Numerical Algorithms

, Volume 42, Issue 2, pp 137–164

Efficient spectral-Galerkin algorithms for direct solution for second-order differential equations using Jacobi polynomials


DOI: 10.1007/s11075-006-9034-6

Cite this article as:
Doha, E.H. & Bhrawy, A.H. Numer Algor (2006) 42: 137. doi:10.1007/s11075-006-9034-6


It is well known that spectral methods (tau, Galerkin, collocation) have a condition number of \(O(N^{4})\) (\(N\) is the number of retained modes of polynomial approximations). This paper presents some efficient spectral algorithms, which have a condition number of \(O(N^{2})\), based on the Jacobi–Galerkin methods 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 \((N-1)^d\) unknowns, while the convergence rates of the algorithms are exponentials with smooth solutions.


Spectral-Galerkin methodJacobi polynomialsPoisson and Helmholtz equations

AMS subject classifications


Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceCairo UniversityGizaEgypt
  2. 2.Department of Mathematics, Faculty of ScienceBeni-Suef UniversityBeni-SuefEgypt