Math 621 - Numerical Methods for Partial Differential Equations
Spring 2013 - Matthias K. Gobbert
This page can be reached via my homepage at
Presentations of the Class Projects
The presentations of the class projects will be on
Friday, May 17, 2013, 01:00 p.m. in MP 104.
Please follow the link to the Program
for the titles and abstracts.
Just like for seminar talks, everybody is welcome to attend!
- Matthias K. Gobbert,
Math/Psyc 416, (410) 455-2404, email@example.com,
office hour: TuTh 02:30-03:30 or by appointment
- Classes: Math/Psyc 104, TuTh 01:00-02:15;
see the detailed schedule for more information.
- Prerequisites: Math 620 and 630; familiarity with a high-level
procedural programming language such as Matlab, C, or Fortran;
or consent of instructor
- 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,
Watkins for linear solvers
(particularly the conjugate gradient method)
Iserles for ODE methods (in the context of time-dependent 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.
Among these, the book by Braess is a particularly
important one, since we will follow it for the rigorous
analysis of the finite element method for the Poisson equation.
See my webpage on
recommended literature for the complete citations.
Copies of some class notes will be handed out.
- Recommended textbook:
A First Course in the Numerical Analysis of Differential
Equations, Cambridge Texts in Applied Mathematics,
Cambridge University Press, second edition, 2008.
Click on "Textbook" in the left column from the webpage
I chose this book as recommended textbook because it does
cover the breadth of material of the course at the right level;
its coverage is in some cases complementary to the approach in class,
and we will follow some other sources as well, as characterized above.
Recommended book on Matlab:
Desmond J. Higham and Nicholas J. Higham,
Matlab Guide, second edition, SIAM, 2005.
The associated webpage
includes updates, code, and a list of errors.
Recommended book on professional issues in mathematics and the sciences:
Nicholas J. Higham,
Handbook of Writing for the Mathematical Sciences,
second edition, SIAM, 1998.
Recommended book on LaTeX:
Leslie Lamport, LaTeX: A Document Preparation System,
second edition, Addison-Wesley, 1994.
Introduction to LaTeX by the author himself.
- Grading policy:
| Homework and Quizzes
|| Class Project
Additional details or changes will be announced as necessary.
See also general policies and procedures
for more information.
The homework includes
the computer assignments that are vital to understanding
the course material.
A late assignment accrues a deduction of
up to 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
will include the use of learning groups formed 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 non-technical in nature.
The graded participation component rewards
your professional behavior and active involvement
in all aspects of the course.
Examples of expected professional behavior include
attending class regularly,
reading assigned material when requested,
cooperating with formal issues such as
submitting requested material on time, and
participating constructively in class, specifically in group work.
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
professional-grade type-set report
and a prepared oral class presentation.
The class project will include all these components:
substantial work on an individual project;
a written report (in the form of a technical report);
and an oral class presentation.
is a course management system that allows for posting
and communicating among registered participants of a course.
We will actively only use the "Course Documents" area
of our course in Blackboard.
I will post class summaries and PDF files of the lecture notes
as well as other material including the homework assignments in this area.
I will also use Blackboard to send e-mail to the class,
which goes to your UMBC account by default.
Therefore, you must either check your UMBC e-mail regularly
or have the mail forwarded to an account that you check frequently.
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.
This course will provide an overview of the types of equations,
their most fundamental mathematical properties, and demonstrate
numerical methods for them.
It will have a strong focus on parabolic problems
such as systems of time-dependent reaction-diffusion equations,
but we will also cover elliptic and hyperbolic prototype problems.
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.
The course use of the state-of-the-art
finite element package COMSOL Multiphysics and include a
pointed but meaningful introduction to it, ranging from
2-D and 3-D meshing and visualization, stationary and transient problems,
and overview of solver options and parameters.
See the COMSOL area of my homepage (link below) or the
webpage of the maker COMSOL.
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 was originally designed
for second-year graduate students in Applied Mathematics,
hence you are ordinarily expected to have knowledge of
Numerical Analysis (Math 620) and
Numerical Linear Algebra (Math 630),
though we will cover all crucial concepts from the background briefly.
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.
The approach of this course is designed to accomodate a varied background
of the audience, and it has been successfully taken both by applied mathematics
students with different ranges of experience and by scientists and engineers
from a variety of departments.
Please contact me if you have any questions about the expectations.
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 methods for and some of their major applications.
--> 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 quizzes.
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 professional-grade 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.
--> 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 professional-grade report
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
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 re-submitting.
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.
UMBC High Performance Computing Facility (HPCF)
including general information, list of projects, publications page,
and resources for users.
Center for Interdisciplinary Research and Consulting (CIRC)
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
UMBC Undergraduate Student Academic Conduct Policy
for undergraduate students,
University of Maryland Graduate School, Baltimore (UMGSB)
Policy and Procedures for Student Academic Misconduct
for graduate students.
Copyright © 2001-2013 by Matthias K. Gobbert. All Rights Reserved.
This page version 1.3, May 2013.