An exponentially convergent method for solving boundary integral equations on polygons

The boundary integral equation of the potential theory in the case of the inner Dirichlet problem for the Laplace operator and the system of boundary integral equations of the Dirichlet boundary value problem for the two-dimensional theory of elasticity in domains with a finite number of corner points are considered. The derivatives of kernels and solutions to the above integral equations on the boundaries of simply connected polygons are estimated. A numerical method based on the application of one and the same family of composite quadrature formulas is proposed. It is proved that the proposed method is exponentially convergent with respect to the number of the quadrature nodes in use.