Algorithm 840
swMATH ID:  4463 
Software Authors:  Boyd, John P. 
Description:  Algorithm 840: Computation of grid points, quadrature weights and derivatives for spectral element methods using prolate spheroidal wave functions – prolate elements High order domain decomposition methods using a basis of Legendre polynomials, known variously as “spectral elements” or “\(p\)type finite elements”, have become very popular. Recent studies suggest that accuracy and efficiency can be improved by replacing Legendre polynomials by prolate spheroidal wave functions of zeroth order. In this article, we explain the practicalities of computing all the numbers needed to switch bases: the grid points \(x_j\), the quadrature weights \(w_j\), and the values of the prolate functions and their derivatives at the grid points. The prolate functions themselves are computed by a LegendreGalerkin discretization of the prolate differential equation; this yields a symmetric tridiagonal matrix. The prolate functions are then defined by Legendre series whose coefficients are the eigenfunctions of the matrix eigenproblem. The grid points and weights are found simultaneously through a Newton iteration. For large \(N\) and \(c\), the iteration diverges from a first guess of the LegendreLobatto points and weights. Fortunately, the variations of the \(x_j\) and \(w_j\) with \(c\) are wellapproximated by a symmetric parabola over the whole range of interest. This makes it possible to bypass the continuation procedures of earlier authors. 
Homepage:  http://dl.acm.org/citation.cfm?id=1055538 
Related Software:  Matlab; DLMF; Differentiation Matrix Suite; mp toolbox; Regularization tools; algorithm 577; Chebfun 
Cited in:  27 Publications 
Standard Articles
1 Publication describing the Software, including 1 Publication in zbMATH  Year 

Algorithm 840: Computation of grid points, quadrature weights and derivatives for spectral element methods using prolate spheroidal wave functions – prolate elements. Zbl 1070.65569 Boyd, John P. 
2005

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Cited by 29 Authors
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