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Mixed Fourier-Jacobi Spectral Method for Two-Dimensional Neumann Boundary Value Problems

Published online by Cambridge University Press:  28 May 2015

Xu-Hong Yu  and
Zhong-Qing Wang
Xu-Hong Yu
Affiliation:
Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China
Zhong-Qing Wang*
Affiliation:
Scientific Computing Key Laboratory of Shanghai Universities, Division of Computational Science of E-institute ofShanghai Universities
*
Corresponding author. Email: [email protected]
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Abstract

In this paper, we propose a mixed Fourier-Jacobi spectral method for two dimensional Neumann boundary value problem. This method differs from the classical spectral method. The homogeneous Neumann boundary condition is satisfied exactly. Moreover, a tridiagonal matrix is employed, instead of the full stiffness matrix encountered in the classical variational formulation. For analyzing the numerical error, we establish the mixed Fourier-Jacobi orthogonal approximation. The convergence of proposed scheme is proved. Numerical results demonstrate the efficiency of this approach.

Keywords

33C4565M7035J25Mixed Fourier-Jacobi orthogonal approximationspectral methodNeumann boundary value problem
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 1 , Issue 3 , August 2011 , pp. 284 - 296
Copyright
Copyright © Global-Science Press 2011

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References

[1]Askey, R., Orthogonal Polynomials and Special Functions, Regional Conference Series in Applied Mathematics, Vol. 21, SIAM, Philadelphia, 1975.Google Scholar
[2]Auteri, F., Parolini, N. and Quartapelle, L., Essential imposition of Neumann condition in Galerkin-Legendre elliptic solvers, J. Comp. Phys., 185(2003), 427–444.Google Scholar
[3]Bergh, J. and Löfström, J., Interpolation Spaces, An Introduction, Spring-Verlag, Berlin, 1976.CrossRefGoogle Scholar
[4]Bernardi, C. and Maday, Y., Spectral Methods, in Handbook of Numerical Analysis, Vol.5, Techniques of Scientific Computing, 209–486, edited by Ciarlet, P. G. and Lions, J. L., Elsevier, Amsterdam, 1997.Google Scholar
[5]Boyd, J. P., Chebyshev and Fourier Spectral Methods, second edition, Dover Publications, Inc., Mineola, NY, 2001.Google Scholar
[6]Canuto, C., Hussaini, M. Y., Quarteroni, A. and Zang, T. A., Spectral Methods: Fundamentals in Single Domains, Springer, Berlin, 2006.CrossRefGoogle Scholar
[7]Canuto, C., Hussaini, M. Y., Quarteroni, A. and Zang, T. A., Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics, Springer, Berlin, 2007.CrossRefGoogle Scholar
[8]Guo, B., Spectral Methods and Their Applications, World Scientific, Singapore, 1998.CrossRefGoogle Scholar
[9]Guo, B. and Wang, L., Jacobi interpolation approximations and their applications to singular differential equations, Adv. Comp. Math., 14(2001), 227–276.Google Scholar
[10]Guo, B. and Wang, T., Composite generalized Laguerre-Legendre spectral method with domain decomposition and its application to Fokker-Planck equation in an finite channel, Math. Comp., 78(2009), 129–151.Google Scholar
[11]Guo, B. and Wang, T., Composite Laguerre-Legendre spectral method for exterior problems, Adv. Comp. Math., 32(2010), 393–429.Google Scholar
[12]Shen, J., Efficient spectral-Galerkin method I. Direct solvers of second- and fourth-order equations using Legendre polynomials, SIAM J. Sci. Comp., 15(1994), 1489–1505.Google Scholar
[13]Shen, J. and Tang, T., Spectral and High-order Methods with Applications, Science Press, Beijing, 2006.Google Scholar
[14]Szego, G., Orthogonal Polynomials, American Mathematical Society, Providence, RI, 1959.Google Scholar
[15]Wang, L. and Guo, B., Mixed Fourier-Jacobi spectral method, J. Math. Anal. Appl., 315(2006), 8–28.Google Scholar
[16]Wang, T. and Guo, B., Composite generalized Laguerre-Legendre pseudospectral method for Fokker-Planck equation in an infinite channel, Appl. Numer. Math., 58(2008), 1448–1466.Google Scholar
[17]Wang, T. and Guo, B., Composite Laguerre-Legendre pseudospectral method for exterior problems, Comm. Comp. Phys., 5(2009), 350–375.Google Scholar
[18]Wang, T. and Wang, Z., Error analysis of Legendre spectral method with essential imposition of Neumann boundary condition, Appl. Numer. Math., 59(2009), 2444–2451.Google Scholar
[19]Yu, X. and Wang, Z., Jacobi spectral method with essential imposition of Neumann boundary condition, submitted.Google Scholar