Quadrature and Collocation Methods for Singular Integral Equations on Curves with Corners

where f is a closed and piecewise smooth curve in the complex plane, c, d and 'kare given continuous functions, it is the unknown solution and the first intigral is to be interpreted as a Cauchy principal value (see,' e.g., [3, 14, 16, 17]). For the numerical solution of this equation spline approximation methods are extensively employed. In fact, collocation and quadrature methods are the most Widely used numerical procedures for solving-the boundary integral equations of the form (0.1) arising from exterior or interior boundar value problems of applications. (See, e.g., [1, 3, 4].) If I' is aclosed smooth curve, a fairly complete stability theory and error analysis of collocation methods for. (0.1) using smooth splines has-been established (see the surveys given in [8, 28, 15: Chap. 17, 26]). A general approach to the stability and error analysis of quadrature methods for (0.1) using equidistant quadrature knots has been developed in [19, 231. In this paper we present a , stability analysis of quadrature 'and spline collocation methods for (0.1) in the case when F is a closed curve with a finite number of corners. -' • For this case, Costabel and Stephan (unpublished) proved the strong ellipticity of the operator A to be sufficient for the L 2-stability of the piecewise linear collocation We establish conditions for the stability of the collocation method with piecewise con-