Trigonometric collocation methods with product integration for boundary integral equations on closed curves

Considering boundary integral equations on smooth closed curves the product integration methods have turned out to be very efficient to define fully discrete schemes with a high convergence rate. Here we give a general approach to product integration methods in connection with the trigonometric collocation. Thr discretization scheme has arbitrarily high convergence rate, even exponential if the solution and the curve are analytic. Applications and numerical experiments confirming our theoretical results are discussed.