Math 621 - Numerical Analysis II

Numerical Methods for Partial Differential Equations

Fall 2007 - Matthias K. Gobbert

Section 0101 - Schedule Number 7565

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Final Projects

The class presentations of the final projects will be held on Friday, December 14, 2007 starting at 02:30 p.m. in MP 401. Please follow the link to the Program for the titles and abstracts. Just like for seminar talks, everybody is welcome to attend!

Basic Information

Course Description

Many models for physical processes in nature and in engineering consist of partial differential equations. The models are as varied as reality itself, but often non-linear and often involving systems of partial differential equations. In all but some textbook examples, an analytic solution is impossible. That necessitates the use of numerical methods for partial differential equations, and this area forms a vast field itself and is one of the major driving forces behind research in many other fields like numerical linear algebra, scientific computing, and the development of parallel computers.

Despite their many forms, many equations share certain fundamental mathematical properties and can be classified into the three basic categories of elliptic, parabolic, and hyperbolic partial differential equations. It makes therefore sense to study the mathematical properties and numerical methods for prototype equations of each type. Classical examples for the three types are the Poisson equation, the heat equation, and the scalar transport equation, respectively. Due to the time limitations of a semester, there will be somewhat of a stronger focus on elliptic and parabolic equations.

This course will provide an overview of the types of equations, their most fundamental mathematical properties, and demonstrate numerical methods for them. Two large classes of methods are finite difference and finite element methods, and we will discuss examples of both methods. We will use this as the basis for discussing the associated issues of discretizing the time-direction and solving large sparse systems of linear equations efficiently with respect to memory and computing time. One specific goal of this course is to understand the method of lines approach to transient reaction-diffusion equations including all numerical techniques necessary to deal with the spatial and time discretizations as well as non-linear and linear solvers.

For the finite difference methods, we will write our own code; we will use MATLAB for this purpose because of its ease of programming. You should have a foundation in using Matlab, equivalent to its Getting Started guide; see a link to my Matlab webpage below. But you should expect to learn additional commands and techniques to get the best resolution and fastest performance. For the computational experiments on the finite element method, we will use COMSOL Multiphysis (formerly known as FEMLAB), a professional-grade finite element package. It has a sophisticated graphical user interface and is sufficiently powerful to allow the solution and visualization in two and three dimensions.

We will focus both on computational experiments and on rigorous mathematical analysis of the numerical methods considered, but with a slant towards the computational side of the subject. This course is designed for second-year graduate students in Mathematics, hence you are ordinarily expected to have knowledge in Math 620 and Math 630, though we will review important concepts briefly; if you do not have this background, you must be ready to review material on your own, as needed. The course is taught without expecting a formal background in the theory of partial differential equations, but you should be ready to learn some background information when necessary. Additionally, you should have a good foundation in mathematical analysis and be ready to learn more. Please contact me if you have any questions about these expectations.

Learning Goals

By the end of this course, you should:

Other Information

UMBC Academic Integrity Policy

By enrolling in this course, each student assumes the responsibilities of an active participant in UMBC's scholarly community in which everyone's academic work and behavior are held to the highest standards of honesty. Cheating, fabrication, plagiarism, and helping others to commit these acts are all forms of academic dishonesty, and they are wrong. Academic misconduct could result in disciplinary action that may include, but is not limited to, suspension or dismissal. To read the full Student Academic Conduct Policy, consult the UMBC Student Handbook, the Faculty Handbook, the UMBC Integrity webpage, or the Graduate School website

Copyright © 2001-2007 by Matthias K. Gobbert. All Rights Reserved.
This page version 1.0, August 2007.