Spring
MATH-6860-01 Finite Element Analysis
2014
homepages.rpi.edu/~lif/S14/Math6860_S14.html
Objectives:

To understand the mathematics behinds the main ingredients of finite element methods: formulating of finite element methods, matrices assembly for implementation, construction of finite element spaces, approximation theories, error estimates

To be able to carry out standard mathematical analysis and derivations related to constructing and analyzing finite element methods

To be able to implement finite element methods for some 1d and 2d problems, as well as to evaluate and to interpret the numerical results
Contents: (tentative)

Finite element methods for elliptic equations: variational /weak formulation for 1d and 2d model problems, formulation of finite element methods, implementation, relation with minimization problem, Lax-Milgram theory and its applications, construction of finite elements, approximation theory, numerical integration, error estimate theory (energy norm, L2 norm), variational crime

Finite element methods for parabolic and hyperbolic equations, and Stokes equation : finite element formulation and/or discontinuous Galerkin methods, mixed finite element methods

Linear solvers: multi-grid methods

Adaptive finite element methods: a posterior error estimates
References: (there is no required textbook)
1. “Numerical Solution of Partial Differential Equations by the Finite Element method”, Claes
Johnson, Cambridge University Press (Chapter 0-5, 9)
2. “Finite Elements, Theory, Fast Solvers, and Applications in Solid Mechanics”, Dietrich
Braess, 3 rd edition, Cambridge (Chapter II, III(4-6, 8))

3. “The Mathematical Theory of Finite Element Methods”, Susanne C. Brenner, L. Ridgway
Scott, 2 nd edition, Springer (Chapter 0-2, 6).
4. “The Finite Element Method for Elliptic Problems”, Philippe G. Ciarlet, SIAM (Chapter
1-3)
5. “Finite Element Analysis”, lecture notes by Dr. Joseph E. Flaherty
(http://www.rpi.edu/~lif/S14/Topics_reading.html)
An e-book of reference [4] is available from the RPI library; References [1,2] can be purchased from RPI bookstore; Selected chapters of [1,2,3] are available from RPI class reserves http://library.rpi.edu/update.do?catcenterkey=19.
Homework: Homework will be assigned regularly and will be collected within one or two weeks.
Your report is expected to be neatly written and well organized.
Collaboration: You are encouraged to discuss the problems with others and to seek assistance from online or library references. However, the work you turn in must be written by yourself and represents your own understanding on the subjects. If your work involves a teammate, or significant help from other people, reference books or papers, explicitly state this in your report to avoid penalty in grade.
Late policy: No late homework is accepted. Under very special circumstances, approval might be granted for extension. Any request for extension needs to be made at least 36 hours before the deadline. Regardless of the reason, any extension will result in a 10%-per day reduction of the grade for that particular homework.
Programming assignments: The report for a programming assignment should consist of

Brief description of the problem

Results such as data, graphs etc. Try to avoid attaching large set of data unless it is necessary.

Discussion, comments, explanation and conclusion on your numerical observations.
This part is equally important as your computer codes and the data you collect. It helps with understanding concepts, algorithms, or other relevant issues not discussed during lectures.

Computer codes printouts. Program text is expected to be well structured so others (at least from the same class) can read it without much difficulty. Add extra comments if needed.
Exams: There is one in-class midterm exam and one take-home final exam, and each will cover all the topics discussed before the exam date. The midterm exam is tentatively scheduled on
March 7, 2014 . Except for unforeseen reasons, one must obtain approval in advance from the

instructor for missing an exam. The instructor reserves the right to change the exam dates if needed.
Mathematical Analysis and Derivation : In both exams and homework assignments, there are questions involving mathematical analysis (proof) and derivation. Regardless of your academic disciplines, your answers to such questions should be mathematically reasonable and have as much rigor as possible in order to receive good grades. Examples of such analysis and derivation will be discussed frequently during lectures.
Your grade for this course is determined by your performance in homework assignments and exams, which are designed based on the Course Objectives.
Homework
Midterm Exam
45%
30%
Take-home Final 25%
Questions regarding the grading of assignments and exams can be discussed with the instructor only on the same day when the papers are returned.
Student-teacher relationships are based on mutual trust. Acts that violate this trust undermine the educational process. The Rensselaer Handbook defines various forms of academic dishonesty and you should make yourself familiar with them. The penalties for cheating in this course include reduction in grade and no grade to the assignments or exams.
Attendance is expected, and be on time. Those who have to miss a class due to unpredictable reasons are responsible for finding out what are discussed or announced in class. If you attend a lecture, you are expected to stay till the end of the meeting unless you inform the instructor before the lecture starts.