Features of the Nyström Method for the Sherman-Lauricella Equation on Piecewise Smooth Contours

Abstract

The stability of the Nyström method for the Sherman-Lauricella equation on contours with corner points $c_j$ , $j = 0, 1,··· , m$ 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 $θ_j$ 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 Nyström method is applied to the Sherman-Lauricella equation on a special model contour that has only one corner point with varying opening angle $θ_j$ . In the interval ($0.1π$, $1.9π$), it is found that there are 8 values of $θ_j$ where the invertibility of the operator $A_{c_j}$ may fail, so the corresponding original Nyström method on any contour with corner points of such magnitude cannot be stable and requires modification.