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Composite generalized Laguerre-Legendre spectral method with domain decomposition and its application to Fokker-Planck equation in an infinite channel

Authors: Ben-yu Guo and Tian-jun Wang
Journal: Math. Comp. 78 (2009), 129-151
MSC (2000): Primary 65M70, 41A30, 82C99
Published electronically: May 30, 2008
MathSciNet review: 2448700
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Abstract: In this paper, we propose a composite generalized Laguerre- Legendre spectral method for partial differential equations on two-dimensional unbounded domains, which are not of standard types. Some approximation results are established, which are the mixed generalized Laguerre-Legendre approximations coupled with domain decomposition. These results play an important role in the related spectral methods. As an important application, the composite spectral scheme with domain decomposition is provided for the Fokker-Planck equation in an infinite channel. The convergence of the proposed scheme is proved. An efficient algorithm is described. Numerical results show the spectral accuracy in the space of this approach and coincide well with theoretical analysis. The approximation results and techniques developed in this paper are applicable to many other problems on unbounded domains. In particular, some quasi-orthogonal approximations are very appropriate for solving PDEs, which behave like parabolic equations in some directions, and behave like hyperbolic equations in other directions. They are also useful for various spectral methods with domain decompositions, and numerical simulations of exterior problems.

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Additional Information

Ben-yu Guo
Affiliation: Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China, Scientific Computing Key Laboratory of Shanghai Universities, Division of Computational Science of E-Institute of Shanghai Universities

Tian-jun Wang
Affiliation: Department of Mathematics and Physics, Henan University of Science and Technology, LuoYang, 471003, China

Keywords: Composite generalized Laguerre-Legendre spectral method, quasi-orthogonal approximation, domain decomposition, Fokker-Planck equation in an infinite channel
Received by editor(s): April 13, 2007
Received by editor(s) in revised form: February 9, 2008
Published electronically: May 30, 2008
Additional Notes: The work of the first author was supported in part by The Grant of Science and Technology Commission of Shanghai Municipality N.75105118, The Shanghai Leading Academic Discipline Project N.T0401, and The Fund for E-institutes of Shanghai Universities N.E03004.
The work of the second author was supported in part by The Doctor Fund of Henan University of Science and Technology N.09001263.
Article copyright: © Copyright 2008 American Mathematical Society

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