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 Phone: 4-1923 Message: 4-1901 E-mail: shen@math.purdue.edu

Course outline:

This is an introduction course on spectral methods for solving partial differential equations (PDEs). We shall present some basic theoretical 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.

Topics:

• 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

Prerequisite:     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.

Requirement:     There will be no exam. Course grades will be based on homework assignments and programming projects.

Textbook:

J. Shen & T. Tang, "Spectral and High-Order Methods with Applications", Science Press of China, 2006; Erratum.

Reference books:
J. Shen, T. Tang and L. Wang, "Spectral Methods: Algorithms, Analysis and Applications" (Springer Series in Computational Mathematics, V. 41, Springer, Aug. 2011), and the associated Matlab codes.

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 2000.

C. Canuto, M. Y. Hussaini, A. Quarteroni & T. A. Zang, ``Spectral Methods. Fundamentals in Single Domains'', Springer-Verlag (2006).