AMS 691: Finite Element Methods: Theory and Computations
Fall 2020
Time: Monday & Wednesday 10:30--11:50am ET
Location: Join via Zoom

Lecture Schedule and Slides

Instructor: Prof. Xiangmin (Jim) Jiao
Phone: 631-632-2339
Office hours: Mon. 7:00pm--8:30pm ET &
                      Wed. 8:30am--10:00am ET
Office: Join via Zoom

[ Description | Outline | Delivery Mode | Tech Requirements | Grading Policy | Homework | References | University Policy ]

Course Description (back to top)

This course is on the finite element methods (FEM) for solving elliptic and time-dependent partial differential equations (PDEs). It covers the mathematical formulation, numerical analysis, and efficient computations of FEM, with applications to continuum mechanics. It also surveys some advanced, research-oriented topics, including shock capturing, high-order and super-convergence, mesh adaptation, multigrid methods, domain decomposition, and hybrid methods. Computing projects will involve programming in Python and MATLAB.

Learning Objectives:

  • Master the fundamentals of the finite element methods
  • Master the mathematical analysis of the finite elements for coercive and non-coercive PDEs
  • Understand the workflow of FEM, including its implementation and pre- and post-processing
  • Understand the computational aspects in linear solvers and domain decomposition
  • Get acquainted with recent developments in FEM, including high-order and hybrid methods


  • AMS 510 & AMS 527 or pre-approval by the instructor. In particular, it is a good idea to familiarize yourself with the following concepts: Gaussian elimination; vector calculus; Green's theorem; initial value and boundary value problems; conditioning and stability; interpolation and approximation; finite difference methods; implicit Runge-Kutta methods; Gaussian quadrature
  • AMS 595 or programming experience in Python

Course Outline (back to top)

The class is composed of six parts:

  1. Fundamentals: finite-element interpolation; L2 projection; inf-sup conditions; Galerkin methods; error analysis; connection with finite differences
  2. Coercive and saddle-point problems: elliptic PDEs; continuum mechanics; Stokes equations; mixed finite elements of biharmonic and Poisson equations; introduction to FEniCS
  3. Noncoercive and time-dependent problems: Helmholtz and Maxwell equations; advection-diffusion equations; least squares formulations; parabolic equations
  4. Implementation and variational crimes: mesh data structure; quadrature; sparse matrix and assembling; curved boundaries
  5. Efficient computation and linear solvers: effective condition numbers; multigrid methods; preconditioned iterative methods; domain decomposition
  6. Advanced topics: a posteriori error estimation; mesh adaptation; discontinuous Galerkin; hybrid methods
Lecture Schedule and Slides

Course Delivery Mode and Structure (back to top)

This course primarily uses Zoom ( for lectures and office hours and uses Microsoft Teams ( to facilitate communications between faculty and students, submission of assignments, and posting grades. All links are available through the Blackboard learning management system. All assignments and course interactions will utilize the Internet. See the “Technical Requirements” section for more information. In Teams, you will access online lessons, course materials, and resources.

Course-related questions should be posted in the General channel in Teams. For personal/private issues, my preferred method of contact is private chat messages on Teams or email listed at the top of this syllabus. I strive to respond to your emails as soon as possible, but please allow between 24-48 hours for a reply. All email communication will be sent to your Stony Brook University email account. You must have an active Stony Brook University e-mail account and access to the Internet. Please plan on checking Teams regularly and your SBU email account for course related messages or set up your SBU email account to forward to your personal email account. To log in to Stony Brook Google Mail, go to and sign in with your NetID and password.

Important announcements will be sent on Blackboard and Teams. These will be posted in the class and may or may not be sent by email I will participate and post regularly in the discussion board and provide feedback on assignments within a week. Regular communication is essential in online classes. When we are participating in class discussions the expectation is that you will respond regularly to your peers and questions posed to your responses. Logging in regularly, checking the discussion board and participating with your colleagues ensures that you are able to remain an active member of the class.

Technical Requirements (back to top)

You are responsible for having a computer and a reliable Internet connection throughout the term. The following lists detail a minimum recommended computer set-up and the software packages you will need to access and use:
  • Hardware:
    • Intel Core i5 processor (or higher).
    • 250 GB hard drive and 8 GB RAM.
    • A reliable high-speed internet connection. Note that public WiFi and internet-service-provider hotspots are not recommended, especially during lectures or exams.
    • Microphone (internal or external); speaker (internal or external) or headphone. A headphone is strongly recommended to reduce the risk of feedback during communications.
    • Webcam for the computer; a second webcam-equipped device (e.g., a smartphone, tablet, or Chromebook) for remote proctoring of exams
    • Optionally a scanner (can be substituted fully by some free software; see below), and optionally a printer (for printing the exams).
  • Software (you may need administrator access to install them on your computer):
All the software with a link above is supported by the University and is available to Stony Brook students at no additional charge. Please make sure that you must sign up for the software using your Stony Brook account, instead of your personal account. Your login password for Teams and Zoom is the same as your NetID password for logging into Blackboard. If you are unsure of your NetID password, visit for more information.
If you need any technical assistance for the University-licensed and support software,
For other questions on computer hardware, software, or Internet connection, post them on Teams to get help from the instructor, the TA, or your peer students.

Grading Policy (back to top)

The course involves five (written & computing) homework assignments, one midterm exam after the first three parts, and a final project with presentation.
  • Home assignments: 25%
  • Midterm exam: 30%
  • Final project: 40% (25% on code and/or report, and 15% on presentation)
  • Class participation: 5% (questions and discussions)
Assignment Policy

Homework assignments are due typically two weeks after they are assigned. You are allowed to discuss course materials and homework problems in small groups, but limited to discussion of general ideas only. You must write your solutions completely independently. Under no circumstances may you copy solutions from any source, including but not limited to other student's solutions, official solutions distributed in past terms, and solutions from courses taught at other universities. Violation of these rules may result in disciplinary actions.


The midterm exam will be closed-book, but you are allowed to bring a single-sided, one-page, letter-size cheat sheet, which you must prepare by yourself. The exams will be proctored using Zoom/Teams. You are also expected to have access to a private and quiet room during the exam. If your room is too noisy, earplugs or soundless hearing-protection headset can be used during the exam.


Each student will have a mini-research project on one of the more advanced topics. The project can be individual or team-based, and it can be theoretical, computational, or application-oriented. The project will be composed of literature review, theoretical analysis or algorithm development, numerical verification or results, a report, and a presentation. A list of potential topics will be posted on Teams.


All students are expected to attend the lectures and the exam synchronously.

Homework (back to top)


You are allowed to discuss the course materials and homework problems in small groups, but the discussions should be limited to the  general ideas only. You must write your solutions completely independently. Under no circumstances may you copy solutions from any source, including but not limited to other students solutions, official solutions distributed in past terms, and solutions from courses taught at other universities. Violation of these rules may result in disciplinary actions.

References and Other Resources (back to top)

Brief Overviews as Book Chapters

Classic Textbooks (Theory Oriented)

Application-Oriented Books

Software-Oriented Books

University Policies and Academic Integrity (back to top)

Student Accessibility Support Center Statement

If you have a physical, psychological, medical or learning disability that may impact your course work, please contact Student Accessibility Support Center, ECC (Educational Communications Center) Building, Room 128, (631) 632-6748. They will determine with you what accommodations, if any, are necessary and appropriate. All information and documentation is confidential. Students who require assistance during emergency evacuation are encouraged to discuss their needs with their professors and Student Accessibility Support Center. For procedures and information go to the following website:

Academic Integrity Statement

Each student must pursue his or her academic goals honestly and be personally accountable for all submitted work. Representing another person's work as your own is always wrong. Faculty are required to report any suspected instances of academic dishonesty to the Academic Judiciary. For more comprehensive information on academic integrity, including categories of academic dishonesty please refer to the academic judiciary website at
Important Note: Any form of academic dishonesty, including cheating and plagiarism, will be reported to the Academic Judiciary.

Critical Incident Management

Stony Brook University expects students to respect the rights, privileges, and property of other people. Faculty are required to report to the Office of University Community Standards any disruptive behavior that interrupts their ability to teach, compromises the safety of the learning environment, or inhibits students' ability to learn. Faculty in the HSC Schools and the School of Medicine are required to follow their school-specific procedures. Further information about most academic matters can be found in the Undergraduate Bulletin, the Undergraduate Class Schedule, and the Faculty-Employee Handbook.