online at the Valley Library with valid ONID access. Specific chapters of these books will be assigned for reading and assignments will be given in the Canvas site for this class.
- (Primary, Reference) The Finite Element Method: Theory, Implementation, and Applications [LB]
Part of the Book Series: Texts in Computational Science and Engineering (TCSE, volume 10), Springer, 2013.
Author: Larson, Mats G; Bengzon, Fredrik
Note: The textbook is available with online access from Valley library for students with a valid ONID account. Here is the permalink for the ebook.
- (Primary, Reference) The Mathematical Theory of Finite Element Methods [BS]
Part of the Book Series: Texts in Applied Mathematics, Vol. 15, Springer, 2008.
Authors: Susanne C. Brenner , L. Ridgway Scott
Note: The textbook is available as an ebook from Valley library for students with a valid ONID account. Here is the permalink for the ebook.
- (Primary, Reference) Theoretical Numerical Analysis: A Functional Analysis Framework [AH]
Part of the Book Series: Texts in Applied Mathematics, TAM Vol. 39, Springer, 2009.
Authors: Weimin Han , Kendall E. Atkinson
Note: The textbook is available as an ebook from Valley library for students with a valid ONID account. Here is the permalink for the ebook.
- (Secondary, Reference) Introduction to Partial Differential Equations A Computational Approach [TW]
Part of the Book Series: Texts in Applied Mathematics, Vol. 29, Springer, 2005.
Authors: Aslak Tveito & Ragnar Winther
Note: The textbook is available as an ebook from Valley library for students with a valid ONID account. Here is the permalink for the ebook.
Reading Assignments:
Reading material will be assigned on CANVAS.
NOTE: Students are expected to read assigned material by deadlines assigned on CANVAS. Questions not addressed
during class time should be asked in office hours. Students are responsible
for any material missed due to absence.
Course Format:
The course format will be lectures combined with collaborative work in class. Students will work in groups during parts of the class.
Learning Outcomes for Math 656/659: A successful student completing Math 656/659 will be able to
- be able to explain what the FEM is, the fundamental differences between the FEM and other numerical methods such as the finite difference method and the finite volume method and also explain the similarities between these methods,
- be able to construct the variational or weak formulation for a linear elliptic PDE in the appropriate Hilbert spaces,
- be able to derive fully discrete schemes using finite element approximations for linear elliptic PDEs in one and two spatial dimensions and assess their accuracy, stability and convergence,
- apply the concepts of stability, consistency and convergence for finite element methods.
- be able to write simple codes in MATLAB or Python to simulate finite element methods in one spatial dimension for simple linear elliptic PDEs, and understand basic computational aspects related to demonstrating accuracy, stability and convergence of the discrete methods,
- be able to use available FEM software in MATLAB or Python to simulate the solution to two- or three-dimensional boundary value or initial boundary value problems based on PDEs.
Course Grading
An important part of the course is computational implementation of finite difference and finite element methods. Thus, the assignments for this course will be partly theoretical and partly computational. Course grading will be based on homework assignments and a final project report along with an in-class presentation. The intended audience for this course is graduate students of mathematics and other science and engineering disciplines.
- Written Assignments (66%): There will be approximately 3 written assignments this term. You will be given at least one week to complete each written assignment. Each student should individually write solutions to the assignment neatly by hand or type these out (see Latex resources in Canvas) and upload to Canvas by the due date and time. Late assignments will not be accepted and will not receive any credit unless prior authorization from Dr. Bokil has been obtained (see policies below).
Every student is expected to complete the writing of each assignment individually. Please see student ethical conduct and policies below.
Students should get their questions asked during office hours or via email. Students are encouraged to get their questions answered in a timely manner.
- Project Report and Presentation (34%): Every student will be assigned a specific project related to the topic of this class. The project can be a survey of a specific topic and must involve both theoretical and computational aspects. Peer reviewed journal papers and available software in MATLAB or Python will be part of this final assignment. A project report of maximum 10 pages is required. In addition, every student must give a presentation of their project on the final exam day for this class, which is Monday, June 9, 2025 at 6:00 pm .
This date and time is set by
OSU. Scheduling conflicts with the final exam must be resolved in advance.
An unexcused absence (i.e., one that is not reported to Dr. Bokil with proper documentation) on the Final Exam will result in a grade of F in the course.
Course Grade Scale: The Grade for the course is determined by the grade on class participation, the written homework assignments, the midterm and final. 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%
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| D | 60 - 64%
|
| D- | 57 - 60%
|
| F | below 57%
|
Also see the OSU online catalog, refer to AR 18 and AR 19 regarding assignment of grades.
Important Policies, Regulations and Information:
Students should check this class website and the CANVAS classpage for general information, announcements, assignments, due dates, exam dates and other policies regarding the class.
Makeup policy:
- Makeup Policy for Assignments and Project: Assignments and project reports will have due dates. There will be no makeups given in general. If you are late by one day you can earn up to 50% of the grade. Beyond that you will not earn any credit unless there is an exceptional reason such as illness or an emergency. Exceptions can be made by reporting issues to Dr. Bokil. Proper documentation will be needed in order to grant a makeup. Dr. Bokil reserves the right to make all decisions in this regard.
- Makeup Policy for Project Presentation: There will be no makeup given without prior request, approval and arrangement with Dr. Bokil. Scheduling conflicts with the project presentation day must be resolved in advance with Dr. Bokil. If you have an emergency, illness and/or hospitalization or are unable to take the project presentation day due to work related reasons, please email Dr. Bokil preferably in advance of the exam or as soon as possible, given the circumstances of the situation. Proper documentation will be needed in order to grant a makeup. Dr. Bokil reserves the right to make all decisions in this regard based on the university guidelines Student Petitions (AR 16: Finals Week) to Change the Time of a Final Examination.
Technology Policy, Unauthorized Recording and Class Materials Use Policy:
I expect that each student will be present and engaged during class
time, unless there are circumstances beyond the student's control.
Recording and/or dissemination of instructional content (for example, using the camera on your cell phones) is prohibited, unless express permission of the instructor is obtained for an approved accommodation.
Acceptable and Unacceptable Use of Artificial Intelligence:
Please see the College of Science Artificial Intelligence Guidelines. For this course, students are not permitted to use AI applications such as ChatGPT, Bard, or Bing for any purpose.
Contacting Dr. Bokil:
The best way to contact me is via direct email or message through CANVAS. Best place/time for questions is during office hours. If you are unable to make it to office hours you may email your questions to me or set up an appointment. You can expect a response within 24 hours. Do not expect an immediate response. I may not be able to answer any homework questions sent via email after 5pm on workdays and between 5pm on Friday and 8am on the subsequent Monday. Please keep this in mind and start working on assignments in a timely manner.
Getting Help:
Short questions can be asked during class. Longer questions should be asked during regular office hours. Appointments can also be made at other times, and you can reach me by email.
Academic Calendar: All students are subject to the registration and refund deadlines as stated in the Academic Calendar: https://registrar.oregonstate.edu/osu-academic-calendar
Office of Student Life: The Office of Student Life is dedicated to providing educational and developmental opportunities for students to make informed decisions that support their success at Oregon State University. In particular, the following sites will be helpful for you:
- Student Ethical Conduct Expectations and Policies:
Students are expected to be familiar with Oregon State University’s Statement of Expectations
for Student Conduct. Please review and consult material at the following sites.
- Statement Regarding Students with Disabilities: 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 (https://ds.oregonstate.edu/) . 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.
- Student Bill of Rights:
OSU has twelve established student rights. They include due process in all university disciplinary processes, an equal opportunity to learn, and grading in accordance with the course syllabus: https://asosu.oregonstate.edu/advocacy/rights
- Student Care: Many students face difficult times during college; unexpected life events, personal crises, illness or injury, hospitalization, loss of a loved one, and mental health challenges, are examples of challenges that can interfere with college success.
The Covid-19 pandemic has caused additional difficulties for many students. Here is an important resource for those of you needing help.
Student Care is designed to help students: Learn and understand Oregon State University policies and procedures, Problem-solve within the university and community, Access referrals for personal matters/emergencies.
Reach Out for Success: University students encounter setbacks from time to time. If you encounter difficulties and need assistance, it’s important to reach out. Consider discussing the situation with an instructor or academic advisor. Learn about resources that
assist with wellness and academic success at oregonstate.edu/Reachout. If you are in immediate crisis, please contact the Crisis Text Line by texting OREGON to 741-741 or call the National Suicide Prevention Lifeline at 1-800-273-TALK (8255).
Student Evaluation of Courses: The online Student Evaluation of Teaching system opens to students the Monday of dead week and closes the Monday following the end of finals. Students will receive notification, instructions and the link through their ONID. They may also log into the system via Online Services. Course evaluation results are extremely important and used to help improve courses and the learning experience of future students. Responses are anonymous (unless a student chooses to “sign” their comments agreeing to relinquish anonymity) and unavailable to instructors until after grades have been posted. The results of scaled questions and signed comments go to both the instructor and their unit head/supervisor. Anonymous (unsigned) comments go to the instructor only.
Course Drop/Add and Other Informational Sites: See
- Office of the Registrar
- OSU's Grading System
- Academic Regulations
- Academic Success Center: Remote & Online Learning
- Learn@OregonState-Keep Learning
- Covid-19 news, resources and updates from OSU Leadership
- Mathematics & Statistics Learning Center