Partial Differential Equations

This page can be reached via my homepage at http://www.math.umbc.edu/~gobbert.

- 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

- Grading policy:

Homework Participation Quizzes Test 1 Test 2 Project Report 20% 20% 20% 20% 20% 20% 20%

- 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.

- The

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.

- Detailed schedule including date for the midterm exam
- Statement of teaching philosophy
- General policies and procedures including grading guidelines
- Guidelines on how to report on computer results

- The recommended literature on my homepage
- A brief introduction to Unix/Linux at UMBC
- My introduction to Matlab at UMBC
- An introduction to LaTeX including a sample file and a template file for project reports.
- Introduction to using MPI under Linux in the Math Department

Copyright © 2001-2003 by Matthias K. Gobbert. All Rights Reserved.

This page version 4.0, November 2003.