Math 621  Numerical Analysis II
Numerical Methods for Partial Differential Equations
Fall 2007  Matthias K. Gobbert
Section 0101  Schedule Number 7565
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
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
 Matthias K. Gobbert,
Math/Psyc 416, (410) 4552404, gobbert@math.umbc.edu,
office hours: TuTh 01:0002:00 or by appointment
 Classes: MP 401, MW 02:3003:45;
see the detailed schedule for more information.
 Prerequisites: Math 620 and 630; familiarity with a highlevel
procedural programming language such as Matlab, C, or Fortran;
or instructor approval.
 Copies of the following books are on reserve in the library.

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 class 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 and Quizzes
 Participation
 Test
 Class Project

30%
 10%
 30%
 30%


The homework includes
the computer assignments that are vital to understanding
the course material.
A late assignment accrues a deduction of 10% 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 quizzes will generally be unannounced and brief
and use learning groups assigned by the instructor.
For instance, they may be designed to initiate class discussion
or to give me feedback on your learning.
They may be technical or nontechnical in nature.

In addition to the formally graded course components,
your professional behavior and active participation
in all aspects of the course are required.
Examples of expected professional behavior include
reading assigned material before class,
submitting material requested on time,
and participating actively in class,
specifically in any group work.

The test is a traditional inclass exam.
It is intended to reinforce only the basic analytic properties
of the methods considered and is 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 test.

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 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 timedirection
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 reactiondiffusion equations
including all numerical techniques necessary to deal with the
spatial and time discretizations as well as nonlinear 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 professionalgrade 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 secondyear 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:

understand and remember the key ideas, concepts, definitions,
and theorems of the subject.
Examples include classification of partial differential equations
and their key properties, the fundamental ideas of
finite difference and finite element methods,
the main error results for both types of methods,
and basic issues of computer implementations of these methods.
More broadly, you should also understand the purpose of
numerical linear algebra and
some of its major applications and methods.
> This information will be discussed in the lecture as well as
in the books, papers, and notes.
You will apply and use them on homework and tests.

have experience using a professional software package,
writing code in it, and understanding how some of its functions work.
We will use both Matlab and COMSOL Multiphysics in this course,
both of which are professionalgrade packages in their fields.
Writing code in this context includes the requirements to deliver code
in a form required, such as writing code to stated specifications,
using a requested method, complying with a required function header, etc.
The knowledge and skills in this item are valuable job skills,
which justifies the emphasis here.
To the same end, gaining significant experience with the Linux operating system
is a declared learning goal of this course
for graduate students in Mathematics.
> This is one of the purposes of the homework and most
learning will take place here.

have some experience how a research paper looks like
and experience in writing a professionalgrade report
in Mathematics.
Reading a paper will give you exposure how a professional report
should look and sound like.
Learning from written material is a crucial skill to develop over time,
thus the requirement of reading assigned material ahead of class and
the fact that you are responsible also for material not discussed in class
in detail.
More specifically, you need to learn how to obtain information
from research papers, thus there will be a first exposure to research papers.
You will then use your observations about structure and style of these
papers to write a technical report yourself and gain experience with
the process of receiving feedback, editing and revising, and resubmitting.
On a technical level, writing the report should expose you to the
relevant software in this field, therefore, the use of the
typesetting system LaTeX is required for graduate students in Mathematics.
> I will supply some papers carefully
selected for their readability and relevance to the course.
The writing, giving and receiving feedback, and editing of the
report will take place during the third part of the semester.

have experience working with peers in a group.
Group work requiring communication for effective collaboration
with peers and supervisors is a vital professional skill,
and the development of professional skills including this networking
is a declared learning goal of this course.
Additionally, getting to know other students as part of learning groups
will prove invaluable for homework, for tests,
and more generally for success in a graduate program.
In this spirit, it also a stated goal to help identify the skills
needed to be first a successful graduate student and then
a responsible scholar in Mathematics.
> The groups will be assigned by the instructor and used
for group quizzes in class, but also for other group work.
We will discuss the skills and expectations explicitly in class, but
you should also try to learn by observing your peers and the instructor.
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 www.umbc.edu/integrity,
or the Graduate School website
www.umbc.edu/gradschool.
Copyright © 20012007 by Matthias K. Gobbert. All Rights Reserved.
This page version 1.0, August 2007.