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A Discrete Collocation Method for a Hypersingular Integral Equation on Curves with Corners

In: Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan

Author

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  • Thomas Hartmann

    (Hochschule Ulm)

  • Ernst P. Stephan

    (Leibniz Universität Hannover, Institut für Angewandte Mathematik)

Abstract

This paper is devoted to the approximate solution of a hypersingular integral equation on a closed polygonal boundary in ℝ 2 $${\mathbb {R}}^2$$ . We propose a fully discrete method with a trial space of trigonometric polynomials, combined with a trapezoidal rule approximation of the integrals. Before discretization the equation is transformed using a nonlinear (mesh grading) parametrization of the boundary curve which has the effect of smoothing out the singularities at the corners and yields fast convergence of the approximate solutions. The convergence results are illustrated with some numerical examples.

Suggested Citation

  • Thomas Hartmann & Ernst P. Stephan, 2018. "A Discrete Collocation Method for a Hypersingular Integral Equation on Curves with Corners," Springer Books, in: Josef Dick & Frances Y. Kuo & Henryk Woźniakowski (ed.), Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan, pages 545-566, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-72456-0_25
    DOI: 10.1007/978-3-319-72456-0_25
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