Some Discrete Methods for Boundary Integral Equations on Smooth Closed Curves

Abstract

In this paper discrete trigonometric methods as well as quadrature methods for solution of boundary integral equations on smooth closed curves are presented and analyzed. The analysis applies the general approach based on the consistency, stability, and convergence properties. The trigonometric methods, including, e.g., variants based on product integration, turn out to have high convergence, even exponential convergence. The family of quadrature methods covers operators with the even symbol, and of order $\beta < 0$. They can be considered as fully discretized Petrov–Galerkin methods that employ approximations both for the integral operator and the inner product. The operator is approximated by using the trapezoidal rule either directly or after subtraction of the kernel singularity. For instance for Symm’s equation $(\beta = - 1)$ a method with the rate $O(h^5 )$ of convergence is presented. Furthermore, the quadrature solution of the Cauchy singular integral equation $(\beta = 0)$ and the hypersingular equation $(\beta = 1)$ is considered, and the exponential convergence is obtained.