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The Nonconvergence of \(h\)-Refinement in Prolate Elements

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Prolate Spheroidal Wave Functions are likely to be a better tool for the design of spectral and pseudo-spectral techniques than the orthogonal polynomials and related functions

— Xiao, Rokhlin and Yarvin (2001), p. 837.

Abstract

Prolate elements are a “plug-compatible” modification of spectral elements in which Legendre polynomials are replaced by prolate spheroidal wave functions of order zero. Prolate functions contain a“bandwidth parameter” \(c \ge 0 \) whose value is crucial to numerical performance; the prolate functions reduce to Legendre polynomials for \(c\,=\,0\). We show that the optimal bandwidth parameter \(c\) not only depends on the number of prolate modes per element \(N\), but also on the element widths \(h\). We prove that prolate elements lack \(h\)-convergence for fixed \(c\) in the sense that the error does not go to zero as the element size \(h\) is made smaller and smaller. Furthermore, the theoretical predictions that Chebyshev and Legendre polynomials require \(\pi \) degrees of freedom per wavelength to resolve sinusoidal functions while prolate series need only 2 degrees of freedom per wavelength are asymptotic limits as \(N \rightarrow \infty \); we investigate the rather different behavior when \(N \sim O(4-10)\) as appropriate for spectral elements and prolate elements. On the other hand, our investigations show that there are certain combinations of \(N,\,h\) and \(c>0\) where a prolate basis clearly outperforms the Legendre polynomial approximation.

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Acknowledgments

This work was supported by the National Science Foundation through grants OCE 1059703 and DMS-1115277. The second author is kindly supported by the Deutsche Forschungsgemeinschaft (DFG) within SPP 1276: MetStroem. We thank the two reviewers for detailed and helpful comments.

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Authors and Affiliations

  1. Department of Atmospheric, Oceanic and Space Science, University of Michigan, 2455 Hayward Avenue, Ann Arbor, MI, 48109, USA

    John P. Boyd

  2. Institute of Aerodynamics and Gas Dynamics, University of Stuttgart, Stuttgart, Germany

    Gregor Gassner

  3. Program in Applied and Interdisciplinary Mathematics, University of Michigan, East Hall, Ann Arbor, MI, 48109, USA

    Burhan A. Sadiq

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Correspondence to John P. Boyd.

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Boyd, J.P., Gassner, G. & Sadiq, B.A. The Nonconvergence of \(h\)-Refinement in Prolate Elements. J Sci Comput 57, 372–389 (2013). https://doi.org/10.1007/s10915-013-9711-2

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