East Asian Journal on Applied Mathematics, 1 (2011), pp. 403-414. Features of the Nystrom method for the Sherman-Lauricella equation on piecewise smooth contours Victor D. Didenko 1*, Johan Helsing 21 Faculty of Science, University of Brunei Darussalam, Bandar Seri Begawan, BE1410 Brunei. 2 Numerical Analysis, Centre for Mathematical Sciences, Lund University, Box 118, SE-221 00 Lund, Sweden. Received 24 June 2011; Accepted (in revised version) 7 August 2011 Available online 23 September 2011 doi:10.4208/eajam.240611.070811a Abstract 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. AMS subject classifications: 65R20, 45L05 Key words: Sherman--Lauricella equation, Nystrom method, stability. *Corresponding author. Email: diviol@gmail.com (V. D. Didenko), helsing@maths.lth.se (J. Helsing)