Commun. Comput. Phys., 5 (2009), pp. 195-241.

Some Recent Advances on Spectral Methods for Unbounded Domains

Jie Shen 1*, Li-Lian Wang 2

1 Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA.
2 Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 637616, Singapore.

Received 13 January 2008; Accepted (in revised version) 8 June 2008
Available online 1 August 2008


We present in this paper a unified framework for analyzing the spectral methods in unbounded domains using mapped Jacobi, Laguerre and Hermite functions. A detailed comparison of the convergence rates of these spectral methods for solutions with typical decay behaviors is carried out, both theoretically and computationally. A brief review on some of the recent advances in the spectral methods for unbounded domains is also presented.

AMS subject classifications: 65N35, 65N22,65F05, 35J05

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Key words: Spectral method, unbounded domain, orthogonal polynomials, rational functions, Hermite functions, Laguerre functions.

*Corresponding author.
Email: (J. Shen), (L. Wang)

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