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Some progress in spectral methods

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Abstract

In this paper, we review some results on the spectral methods. We first consider the Jacobi spectral method and the generalized Jacobi spectral method for various problems, including degenerated and singular differential equations. Then we present the generalized Jacobi quasi-orthogonal approximation and its applications to the spectral element methods for high order problems with mixed inhomogeneous boundary conditions. We also discuss the related spectral methods for non-rectangular domains and the irrational spectral methods for unbounded domains. Next, we consider the Hermite spectral method and the generalized Hermite spectral method with their applications. Finally, we consider the Laguerre spectral method and the generalized Laguerre spectral method for many problems defined on unbounded domains. We also present the generalized Laguerre quasi-orthogonal approximation and its applications to certain problems of non-standard type and exterior problems.

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Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

    BenYu Guo

  2. Scientific Computing Key Laboratory of Shanghai Universities, Shanghai, 200234, China

    BenYu Guo

  3. Division of Computational Science, E-institute of Shanghai Universities, Shanghai, 200234, China

    BenYu Guo

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  1. BenYu Guo
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Correspondence to BenYu Guo.

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Dedicated to Professor Shi Zhong-Ci on the Occasion of his 80th Birthday

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Guo, B. Some progress in spectral methods. Sci. China Math. 56, 2411–2438 (2013). https://doi.org/10.1007/s11425-013-4660-7

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  • DOI: https://doi.org/10.1007/s11425-013-4660-7

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