AMS 529/691: Finite Element Methods: Fundamentals, Applications, and New Trends
Fall 2017
Time: Monday & Wednesday 7:00--8:20pm
Location: Frey Hall 222

Lecture Schedule and Slides

Instructor: Prof. Xiangmin (Jim) Jiao
Email:  Phone: 631-632-2339
Office hours: Mon. 1:30-3:30pm & Wed. 2:30-3:30pm
Office: Math Tower P-137

[ Course Description | Course Outline | Grading Policy | Homework | References | University Policy ]

Course Description (back to top)

This is an intermediate-level graduate course on the finite element methods (FEM) for solving partial differential equations. It will introduce the mathematical formulation, numerical analysis, and computational aspects of FEM, applications in solid mechanics fluid mechanics, and multiphysics phenomena, as well as the recent trends in improving their stability, accuracy, efficiency, and generality. Computing projects will involve programming in Python and MATLAB/Octave, as well as using software FEniCS and ANSYS for understanding the typical workflow of finite element analysis for solving real-world problems.

Learning Objectives:

  • Master the fundamentals of the finite element methods
  • Understand numerical analysis of the finite element methods
  • Familiarize with the workflow of FEA for real-world problems, including pre- and post-processing tools
  • Understand the computational aspects in efficient implementation of the finite element methods
  • Get acquainted with some new trends in FEM and related research areas


  • AMS 510 (linear algebra and multivariable calculus) and AMS 527 (numerical analysis) or pre-approval by the instructor. In particular, it is a good idea to familiarize yourself with the following concepts: Gaussian elimination, eigenvalue problems; vector calculus, Green's theorem; initial value and boundary value problems; conditioning and stability, interpolation and approximation, numerical quadrature, finite difference methods)
  • Programing experience in MATLAB or Python

Computing Resources

  • A laptop (64-bit Windows, Mac, or Linux) is required

Course Outline (back to top)


  • Fundamentals: boundary value problems and finite elements; Galerkin methods; discrete minimum principle; introduction to FEniCS; accuracy and stability
  • Applications of FEA: linear elasticity; initial value problems and heat conduction; FEA for structural dynamics; FEA for fluid mechanics and fluid dynamics
  • Computational issues: control flow and element library of FEM; mesh generation and adaptation; variational crimes for curved boundaries; linear solvers and multigrid methods; domain decomposition and parallel computing
  • Select topics: multiphysics problems; discontinuous Galerkin method; other extensions of FEM (if time permits)
Lecture Schedule and Slides

Grading Policy (back to top)

The course involves five (written/computing) homework assignments, an individual project, and a team project (23 members per team). You can use a full-length presentation to substitute the individual project.
  • Home assignments: 30%
  • Individual project (or full-length presentation): 20%
  • Team project: 30% (code and report) + 10% (presentation)
  • Class participation (questions and discussions): 10%

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.

  • Homework #1 (due Sept. 18th. You can also download the lyx file and type your answers in LyX.)
  • Homework #2 (due Oct. 4th. You can also download the lyx file and type your answers in LyX.)
  • Homework #3 (due Oct. 18th)
  • Homework #4 (due Nov. 13th)
  • Project
    • proposal: due Nov. 20th
    • presentations: Dec. 4--6th
    • report: Dec. 19th

References and Useful Links (back to top)

Overviews on Finite Element Methods

Finite Element Methods (Mathematics Oriented)

Finite Element Analysis (Mechanics Oriented)

Finite Element Implementations (Software Oriented)

Policies and Academic Integrity (back to top)

Americans with Disabilities Act

If you have a physical, psychological, medical or learning disability that may impact your course work, please contact Disability Support Services, ECC (Educational Communications Center) Building, room128, (631) 632-6748. They will determine with you what accommodations, if any, are necessary and appropriate. All information and documentation is confidential.

Academic Integrity

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

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 Judicial Affairs any disruptive behavior that interrupts their ability to teach, compromises the safety of the learning environment, or inhibits students' ability to learn.