A fully diagonalized spectral method using generalized Laguerre functions on the half line
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A fully diagonalized spectral method using generalized Laguerre functions on the half line Fu-Jun Liu1,2 · Zhong-Qing Wang1 · Hui-Yuan Li3
Received: 12 April 2016 / Accepted: 9 February 2017 © Springer Science+Business Media New York 2017
Abstract A fully diagonalized spectral method using generalized Laguerre functions is proposed and analyzed for solving elliptic equations on the half line. We first define the generalized Laguerre functions which are complete and mutually orthogonal with respect to an equivalent Sobolev inner product. Then the Fourier-like Sobolev orthogonal basis functions are constructed for the diagonalized Laguerre spectral method of elliptic equations. Besides, a unified orthogonal Laguerre projection is established for various elliptic equations. On the basis of this orthogonal Laguerre projection, we obtain optimal error estimates of the fully diagonalized Laguerre spectral method for both Dirichlet and Robin boundary value problems. Finally, numerical experiments, which are in agreement with the theoretical analysis, demonstrate the effectiveness and the spectral accuracy of our diagonalized method. Keywords Spectral method · Sobolev orthogonal Laguerre functions · Elliptic boundary value problems · Error estimates Communicated by: Jan Hesthaven Zhong-Qing Wang
[email protected] Fu-Jun Liu [email protected] Hui-Yuan Li [email protected] 1
School of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China
2
School of Science, Henan Institute of Engineering, Zhengzhou, 451191, China
3
State Key Laboratory of Computer Science/Laboratory of Parallel Computing, Institute of Software, Chinese Academy of Sciences, Beijing 100190, China
F.-J. Liu et al.
Mathematics Subject Classification (2010) 76M22 · 33C45 · 35J25 · 65L70
1 Introduction Spectral methods for solving partial differential equations on unbounded domains have gained a rapid development during the last few decades. An abundance of literature on this research topic has emerged, and their underlying approximation approaches can be essentially classified into three catalogues [1, 2]: (i) (ii)
(iii)
truncate an unbounded domain to a bounded one and solve the problem on the bounded domain subject to artificial or transparent boundary conditions [3, 4]; map the original problem on an unbounded domain to one on a bounded domain and use classic spectral methods to solve the new problem [5]; or equivalently, approximate the original problem by some non-classical functions mapped from the classic orthogonal polynomials/functions on a bounded domain [2, 6–12]; directly approximate the original problem by genuine orthogonal functions such as Laguerre polynomials or functions on the unbounded domain [13–27].
The third approach is of particular interest to researchers, and has won an increasing popularity in a broad class of applications, owing to its essential advantages over other two approaches. These direct approximation schemes constitute an initial step towards the efficient spectral methods,
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