Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-24T14:10:00.306Z Has data issue: false hasContentIssue false
  • English
  • Français

Efficient Chebyshev Spectral Method for Solving Linear Elliptic PDEs Using Quasi-Inverse Technique

Published online by Cambridge University Press:  28 May 2015

Fei Liu ,
Xingde Ye  and
Xinghua Wang
Fei Liu*
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310027, China
Xingde Ye*
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310027, China
Xinghua Wang*
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310027, China
*
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Get access

Abstract

We present a systematic and efficient Chebyshev spectral method using quasi-inverse technique to directly solve the second order equation with the homogeneous Robin boundary conditions and the fourth order equation with the first and second boundary conditions. The key to the efficiency of the method is to multiply quasi-inverse matrix on both sides of discrete systems, which leads to band structure systems. We can obtain high order accuracy with less computational cost. For multi-dimensional and more complicated linear elliptic PDEs, the advantage of this methodology is obvious. Numerical results indicate that the spectral accuracy is achieved and the proposed method is very efficient for 2-D high order problems.

Keywords

65N3565N2265F0535J05Chebyshev spectral methodquasi-inverseHelmholtz equationRobin boundary conditionsgeneral biharmonic equation
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 4 , Issue 2 , May 2011 , pp. 197 - 215
Copyright
Copyright © Global Science Press Limited 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]Canuto, C., Hussaini, M. Y., Quarteroni, A., Zang, T. A., Spectral Methods in Fluid Dynamics, Springer-Verlag, New York, 1987.Google Scholar
[2]Canuto, C., Hussaini, M. Y., Quarteroni, A., Zang, T. A., Spectral Methods: Fundamentals in Single Domains, Springer-Verlag, New York, 2006.CrossRefGoogle Scholar
[3]Doha, E. H. and Bhrawy, A. H., Efficient spectral-Galerkin algorithms for direct solution for second-order differential equations using Jacobi polynomials, Numer. Algorithms., 42 (2006), pp. 137–164.CrossRefGoogle Scholar
[4]Dang-Vu, H. and Delcarte, C., An accurate solution of the Poisson equation by the Chebyshev collocation method, J. Comput. Phys., 104 (1993), pp. 211–220.CrossRefGoogle Scholar
[5]Doha, E. H. and Abd-Elhameed, W. M., Efficient spectral-Galerkin algorithms for direct solution for second-order equations using ultraspherical polynomials, SIAM J. Sci. Comput., 24: 2 (2002), pp. 548–571.CrossRefGoogle Scholar
[6]Heinrichs, W., Improved Condition Number for Spectral Methods, Math. Comput, 53: 187 (1989), pp. 103–119.CrossRefGoogle Scholar
[7]Julien, K. and Watson, M., Efficient multi-dimensional solution of PDEs using Chebyshev spectral methods, J. Comput. Phys., 228 (2009), pp. 1480–1503.CrossRefGoogle Scholar
[8]Mai-Duy, N. and Tanner, R. I., A spectral collocation method based on integrated Chebyshev polynomials for two-dimensional biharmonic boundary-value problems, J. Comput. Appl. Math., 201 (2007), pp. 30–47.CrossRefGoogle Scholar
[9]Shen, J., Efficient spectral-Galerkin method I. Direct solvers for the second and fourth order equations using Legendre polynomials, SIAM J. Sci. Comput., 15: 6 (1994), pp. 1489–1505.CrossRefGoogle Scholar
[10]Shen, J., Efficient spectral-Galerkin method II. Direct solvers of second and fourth order equations by using Chebyshev polynomials, SIAM J. Sci. Comput., 16: 1 (1995), pp. 74–87.CrossRefGoogle Scholar
[11]Shen, J. and Tang, T., Spectral and High-Order Methods with Applocations, Science Press, Beijing, 2006.Google Scholar
[12]Trefethen, L. N., Spectral Methods in MATLAB, PA: SIAM, Philadelphia, 2000.CrossRefGoogle Scholar
[13]Weideman, J. A.C. and Reddy, S. C., A MATLAB differentiation matrix suite, ACM Trans. Math. Softw., 26: 4 (2000), pp. 465–519.CrossRefGoogle Scholar