Math 621 - Numerical Analysis II
Numerical Methods for
Partial Differential Equations
Fall 2003 - Matthias K. Gobbert
Section 0101 - Schedule Number 3842
This page can be reached via my homepage at
http://www.math.umbc.edu/~gobbert.
Project Presentations
The project presentations will be held on
Monday, November 24, 2003 and Wednesday, November 26, 2003
starting at 05:30 p.m. in MP 401.
Please follow the link to the Program
for the titles and abstracts.
Grading Information
Scores and grades will be posted here at the end of the semester.
Basic Information
- Matthias K. Gobbert,
Math/Psyc 416, (410) 455-2404, gobbert@math.umbc.edu,
office hours: MW 03:00-04:00 or by appointment
- Classes: MP 401, MW 05:30-07:00
from August 27 to November 17 and
from November 24 to December 01;
note the lengthened time slot!
See the detailed schedule for more information.
Notice that the above dates have been changed compared to
the printed syllabus!
- Prerequisites: Math 620, Math 630,
familiarity with Matlab or other high-level programming languages and
basic knowledge of the Unix/Linux operating system,
or instructor approval.
See the Course Description below for more information.
- The following books are recommended for the course;
there is no required textbook.
-
Arieh Iserles, A First Course in the Numerical Analysis
of Differential Equations, Cambridge University Press, 1996.
Associated webpage:
Click on "Textbook" in the left column from the webpage
http://www.amtp.cam.ac.uk/user/na/people/Arieh
A copy of this book is on reserve in the library.
-
Recommended book on Matlab:
Desmond J. Higham and Nicholas J. Higham,
Matlab Guide, SIAM, 2000.
Associated webpage:
http://www.ma.man.ac.uk/~higham/mg
Additional recommended books include the ones by
Hall and Porsching (1990) and by Strikwerda (1989)
on finite differences and the ones
by Braess (2001), by Thomée (1997), and by
Quarteroni and Valli (1994) on the finite element method.
The book by LeVeque (1992) is a well-known book
on the theory and methods for hyperbolic equations.
See the list of
recommended literature for the details of the references.
- Grading policy:
Homework
| Participation
| Quizzes
| Test 1
| Test 2
| Project
| Report
|
20%
| 20%
| 20%
| 20%
| 20%
| 20%
| 20%
|
This class is taught following a learner-centered teaching philosophy.
As part of the empowerment of your own educational decision-making,
you will be asked, at the beginning of the semester,
to select the five categories from the list above
whose grades you wish to have counted; each will count 20% towards your grade.
Note that you are still expected to partake in the in-class activities,
e.g., the quizzes, even if you do not select them for counting
towards your grade.
- The homework includes the computer assignments that are
vital to understanding the course material.
- Class participation measures your active participation
in the classroom, including from answering questions, posing questions,
and taking part in discussions.
- The quizzes will generally be unannounced and extremely brief
(5 minutes) at the beginning or end of class. They are designed to
initiate class discussion or to give me feedback on your learning.
Many will not be technical in nature. We will drop a sufficient number
of quizzes in order to avoid penalizing infrequent excused absences.
- There will be two in-class tests;
see the detailed schedule for the dates.
- The project includes the independent work as well as a short
class presentation.
- The report is a written report on the project; this score
only refers to the report and not the presentation.
Additional details or changes will be announced if necessary.
See also the
teaching philosophy and the
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 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 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 time-direction
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;
I suggest Matlab for this purpose because of its ease of programming,
but you will probably need 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 commercial package based on Matlab and available across campus.
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.
Hence, you are expected to have background knowledge equivalent to
our first-year graduate courses, in particular Math 620 and Math 630.
Additionally, you should have a good foundation in mathematical analysis
and be ready to learn more.
Information to Download
This area might be used later to distribute some computer files.
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 Policies section of the UMBC Directory for undergraduate students,
or the Graduate School website for graduate students.
Copyright © 2001-2003 by Matthias K. Gobbert. All Rights Reserved.
This page version 4.0, November 2003.