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An Unconventional Quadrature Method for Logarithmic-Kernel Integral Equations Equations on Closed Curves

1992; Volume: 4; Issue: 1 Linguagem: Inglês

10.1216/jiea/1181075670

1938-2626

Ian H. Sloan, Bob Burn,

Matrix Theory and Algorithms

A new, fully discrete method is proposed for the logarithmic-kernel integral equation of the first kind on a smooth closed curve.The method uses two levels of numerical quadrature: a trapezoidal rule for the integral containing the logarithmic singularity; and a special quadrature rule for the outer integral, which compensates, in part, for the errors in the first integral.A convergence and stability analysis is given, and the predicted orders of convergence verified in a numerical example.A numerical experiment suggests that the method can be useful even for a curve with corners.