Two-Grid Solution of Symm's Integral Equation

We propose two-grid iteration methods for Symm's integral equation discretized by quadrature-collocation or quadrature methods. Asymptotically the optimal order of error estimate is achieved already on the first iteration, for some modifications on the second iteration. This enables us to introduce some solvers which are of the optimal convergence order and cheap in a practical implementation; the cost varies between O(N²) and O(N log N) arithmetic operations. Numerical experiments confirm the approximation properties of the schemes.