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

Numerical solution of boundary integral equations by means of attenuation factors

“…Other effective methods for solving (1.1) have been presented, among them qualocation methods (see Chandler & Sloan, 1990;Sloan & Burn, 1992;Sarane & Sloan, 1992, for smooth curves and Elschner & Graham, 1994, for curves with corners). Another efficient method for piecewise analytic curves was developed in Hough (1990), where a very elaborate program package was also given.…”

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

“…Computationally efficient quadrature methods for the solution of (1.1) have recently been introduced and analysed in the case of a closed curve. In [13] Sloan and Burn introduced a family of "unconventional" quadrature methods. This approach was further developed and analysed by Saranen and Sloan [10].…”

“…Here we propose a quadrature method which is an adaptation of the quadrature method introduced by Sloan and Burn [13] for the closed-curve case. To be more precise, the quadrature method is here applied after the well-known cosine substitution.…”

“…The cosine substitution removes this bad behaviour of the solution and therefore enables us to construct numerical schemes with good stability and convergence properties. As in the case of a smooth closed curve (see [10], [13]), the family of quadrature methods in this paper includes schemes of arbitrarily high, but fixed order O{h p ) of convergence. In particular, we describe explicitly a simple method (see Example 4.1) which is stable and has the order O(h 3 ) of convergence; the discretisation parameter h is inversely proportional to the number of the meshpoints on T.…”

On the numerical solution of a logarithmic integral equation of the first kind for the Helmholtz equation

“…For a review and an extensive bibliography we refer to [11]. In addition, some of the more practical quadrature methods have been considered by Sloan and Burn [12], by Saranen and Sloan [9] and by Saranen [8]. Quadrature methods are generally considered to be more practical since in their numerical implementation the computation of the matrix elements is less costly than in the corresponding collocation and Galerkin methods.…”