Convergence results for discrete trigonometric collocation methods with product integration in Hölder-Zygmund spaces (Q1368789)

Convergence results with respect to Hölder-Zygmund norms and maximum norm error estimates are derived for the fully discrete trigonometric collocation method presented earlier by \textit{J. Saranen} and \textit{G. Vainikko} [SIAM J. Numer. Anal. 33, No. 4, 1577-1596 (1996; Zbl 0855.65119)] for the solution of boundary integral equations on smooth closed curves. The approximation of the integral operator is based on product integration for which the explicit Fourier representation of the main part is not needed. Convergence of arbitrarily high rate for smooth solutions can be achieved. Saranen and Vainikko's method does not give pointwise error estimate of optimal order (the results are yield with respect to Sobolev norms). Optimal order maximum norm estimates are accomplished in this paper.