John P. Boyd
Abstract
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 xj, the quadrature weights wj, and the values of the prolate functions and their derivatives at the grid points. The prolate functions themselves are computed by a Legendre-Galerkin 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 Legendre-Lobatto points and weights. Fortunately, the variations of the xj and wj with c are well-approximated by a symmetric parabola over the whole range of interest. This makes it possible to bypass the continuation procedures of earlier authors.
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Software for "Computation of grid points, quadrature weights and derivatives for spectral element methods using prolate spheroidal wave functions---prolate elements"
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Index Terms
- Algorithm 840: computation of grid points, quadrature weights and derivatives for spectral element methods using prolate spheroidal wave functions---prolate elements
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Reviews
Basem S. Attili
Boyd considers the use of prolate spheroidal wave functions: prolate elements to compute the grid points, quadrature weights, and derivatives for spectral element methods. This is done by first computing the prolate nodal basis, and the appropriate quadrature and weights that replace the Legendre-Lobatto grid points, quadrature weights, and cardinal function derivative matrices. The logic of the spectral element method is not modified. The resulting method is the so-called prolate element method. The author developed the method as a library, resulting in software that can be applied to various classes of partial differential equations, linear and nonlinear. The author successfully introduces the transformation from the modal to nodal basis, and also reports some results on its condition number. Another area of improvement is the initialization for the computation of weights and grid points that replace the continuation algorithm used previously. Boyd is still testing the software on the shallow water problem. Online Computing Reviews Service
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