Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-23T01:19:54.370Z Has data issue: false hasContentIssue false
  • English
  • Français

Efficient spectral-Galerkin algorithms for direct solution of the integrated forms of second-order equations using ultraspherical polynomials

Published online by Cambridge University Press:  17 February 2009

E. H. Doha  and
A. H. Bhrawy
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is well known that spectral methods (tau, Galerkin, collocation) have a condition number of O(N4) where N is the number of retained modes of polynomial approximations. This paper presents some efficient spectral algorithms, which have a condition number of O(N2), based on the ultraspherical-Galerkin methods for the integrated forms of second-order elliptic equations in one and two space variables. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to systems with specially structured matrices that can be efficiently inverted. The complexities of the algorithms are a small multiple of Nd+1 operations for a d-dimensional domain with (N – 1)d unknowns, while the convergence rates of the algorithms are exponentials with smooth solutions.

Keywords

spectral-Galerkin methodultraspherical polynomialsPoisson and Helmholtz equations.
Type
Research Article
Information
The ANZIAM Journal , Volume 48 , Issue 3 , January 2007 , pp. 361 - 386
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Ben-Yu, G., “Gengenbauer approximation and its applications to differential equations on the whole line”, J. Math. Anal. Appl. 226 (1998) 180–206.CrossRefGoogle Scholar
[2]Ben-Yu, G., Spectral methods and their applications (World Scientific, London, 1998).CrossRefGoogle Scholar
[3]Buzbee, B. L., Golub, G. H. and Neilson, C. W., “On direct methods for solving Poisson's equations”, SIAM J. Numer. Anal. 7 (1970) 627–656.CrossRefGoogle Scholar
[4]Canuto, C., Hussaini, M. Y., Quarteroni, A. and Zang, T. A., Spectral methods in fluid dynamics (Springer, New York, 1989).Google Scholar
[5]Doha, E. H., “An accurate double Chebyshev spectral approximation for Poisson's equation”, Ann. Univ. Sci. Budapest Sect. Comp. 10 (1990) 243–276.Google Scholar
[6]Doha, E. H., “An accurate solution of parabolic equations by expansion in ultraspherical polynomials”, J. Comput. Math. Appl. 19 (1990) 75–88.CrossRefGoogle Scholar
[7]Doha, E. H., “The coefficients of differentiated expansions and derivatives of ultraspherical polynomials”, J. Comput. Math. Appl. 21 (1991) 115–122.Google Scholar
[8]Doha, E. H., “The ultraspherical coefficients of the moments of a general-order derivative of an infinitely differentiable function”, J. Comput. Appl. Math. 89 (1998) 53–72.CrossRefGoogle Scholar
[9]Doha, E. H., “On the coefficients of integrated expansions and integrals of ultraspherical polynomials and their applications for solving differential equations”, J. Comput. Appl. Math. 139 (2002) 275–298.CrossRefGoogle Scholar
[10]Doha, E. H. and Abd-Elhameed, W. M., “Efficient spectral-Galerkin algorithms for direct solution of second-order equations using ultraspherical polynomials”, SIAM J. Sci. Comput. 24 (2002) 548–571.CrossRefGoogle Scholar
[11]Fox, L. and Parker, I. B., Chebyshev polynomials in numerical analysis (Oxford University Press, Oxford, 1972).Google Scholar
[12]Gottlieb, D. and Orszag, S. A., Numerical analysis of spectral methods: theory and applications (SIAM, Philadelphia, 1977).CrossRefGoogle Scholar
[13]Haidvogel, D. B. and Zang, T., “The accurate solution of Poisson's equation by expansion in Chebyshev polynomials”, J. Comput. Phys. 30 (1979) 167–180.Google Scholar
[14]Heinrichs, W., “Improved condition number of spectral methods”, Math. Comp. 53 (1989) 103–119.CrossRefGoogle Scholar
[15]Heinrichs, W., “Spectral methods with sparse matrices”, Numer. Math. 56 (1989) 25–41.CrossRefGoogle Scholar
[16]Heinrichs, W., “Algebraic spectral multigrid methods”, Comput. Methods Appl. Mech. Engrg. 80 (1990) 281–286.Google Scholar
[17]Lanczos, C., Applied analysis (Pitman, London, 1957).Google Scholar
[18]Luke, Y., The special functions and their approximations, Vol. 1 (Academic Press, New York, 1969).Google Scholar
[19]Orszag, S. A., “Spectral methods for problems in complex geometries”, J. Comput. Phys. 37 (1980) 70–92.Google Scholar
[20]Phillips, T. N. and Karageorghis, A., “On the coefficients of integrated expansions of ultraspherical polynomials”, SIAM J. Numer. Anal. 27 (1990) 823–830.CrossRefGoogle Scholar
[21]Shen, J., “Efficient spectral-Galerkin method I. Direct solvers of second- and fourth-order equations using Legendre polynomials”, SIAM J. Sci. Comput. 15 (1994) 1489–1505.Google Scholar
[22]Shen, J., “Efficient spectral-Galerkin method II. Direct solvers of second- and fourth-order equations using Chebyshev polynomials”, SIAM J. Sci. Comput. 16 (1995) 74–87.Google Scholar
[23]Siyyam, H. I. and Syam, M. I., “An accurate solution of Poisson equation by the Chebyshev-Tau method”, J. Comput. Appl. Math. 85 (1997) 1–10.Google Scholar