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Exponential node clustering at singularities for rational approximation, quadrature, and PDEs

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Abstract

Rational approximations of functions with singularities can converge at a root-exponential rate if the poles are exponentially clustered. We begin by reviewing this effect in minimax, least-squares, and AAA approximations on intervals and complex domains, conformal mapping, and the numerical solution of Laplace, Helmholtz, and biharmonic equations by the “lightning” method. Extensive and wide-ranging numerical experiments are involved. We then present further experiments giving evidence that in all of these applications, it is advantageous to use exponential clustering whose density on a logarithmic scale is not uniform but tapers off linearly to zero near the singularity. We propose a theoretical model of the tapering effect based on the Hermite contour integral and potential theory, which suggests that tapering doubles the rate of convergence. Finally we show that related mathematics applies to the relationship between exponential (not tapered) and doubly exponential (tapered) quadrature formulas. Here it is the Gauss–Takahasi–Mori contour integral that comes into play.

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Notes

  1. Stenger considered rational approximations of this kind, though not in this precise setting of a finite interval with just one endpoint singularity.

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Acknowledgements

We have benefited from helpful advice from Bernd Beckermann, Pablo Brubeck, Silviu Filip, Abi Gopal, Stefan Güttel, Leonid Knizhnerman, Arno Kuijlaars, Andrei Martínez-Finkelshtein, Ed Saff, Kirill Serkh, Alex Townsend, Heather Wilber, and an anonymous referee.

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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK

    Lloyd N. Trefethen & Yuji Nakatsukasa

  2. Applied Mathematics, Stellenbosch University, Private Bag X1, Matieland, 7602, South Africa

    J. A. C. Weideman

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  1. Lloyd N. Trefethen
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Correspondence to Lloyd N. Trefethen.

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Trefethen, L.N., Nakatsukasa, Y. & Weideman, J.A.C. Exponential node clustering at singularities for rational approximation, quadrature, and PDEs. Numer. Math. 147, 227–254 (2021). https://doi.org/10.1007/s00211-020-01168-2

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