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BIT Numerical Mathematics

, Volume 49, Issue 2, pp 249–280 | Cite as

Univariate modified Fourier methods for second order boundary value problems

  • Ben Adcock
Article

Abstract

We develop and analyse a new spectral-Galerkin method for the numerical solution of linear, second order differential equations with homogeneous Neumann boundary conditions. The basis functions for this method are the eigenfunctions of the Laplace operator subject to these boundary conditions. Due to this property this method has a number of beneficial features, including an \(\mathcal{O}(N^{2})\) condition number and the availability of an optimal, diagonal preconditioner. This method offers a uniform convergence rate of \(\mathcal{O}(N^{-3})\) , however we show that by the inclusion of an additional 2M basis functions, this figure can be increased to \(\mathcal{O}(N^{-2M-3})\) for any positive integer M.

Keywords

Spectral methods Neumann boundary value problems Generalized Fourier expansions Convergence acceleration 

Mathematics Subject Classification (2000)

65N35 42C15 65B99 

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Copyright information

© Springer Science + Business Media B.V. 2009

Authors and Affiliations

  1. 1.DAMTP, Centre for Mathematical SciencesUniversity of CambridgeCambridgeUK

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