Spline collocation for a boundary integral equation on polygons with cuts

In this paper we construct collocation methods for a modified integral equation of the second kind on the boundary of a polygon with a finite number of cuts. Cuts are often met in practice and are not treated by the classical methods. The double layer potential which degenerates in this situation is modified by the addition of two kinds of terms on each cut. They take into account the highly singular functions generated by the cut and lead to a bijective boundary operator. This operator remains bijective between spaces of Sobolev types with high orders of regularity. This allows the construction of collocation methods with quasi-optimal estimates and high order of convergence with respect to strong norms. They use smoothest splines of odd degree. Numerical examples of the methods are presented.