A QUADRATURE METHOD FOR THE SINGULAR INTEGRAL-EQUATION ON CURVES WITHCORNER POINTS

This paper is concerned with a quadrature method for the approximate s olution of the singular integral equation a(t)x(t) + b(t)/pii integral -GAMMA x(tau)/tau-t d(tau) + integral-GAMMA k(t, tau)x(tau)dtau = y(t) , t is-an-element-of GAMMA, on the closed curve GAMMA with a finite nu mber of corner points. Here a and b are piecewise continuous functions on GAMMA. We establish necessary and sufficient conditions for the st ability of the quadrature method and, in addition, derive the converge nce rates. For this we shall apply techniques similar to those used in the case where GAMMA is an interval (cf. [8]). The crucial point is a change of the variables depending on a parameter and a subsequent app lication of a simple quadrature rule on a uniform grid.