TAM 574

Advanced theory and applications of the finite-element method, as needed for research in computational science and engineering: applications to mechanics of solids and fluids, thermal problems, etc.; variational foundations of the finite-element method, error estimates, and adaptive analysis; finite-element methods for parabolic and hyperbolic problems; mixed finite-element methods; applications to systems of equations. Course Information: Same as CSE 517. Prerequisite: One of TAM 470, CEE 570, CS 555, ME 471.

Objectives: This course is intended to serve as a sequel to an introductory finite element course, such as AAE 420, CEE 570, or ME 471 or a survey numerical methods course such as TAM 470. It is designed to deepen the students understanding of the mathematical background of the finite element method and to introduce topics that are typically not covered in a first course. Among the topics to be covered are: an introduction to functional analysis and other mathematical theory that supports the finite element method, FEM for elliptic and parabolic equations, error estimates,
adaptive analysis, C1 finite elements, nonconforming finite finite elements, and mixed finite element methods for the Poisson and the Stokes equations. Applications to problems of current interest in mechanics and other sciences will be developed. We will also look at computational examples using the FEniCS finite element software environment.

Note: This course does not cover topics such as programming the finite element method, isoparametric elements, numerical quadrature etc.

Course Website and Lecture Notes: The TAM 574 website is on Piazza. There you will find the syllabus, announcements, homework assignments, and grade postings. In addition, my scanned lecture notes will be made available for download as the course progresses. These lecture notes are largely derived from the following references:

• Lecture notes of Prof. Douglas Arnold at the University of Minnesota. See http://www-users.math.umn.edu/ arnold/8445-8446.17-18/index.html.
• Notes on functional analysis are based on Prof. Robert Haber’s lecture notes of TAM 574.
• The Mathematical Theory of Finite Element Methods, Susanne Brenner and Ridgway Scott, third edition, Springer, 2008.
• Numerical Solutions of Partial Differential Equations by the Finite Element Method, Claes Johnson, Dover Publications, 2009.

Exams: Students will be tested based on homeworks, an in-class midterm, and a final exam. The following distribution will be used to decide the course grade.

Homework 60%
Midterm 20%
Finals 20%

I encourage discussing homework problems with others or with me. However, the solutions must be written independently. The same applies to computer assignments.

Academic integrity: Carefully review the following links on academic integrity:

• Students Quick Reference Guide to Academic Integrity
• University of Illinois Student Code: Part 4. Academic Integrity and Procedure

1/15/2019