Math 621  Numerical Analysis II
Numerical Methods for Partial Differential Equations
Spring 2006  Matthias K. Gobbert
Section 0101  Schedule Number 3963
This page can be reached via my homepage at
http://www.math.umbc.edu/~gobbert.
Final Projects
The class presentations of the final projects will be held on
Thursday, May 18, 2006 starting at 01:00 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
 Matthias K. Gobbert,
Math/Psyc 416, (410) 4552404, gobbert@math.umbc.edu,
office hours: TuTh 04:0005:00 or by appointment
 Classes: MP 401, TuTh 02:3003:45;
see the detailed schedule for more information.
 Prerequisites: Math 620; familiarity with a highlevel
procedural programming language such as Matlab, C, or Fortran;
or instructor approval.
Corequisite: Math 630.
 Textbooks:

There is no required textbook.
Several books are recommended for various parts of the course, namely
Evans for the mathematical background on PDEs,
Iserles for ODE methods (in the context of timedependent PDEs)
and finite differences for parabolic problems,
Braess for finite elements for elliptic problems,
Thomée for finite elements for parabolic problems, and
Strikwerda for finite differences for hyperbolic problems.
See my webpage on
recommended literature for the complete citations.
Copies of some important sections and additional notes will
be handed out.

Recommended book on Matlab:
Desmond J. Higham and Nicholas J. Higham,
MATLAB Guide, second edition, SIAM, 2005.
Associated webpage:
http://www.ma.man.ac.uk/~higham/mg
 Grading policy:
Homework
 Test 1
 Test 2
 Class Project

25%
 25%
 25%
 25%

In addition to these formally graded course components,
your professional behavior and
active participation in all aspects of the course are required.
In particular, you are required to read assigned material before class
as well as to participate actively in class.

The homework includes
the computer assignments that are vital to understanding
the course material.
A late assignment accrues a deduction of 5% of the possible score
for each day late until my receiving it; I reserve the right
to exclude any problem from scoring on late homework,
for instance, if we discuss it in class.

The tests are traditional inclass exams.
They are intended to reinforce only the basic analytic properties
of the methods considered and are therefore scheduled rather
early throughout the semester, so as to allow for a focus on
the class project afterwards.
See the detailed schedule for the dates
of the exams.

It is increasingly important
at this point in your education to learn
how to work on a larger project on your own
(with guidance by the instructor)
and to present your results in the form of a
professionalgrade typeset report
and an oral class presentation.
The class project will include all these components:
substantial work on an individual project;
a written report; and an oral class presentation.
Additional details or changes will be announced as necessary.
See also general policies and procedures
for more 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 nonlinear
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 linear prototype
equations of each type. Classical examples for the three types
are the Poisson equation, the heat equation,
and the scalar transport equation, respectively.
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 for each prototype equation. We will use this as the basis for
discussing the associated issues of discretizing the timedirection
and solving large sparse systems of linear equations efficiently
with respect to memory and computing time.
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 FEMLAB,
a professionalgrade finite element package originally based on Matlab.
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 simultaneously and equivalently on computational experiments
and on rigorous
mathematical analysis of the numerical methods considered.
This course is designed to be accessible to firstyear graduate students,
hence we will fill in the knowledge necessary from Math 630,
but an excellent knowledge of Math 620 is expected.
The course is taught without expecting a formal background in
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.
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,
and the UMBC Academic Integrity webpage at
www.umbc.edu/integrity.
Copyright © 20012006 by Matthias K. Gobbert. All Rights Reserved.
This page version 1.0, January 2006.