Efficient Spectral-Galerkin Algorithms for Direct Solution of Second-Order Equations Using Ultraspherical Polynomials

Authors:
Eid H. Doha;Waleed M. Abd-Elhameed
Affiliations:
-;-
Venue:
SIAM Journal on Scientific Computing
Year:
2002

Citing 0
Cited 8

On the coefficients of integrated expansions and integrals of ultraspherical polynomials and their applications for solving differential equations

Journal of Computational and Applied Mathematics
On the coefficients of integrated expansions of Bessel polynomials

Journal of Computational and Applied Mathematics
Efficient spectral-Galerkin methods for polar and cylindrical geometries

Applied Numerical Mathematics
Efficient spectral ultraspherical-dual-Petrov-Galerkin algorithms for the direct solution of (2n+1)th-order linear differential equations

Mathematics and Computers in Simulation
On the coefficients of integrated expansions of Bessel polynomials

Journal of Computational and Applied Mathematics
Matrix decomposition algorithms for elliptic boundary value problems: a survey

Numerical Algorithms
A spectral method for elliptic equations: the Neumann problem

Advances in Computational Mathematics
New algorithms for solving high even-order differential equations using third and fourth Chebyshev-Galerkin methods

Journal of Computational Physics

Quantified Score

Hi-index	0.01

Visualization

Abstract

It is well known that spectral methods (tau, Galerkin, collocation) have a condition number of O(N4) (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 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 exponential for problems with smooth solutions.