Abstract
This paper is devoted to the approximate solution of a hypersingular integral equation on a closed polygonal boundary in \({\mathbb {R}}^2\). 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.
Dedicated to Ian H. Sloan on the occasion of his 80th birthday.
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Hartmann, T., Stephan, E.P. (2018). A Discrete Collocation Method for a Hypersingular Integral Equation on Curves with Corners. In: Dick, J., Kuo, F., Woźniakowski, H. (eds) Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan. Springer, Cham. https://doi.org/10.1007/978-3-319-72456-0_25
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