Victor D. Didenko ^1*, Johan Helsing ²

The stability of the Nystr\"om method for the Sherman-Lauricella equation on contours with corner points $c_j$, $j=0,1,\cdots,m$ relies on the invertibility of certain operators $A_{c_j}$ belonging to an algebra of Toeplitz operators. The operators $A_{c_j}$ do not depend on the shape of the contour, but on the opening angle $\theta_j$ of the corresponding corner $c_j$ and on parameters of the approximation method mentioned. They have a complicated structure and there is no analytic tool to verify their invertibility. To study this problem, the original Nystr\"om method is applied to the Sherman-Lauricella equation on a special model contour that has only one corner point with varying opening angle $\theta_j$. In the interval $(0.1\pi,1.9\pi)$, it is found that there are $8$ values of $\theta_j$ where the invertibility of the operator $A_{c_j}$ may fail, so the corresponding original Nystr\"om method on any contour with corner points of such magnitude cannot be stable and requires modification.