AMATH 585
SLN 1207, MWF 2:30-3:20, Loew Hall 113

Approximate and Numerical Analysis II


Instructor:

 Professor Anne Greenbaum
Padelford C-434
tel: 543-1175
fax: 543-0397
greenbau@math.washington.edu
office hours: MW 4:00-5:00 PM

Homework Handouts Grades Message Board 2002 Web Page
Course description Textbook Syllabus Objectives Schedule

Course Description

Numerical methods for steady-state differential equations. Numerical differentiation and integration. Two-point boundary value problems and elliptic equations. Finite difference, spectral, and finite element methods. The fast Fourier transform. Iterative methods for sparse linear systems: conjugate-gradients, preconditioners, and methods for nonsymmetric linear systems.

Textbook

There are no required textbooks for this class. Some of the material is covered by course notes written by Prof. Randy LeVeque, which you may download as a ps file and print. These will be supplemented by other handouts. A good reference for the numerical solution of differential equations is: A First Course in the Numerical Analysis of Differential Equations by A. Iserles, Cambridge University Press, 1996. A recommended text on iterative methods is: Iterative Methods for Solving Linear Systems by A. Greenbaum, SIAM, 1997.

Syllabus

(1) Numerical differentiation and integration: (2 weeks) Finite difference approximations to derivatives; polynomial interpolation; numerical integration via trapezoidal rule, composite trapezoidal rule, Simpson's rule, Romberg integration, Gaussian quadrature.

(2) Two-point boundary value problems: (3 weeks) The heat equation; finite difference methods; stability, consistency, and convergence; nonlinear equations; singular perturbations and boundary layers; adaptive mesh refinement; spectral and finite element methods.

(3) Multidimensional boundary value problems for PDE's: (2 weeks) Poisson's equation; finite difference and finite element methods; accuracy and stability.

(4) The Fast Fourier transform: (1 week) Solution of Poisson's equation via the FFT.

(5) Iterative methods for solving linear systems: (2 weeks) The conjugate gradient method; preconditioners; methods for solving nonsymmetric linear systems.

Learning Objectives and Instructor Expectations

The course will be a combination of computation and theoretical analysis. There will be homework assignments every week or two that will involve MATLAB programming and written exercises. You are encouraged to consult with your classmates about how to do the homework, but each person should write his own code and express the answers to the written questions in his own words. In the middle of the quarter there will be one homework assignment which you will be required to do without help from others (take-home midterm), although you may consult the course notes or other books. There will be an in-class final.

The goal is to develop computational expertise and understanding of numerical methods for solving steady-state differential equations and related problems.

Schedule and Homework

Follow links in the table below to obtain a copy of the homework in PostScript (.ps) or Adobe Acrobat (.pdf) format. You may also obtain here solutions to some of the homework and exam problems. An item shown below in plain text is not yet available. For additional information regarding viewing and printing the homework and solution sets, click here.

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Homework and Exams Homework Due Date Homework Problem Sets Homework Solutions
First day of classes Monday, January 6
Homework#1 due Wednesday, 1/22 Homework #1 (.ps, .pdf)
Martin Luther King Day Monday, January 20 No class
Homework#2 due Wednesday, 1/29 Homework #2 (.ps, .pdf)
Homework#3 due Friday, 2/7 Homework #3 (.ps, .pdf)
Take-home Midterm due Wednesday, Feb. 19 Homework #4 (.ps, .pdf)
President's Day Monday, February 17 No class
Homework#5 due Friday, Feb. 28 Homework #5 (.ps, .pdf)
Homework#6 (Last Homework!) due Friday, Mar. 14 Homework #6 (.ps, .pdf)
Last day of classes Friday, March 14

Handouts

Notes on Polynomial Interpolation: (.ps,.pdf)
Notes on Numerical Integration (Part I): (.ps,.pdf)
Notes on Numerical Integration (Part II): (.ps,.pdf)
Notes on Finite Element Methods in One Dimension: (.ps,.pdf)
Notes on Fast Poisson Solvers and the FFT: (.ps,.pdf)
Notes on Iterative Methods for Solving Linear Systems (excerpt from book published by SIAM): (.ps)
MATLAB code for composite trapezoid rule: (trap.m)
MATLAB code for composite Simpson's rule: (simp.m)
MATLAB code for Romberg integration: (romberg.m)
MATLAB code for the 1-D steady state heat equation (finite differences): (heat1d.m)
MATLAB code for 2-D Poisson's equation (five-point formula): (fivept.m)
MATLAB code for 2-D Poisson's equation (modified nine-point formula): (ninept.m)
MATLAB code for the 1-D steady state heat equation (finite elements): (fem1d.m)

Grading

Homework will count 50%, the midterm will count 25%, and the final will count 25%. You may view your homework and exam grades on-line. Before doing so for the first time, you must request a password.

Tutorials

No on-line tutorials have been assigned for AMATH 585.

<greenbau@math.washington.edu> Thu Dec 12 09:59:13 2002