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

ANZIAM J. 48 (2007), 361-386

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

E. H. Doha
Department of Mathematics
Faculty of Science
Cairo University
Giza
Egypt
eiddoha@frcu.eun.eg

and

A. H. Bhrawy
Department of Mathematics
Faculty of Science
Beni-Suef University
Beni-Suef
Egypt
alibhrawy@yahoo.co.uk

Abstract

It is well known that spectral methods (tau, Galerkin, collocation) have a condition number of

where N is the number of retained modes of polynomial approximations. This paper presents some efficient spectral algorithms, which have a condition number of

, 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 $N^{d+1}$ operations for a d-dimensional domain with

unknowns, while the convergence rates of the algorithms are exponentials with smooth solutions.

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