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Commun. Comput. Phys., 5 (2009), pp. 195-241. |
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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 Abstract 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, 35J05Key words: Spectral method, unbounded domain, orthogonal polynomials, rational functions, Hermite functions, Laguerre functions. *Corresponding author. Email: shen@math.purdue.edu (J. Shen), lilian@ntu.edu.sg (L. Wang) |