UNIVERSITY OF TENNESSEE

SYLLABUS

Math 574 - Finite Element Methods

3 Credit Hours

Section 001, CRN 43244

Meetings: TR 9:40am--10:55am Ayres Hall 111.

Fall 2017

**Course Description**: Finite element techniques for solution of boundary and initial-boundary value problems. Variational formulation. Finite dimensional subspaces and their approximating properties; rates of convergence. Computer implementation.

**Instructor Name**: Abner J. Salgado

**Office Hours and Location**: Wednesdays 1--2pm in Ayres 204. Office hours can also be arranged by appointment.

**Email**: `asalgad1[at]utk[dot]edu`

**Course Communications**: You may write to me at `asalgad1@utk.edu` with questions, comments, etc. To avoid confusion, please write the course name (MATH574) in the Subject line. You should use your university e-mail account when sending me emails.

**Course web page**: `Here`

**Goals**: This is a graduate level class on the **mathematical** theory of finite element methods. Students, after successfully completing this class, will be able to apply finite element techniques to a variety of model problems, and analyze their stability and convergence properties.

The analysis of finite element methods relies on the understanding of fundamental properties of the PDE in question. For this reason, knowledge of classical topics in numerical analysis (at the level of MATH 572) will be largely assumed, and the theory of partial differential equations will be recalled as needed.

**Textbook**:
*Required:*

- Theory and Practice of Finite Elements, A. Ern and J.-L. Guermond, Springer, 2004.

*Supplementary:*

- The Mathematical Theory of Finite Element Methods. S.C. Brenner and L.R. Scott, Springer, 2008.
- The Finite Element Method for Elliptic Problems. P.G. Ciarlet, SIAM, 2002.
- Galerkin Finite Element Method for Parabolic Problems. V. Thomeé, Springer, 2006.

**Course Requirements, Assessment and Evaluations**:

- There will be a short (10-15 minutes) quiz approximately every other week. They will generally cover the material of the previous lectures.
- There will be one midterm and a
*comprehensive*final exam. A tentative date for the midterm is**October 19**. The official date will be announced*at least*two weeks in advance. The University Calendar sets the final examination date to be**Tuesday, December 12 8:00am--10:00am**. - The midterm is worth
**35**points, the final is worth**35**points, quizzes are worth**30**points. The cutoffs will be approximately as follows: 90% or higher is an A, 80% - B, 70% - C and 60% - D.

**Makeup Policy**: Makeups for the quizzes, midterm and final will be given only if a student can present evidence that an absence was caused by serious illness, a death in the immediate family, religious observance, or participation in University activities at the request of University authorities. For an illness, you must present a signed statement from a doctor that your illness was sufficiently serious to make you miss class. A note saying only that you visited the doctor or the Health Center will not suffice.

**Course Outline**:
This is a tentative list of topics to be covered in class.

- Weak formulation of PDEs: well-posedness, the Banach-Necas-Babuška theorem.
- Approximation in Banach spaces by Galerkin methods: discrete inf-sup conditions, Strang's lemmas.
- Elliptic equations: polynomial interpolation in Sobolev spaces, construction of finite element spaces, nonconforming approximations.
- Mixed problems: The Stokes and Darcy problems, compatibility conditions between discrete spaces, linear algebra issues.
- First order problems: First order PDEs in \(L^2\), Bubnov-Galerkin methods, discontinuous Galerkin methods.
- Time dependent problems: Parabolic problems, the method of lines, backward Euler and BDF2 methods.

The last two topics are optional. Emphasis on either one of them will be placed based on the interests of the audience.

*If the instructor finds it necessary to make informational changes (e.g. office hours, schedule adjustments) due to students' needs or unforeseen circumstances, students will be notified in writing/email of any such changes.*

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Abner Salgado 2017-08-09