MA69200.001, Fall 2012
INTRODUCTION TO SPECTRAL METHODS FOR SCIENTIFIC COMPUTING
Instructor: Jie Shen
TTh 10:30-11:45 at REC 317
|Office: MATH 406
Office Hours: M,Th 3:00-4:20pm
or by appointment
This is an introduction course on spectral methods for solving
partial differential equations (PDEs). We shall present some basic
results on spectral approximations as well as practical algorithms for
implementing spectral methods. We shall specially emphasize on how
to design efficient and accurate spectral algorithms for solving PDEs
of current interest.
- Fourier-spectral methods
- basic results for polynomial approximations
- Galerkin method using Legendre and Chebyshev polynomials
- Collocation method using Legendre and Chebyshev polynomials
- Fast elliptic solvers using the spectral method
- Applications to various PDEs of current interest
A good knowledge of calculus, linear algebra, numerical
analysis and some basic programming skills
are essential. Some knowledge of real
analysis and functional analysis will be helpful but not necessary.
There will be no exam. Course grades will be based on homework
assignments and programming projects.
- J. Shen & T. Tang, "Spectral and High-Order Methods with
Applications", Science Press of China, 2006; Erratum.
T. Tang and L. Wang, "Spectral Methods: Algorithms, Analysis and Applications" (Springer
Series in Computational Mathematics, V. 41, Springer, Aug. 2011), and the associated
- C. Bernardi & Y. Maday, Spectral Method, in ``Handbook
of Numerical Analysis, V. 5 (Part 2)"
eds. P. G. Ciarlet and L. L. Lions, North-Holland, 1997.
- L. N.
Trefethen, Spectral Methods in Matlab, SIAM
- C. Canuto, M. Y. Hussaini, A. Quarteroni & T. A. Zang,
``Spectral Methods. Fundamentals in Single Domains'', Springer-Verlag (2006).