## MATH 679-002

Fall 2012

Analysis of the Finite Element Method

### ABOUT THE COURSE

Partial differential equations arise in the mathematical modeling of many physical, chemical and biological phenomena and many diverse subject areas such as fl
uid dynamics, electromagnetism, material science, astrophysics, economy, financial modelling, etc. Often the equations under consideration are so complicated that finding their analytical solutions in closed form is either impossible or impractical, and one has to resort to seeking numerical approximations to the unknown analytical solution. This course is devoted to a special class of numerical techniques for the approximate solution of partial differential equations: finite element methods (FEM). The
objective of this course is to provide a mathematical background that explains the working of finite elements, their construction, capabilities and limitations, stability and convergence. An introduction to various new advances in the field will also be provided along with an opportunity for the students to implement the method in practice. The viewpoint will be modern, with connections made between each topic and a variety of applications. By the end of the course, the student should not only be familiar, but more confident, in effectively using finite element methods to solve problems in their own field of interest.
- Lectures: Thursday (7:20 pm - 10:00 pm)
- Venue: Planetary Hall 242

### INSTRUCTOR

** Dr. Padmanabhan Seshaiyer **
- Office:MATH 222B
- Office Hours: Thursday (4:30 pm - 6:30 pm) and by appointment

- E-mail:
pseshaiy@gmu.edu

### PREREQUISITES

Sufficient recall of Calculus, Linear Algebra, Differential Equations and computer literacy including familiarity with MATLAB. A course in Numerical Analysis or knowledge of
numerical integration, solution to linear systems of equations, interpolation and approximation is
recommended.
### EXPECTED LEARNING OUTCOMES

In this course, the emphasis will be to expose the students to the mathematical and computational aspects of the finite element methods. The objective will be to train students to understand why the methods work, how to evaluate stabilty and convergence of these methods along with what type of errors to expect when this method is implemented in practice. The expected learning outcomes for the course will be assessed through: Exams,
homeworks, projects, in-class activities and class discussions. Problem-based learning will be an
integral part of the course.
### COURSE EVALUATION

Evaluation for the course will be based on the following criteria:

Homework |
50% |

Course Project |
25% |

Final Exam |
25% |

TOTAL |
100% |

There will be five homework assignments during the semester each worth 10 %. There will also be a course project which is worth 25 % and will involve using the finite element method (with some programming) to solve a problem of your choice, reading, reporting on a paper concerning aspects of the finite element method beyond those covered in class. There will be one comprehensive final exam in this course. Make-up exams may be possible only in the case of documented emergencies. The Final Exam will be on Thursday, Dec 13, 2012 from 7:30 PM - 10:15 PM and will be comprehensive. Make-up exams may be possible only in the case of documented emergencies.

### COURSE OUTLINE

We plan to cover the following topics in this class. Please note that topics
covered may vary slightly from those below depending on class interest and time.
- Review of Finite Difference Methods
- Introduction to FEM for elliptic problems
- Abstract formulation of the FEM for elliptic problems
- Approximation theory for the FEM
- Application of the FEM to structural mechanics and
uid dynamics problems
- Direct and Indirect methods for solving linear systems of equations arising from the FEM
- FEM for hyperbolic and parabolic problems

The primary reference will be lecture notes provided by the instructor that will be posted on the course website on a regular basis.
### LECTURES/HOMEWORK/PROJECT

- A review of Finite Difference Methods for PDEs (Aug 30, 2012). Homework I is included and is due on Sept 20, 2012.
- Numerical approximation of the Burger's Equation via FEM (Sept 6, 2012). Includes notes from the class without the source term.
- Mathematical Preliminaries of the FEM (Sept 13, 2012).
- FEM Basic Theory Summary (Sept 20, 2012)
- Quantitative Estimates for FEM (Sept 27, 2012)
- Homework II due on Oct 11, 2012.
- FEM 1D for axial bar (Oct 4, 2012)
- Homework III due on Nov 1, 2012.
- Nonlinear FEM 1D for axial bar (Oct 11, 2012)

### MATLAB

MATLAB is an interactive programming language for general scientific and technical computation with powerful graphics and library functions. MATLAB will be used for scientific computation, analysis and presentation of data in the course.

### ACADEMIC INTEGRITY

All students will be expected to abide by the Honor Code: ** Student
members of the George Mason University
community pledge not
to cheat, plagiarize, steal, or lie in matters related to academic
work **.
### DISABILITY ACCOMODATION

Any student who, because of a disability, may require some special
arrangements in order to meet course requirements should contact the
instructor as soon as possible to make such accommodations as may be
necessary.