A discrete method for the logarithmic-kernel integral equation on an open arc

Prößdorf, Siegfried; Saranen, Jukka; Sloan, Ian H.

doi:10.1017/s0334270000009000

Cited by 4 publications

(1 citation statement)

References 15 publications

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“…For a special case when S is an interval, a collocation method using Chebyshev polynomials is analyzed in [20] and a piecewise constant collocation method using cosine mesh grading was studied in [14]. A quadrature method was discussed in [18]. A fully discrete quadrature method for the equivalent second kind equations with kernels defined by Cauchy singular integral was proposed in [24].…”

Section: Introductionmentioning

confidence: 99%

Fast Fourier-Galerkin methods for first-kind logarithmic-kernel integral equations on open arcs

Wang

2010

Sci. China Ser. A-Math.

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We propose a fully discrete fast Fourier-Galerkin method for solving an integral equation of the first kind with a logarithmic kernel on a smooth open arc, which is a reformulation of the Dirichlet problem of the Laplace equation in the plane. The optimal convergence order and quasi-linear complexity order of the proposed method are established. A precondition is introduced. Combining this method with an efficient numerical integration algorithm for computing the single-layer potential defined on an open arc, we obtain the solution of the Dirichlet problem on a smooth open arc in the plane. Numerical examples are presented to confirm the theoretical estimates and to demonstrate the efficiency and accuracy of the proposed method. Keywords Dirichlet problem, open arc, singular boundary integral equations, Fourier-Galerkin methods, logarithmic potentials MSC(2000): 65R20, 45E05, 41A55, 65F35 Citation: Wang B, Wang R, Xu Y S. Fast Fourier-Galerkin methods for first-kind logarithmic-kernel integral equations on open arcs.

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