Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-19T08:54:40.556Z Has data issue: false hasContentIssue false
  • English
  • Français

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: zqwang@shnu.edu.cn
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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