Convergence Results for Discrete Trigonometric Collocation Methods with Product Integration in Hölder-Zygmund Spaces

In this paper convergence results with respect to Hölder-Zygmund norms including also maximum norm error estimates are derived for the fully discrete trigonometric collocation method presented earlier by Saranen and Vainikko for solution of boundary integral equations on smooth closed curves. Approximation of the integral operator is based on product integration for which the explicit Fourier representation of the main part is not needed, and still the convergence of arbitrarily high rate for smooth solutions can be achieved. Saranen and Vainikko have given their analysis with respect to Sobolev norms yielding results that do not imply pointwise error estimates of optimal order. In this work the approach is based on the use of Hölder-Zygmund norms, and the optimal order maximum norm estimates are accomplished.