The stability of the Nystr\"om method for double layer potential equations

The stability of the Nystr\"om method for the double layer potential equation on simple closed piecewise smooth contours is studied. Necessary and sufficient conditions of the stability of the method are established. It is shown that the method under consideration is stable if and only if certain operators associated with the opening angles of the corner points are invertible. Numerical experiments show that there are opening angles which cause instability of the method.